Representation of Curves amp Surfaces

Prof Lizhuang Ma

Shanghai Jiao Tong University

Contents

bull Specialized Modeling Techniques

bull Polygon Meshes

bull Parametric Cubic Curves

bull Parametric Bi-Cubic Surfaces

bull Quadric Surfaces

bull Specialized Modeling Techniques

The Teapot

Representing Polygon Meshes

bull Explicit representationbull By a list of vertex coordinates

bull Pointers to a vertex listbull Pointers to an edge list

))()()(( 222111 nnn zyxzyxzyxP

Pointers to A Vertex List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)324(

)421(

2

1

P

P

Pointers to An Edge List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)(

)(

)(

)(

)(

)(

)(

4322

5411

1145

21244

2433

2322

1211

EEEP

EEEP

PVVE

PPVVE

PVVE

PVVE

PVVE

Parametric Cubic Curves

bull The cubic polynomials that define a curve segment are of the form

10)(

)(

)(

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Contents

bull Specialized Modeling Techniques

bull Polygon Meshes

bull Parametric Cubic Curves

bull Parametric Bi-Cubic Surfaces

bull Quadric Surfaces

bull Specialized Modeling Techniques

The Teapot

Representing Polygon Meshes

bull Explicit representationbull By a list of vertex coordinates

bull Pointers to a vertex listbull Pointers to an edge list

))()()(( 222111 nnn zyxzyxzyxP

Pointers to A Vertex List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)324(

)421(

2

1

P

P

Pointers to An Edge List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)(

)(

)(

)(

)(

)(

)(

4322

5411

1145

21244

2433

2322

1211

EEEP

EEEP

PVVE

PPVVE

PVVE

PVVE

PVVE

Parametric Cubic Curves

bull The cubic polynomials that define a curve segment are of the form

10)(

)(

)(

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

The Teapot

Representing Polygon Meshes

bull Explicit representationbull By a list of vertex coordinates

bull Pointers to a vertex listbull Pointers to an edge list

))()()(( 222111 nnn zyxzyxzyxP

Pointers to A Vertex List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)324(

)421(

2

1

P

P

Pointers to An Edge List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)(

)(

)(

)(

)(

)(

)(

4322

5411

1145

21244

2433

2322

1211

EEEP

EEEP

PVVE

PPVVE

PVVE

PVVE

PVVE

Parametric Cubic Curves

bull The cubic polynomials that define a curve segment are of the form

10)(

)(

)(

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Representing Polygon Meshes

bull Explicit representationbull By a list of vertex coordinates

bull Pointers to a vertex listbull Pointers to an edge list

))()()(( 222111 nnn zyxzyxzyxP

Pointers to A Vertex List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)324(

)421(

2

1

P

P

Pointers to An Edge List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)(

)(

)(

)(

)(

)(

)(

4322

5411

1145

21244

2433

2322

1211

EEEP

EEEP

PVVE

PPVVE

PVVE

PVVE

PVVE

Parametric Cubic Curves

bull The cubic polynomials that define a curve segment are of the form

10)(

)(

)(

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Pointers to A Vertex List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)324(

)421(

2

1

P

P

Pointers to An Edge List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)(

)(

)(

)(

)(

)(

)(

4322

5411

1145

21244

2433

2322

1211

EEEP

EEEP

PVVE

PPVVE

PVVE

PVVE

PVVE

Parametric Cubic Curves

bull The cubic polynomials that define a curve segment are of the form

10)(

)(

)(

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Pointers to An Edge List

)( 4321 VVVVV

))()(( 444111 zyxzyx

)(

)(

)(

)(

)(

)(

)(

4322

5411

1145

21244

2433

2322

1211

EEEP

EEEP

PVVE

PPVVE

PVVE

PVVE

PVVE

Parametric Cubic Curves

bull The cubic polynomials that define a curve segment are of the form

10)(

)(

)(

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Parametric Cubic Curves

bull The cubic polynomials that define a curve segment are of the form

10)(

)(

)(

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Parametric Cubic Curves

bull The curve segment can be rewrite as

bull Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Parametric Cubic Curves

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()(

TttCTCdt

d]123[ 2

Tzzzyyyxxx ctbtactbtactbta ]232323[ 222

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

ContinuityBetween Curve Segments

bull G0 geometric continuityndash Two curve segments join together

bull G1 geometric continuityndash The directions (but not necessarily the magnitudes)

of the two segmentsrsquo tangent vectors are equal at a join point

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

ContinuityBetween Curve Segments

bull C1 continuousndash The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at the segmentsrsquo joint point1048710

bull Cn continuousndash The direction and magnitude of through

the nth derivative are equal at the joint point

)]([ tQdtd nn

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

ContinuityBetween Curve Segments

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

ContinuityBetween Curve Segments

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Three Types ofParametric Cubic Curves

bull Hermite Curvesndash Defined by two endpoints and two endpoint tangent

vectors

bull Beacutezier Curvesndash Defined by two endpoints and two control points w

hich control the endpointrsquo tangent vectors

bull Splinesndash Defined by four control points

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Parametric Cubic Curves

bull Representationbull Rewrite the coefficient matrix as

ndash where M is a 4x4 basis matrix G is called the geometry matrix

ndash So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Parametric Cubic Curves

ndash Where is called the blending functions

BGTMGtQ )(

TMB

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Hermite Curves

bull Given the endpoints P1 and P4 and tangent vectors at them R1 and R4

bull What isndash Hermite basis matrix MH

ndash Hermite geometry vector GH

ndash Hermite blending functions BH

bull By definition

GH=[P1 P4 R1 R4]

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Hermite Curves

bull Since THH

THH

THH

THH

MGPQ

MGRQ

MGPQ

MGPQ

0123)1(

0100)0(

1111)1(

1000)0(

4

1

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Hermite Curves

bull So

bull And

0011

0121

0032

1032

0011

1110

2010

30101

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpointsrsquo tangent vectors such that

bull What is1048710ndash Beacutezier basis matrix MB

ndash Beacutezier geometry vector GB

ndash Beacutezier blending functions BB

Beacutezier Curves

)(3)1(

)(3)0(

34

4

12

1

PPQR

PPQR

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Beacutezier Curves

bull By definitionbull Then

bull So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQ HHBBHH )()(

TMGTMMG BBHHBB )(

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Beacutezier Curves

bull And

HBBHHBB MGMMM

0001

0033

0363

1331

43

32

22

13 )1(3)1(3)1()( PtPttPttPttQ

TttttttBB 3223 )1(3)1(3)1( (Bernstein polynomials)

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Convex Hull

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull The polynomial coefficients for natural cubic splines are dependent on all n control pointsndash Has one more degree of continuity than is inherent in t

he Hermite and Beacutezier formsndash Moving any one control point affects the entire curvendash The computation time needed to invert the matrix can i

nterfere with rapid interactive reshaping of a curve

Spline

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

B-Spline

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull Cubic B-Splinebull Has m+1 control pointsbull Has m-2 cubic polynomial curve segments

bull Uniformndash The knots are spaced at equal intervals of the paramet

er t1048710bull Non-rational

ndash Not rational cubic polynomial curves

Uniform NonRational B-Splines

3 10 mPPP m

mQQQ 43

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull Curve segment Qi is defined by pointsbull B-Spline geometry matrix

bull If bull Then

Uniform NonRational B-Splines

iiii PPPP 223

miPPPPG iiiiBSi 3123

Tiiii ttttttT 1)()()( 23

1)( iiiBsBsii tttTMGtQ

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull So B-Spline basis matrix

bull B-Spline blending functions

Uniform NonRational B-Splines

TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull The knot-value sequence is a nondecreasing sequence

bull Allow multiple knot and the number of identical parameter is the multiplicityndash Ex (00001123445555)

bull So

Uniform NonRational B-Splines

)()()()()( 4411422433 tBPtBPtBPtBPtQ iiiiiiiii

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Uniform NonRational B-Splines

bull Where is j th-order blending function for weighting control point Pi

)()()(

)()()(

)()()(

0

1)(

3114

43

34

2113

32

23

1112

21

12

11

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

iii

ii

ii

ii

iii

ii

ii

ii

iii

ii

ii

ii

iii

)( tB ji

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Knot Multiplicity amp Continuity

bull Since is within the convex hull of and

bull If is within the convex hull of and and the convex hull of and so it will lie on

bull If will lie on bull If will lie on both and and th

e curve becomes broken

)( itQ 23 ii PP 1iP

)(1 iii tQtt 23 ii PP

1iP 12 ii PP1iP

12 ii PP

)(21 iiii tQttt 1iP

)(321 iiiii tQtttt 1iP iP

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull Multiplicity 1 C2 continuitybull Multiplicity 2 C1 continuitybull Multiplicity 3 C0 continuitybull Multiplicity 4 no continuity

Knot Multiplicity amp Continuity

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

NURBSNonUniform Rational B-Splines

bull Rational bull and are defined as the ratio of two cubi

c polynomialsbull Rational cubic polynomial curve segments are rati

os of polynomials

bull Can be Beacutezier Hermite or B-Splines

)()( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Parametric Bi-Cubic Surfacesbull Parametric cubic curves are bull So parametric bi-cubic surfaces are bull If we allow the points in G to vary in 3D along so

me path then

bull Since are cubicsbull where

SMtGtGtGtGtsQ )()()()(( 4321）＝

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG＝

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Parametric Bi-Cubic Surfacesbull So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

)(

10 tsSMGMT TT

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Hermite Surfaces

SMGMTtsQ HHTH

T )( TMGtRtRtPtP HH ）()()( 4141

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Beacutezier Surfaces

SMGMTtsQ BsBsTBs

T )(

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Normals to Surfaces

Ss

MGMTtsQs

TT

)(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)()(

SMGMss TT 0123 2

)()( tsQt

tsQs

normal vector

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Quadric Surfaces

bull Implicit surface equation

bull An alternative representation

bull with

0 PQPT

0222222)( 22

2 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

y

x

P

kjhg

jcef

hebd

gfda

Q

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

bull Advantagesndash Computing the surface normalndash Testing whether a point is on the surfacendash Computing z given x and yndash Calculating intersections of one surface with another

Quadric Surfaces

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Fractal Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models

Grammar-Based Models

bull L-grammarsndash A -gt AAndash B -gt A[B]AA[B]

• Representation of Curves amp Surfaces
• Contents
• The Teapot
• Representing Polygon Meshes
• Pointers to A Vertex List
• Pointers to An Edge List
• Parametric Cubic Curves
• Slide 8
• Slide 9
• Tangent Vector
• Continuity Between Curve Segments
• Slide 12
• Slide 13
• Slide 14
• Three Types of Parametric Cubic Curves
• Slide 16
• Slide 17
• Hermite Curves
• Slide 19
• Slide 20
• Beacutezier Curves
• Slide 22
• Slide 23
• Convex Hull
• Spline
• B-Spline
• Uniform NonRational B-Splines
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Knot Multiplicity amp Continuity
• Knot Multiplicity amp Continuity
• NURBS NonUniform Rational B-Splines
• Parametric Bi-Cubic Surfaces
• Slide 36
• Hermite Surfaces
• Beacutezier Surfaces
• Normals to Surfaces
• Quadric Surfaces
• Quadric Surfaces
• Fractal Models
• Grammar-Based Models