Curves and Surface I

Angel Ch.10

Representation of Curves and Surfaces

Piece-wise linear representation is inefficient - line segments to approximate curve - polygon mesh to approximate surfaces - can not approximate general curves exactly - only continous in position (not derivatives)

Modelling of smooth curves and surface - ‘smooth’ continuous in position and derivatives (1st/2nd…) - exact representation of non-planar objects

Representation of curves and surfaces: (1) Explicit: y=g(x) (2) Implicit: f(x,y)=0 (3) Parametric: x=x(u), y=y(u)

Parametric forms commonly used in computer graphics/CAD

Explicit Representation of Curves

Curves in 2D y=g(x) is a general curve in x,y space x independent variable y dependent variable

Inverse relation: x=h(y) - inverse may not exist

Example: 2D Line y=ax+b a-slope b-intersection with y-axis Problem with representing a vertical line a=infinity

Example: 2D Circle radius r

- explicit equation only represent half the circle

22 xry −=

Explicit Curves in 3D The 3D explicit representation of a curve requires 2 equations

y=g(x) z=f(x) - y,z are both dependent variables

Example: Line in 3D

y=ax+b z=cx+d - cannot represent a line in a plane of x=const

Explicit representation of a surface z=f(x,y) - 2 independent variables x,y - z dependent variable

Cannot represent a full sphere 222 yxrz −−=

Explicit representations are co-ordinate system dependent - can not represent all curves such as lines/circles - Curves exist independent of any representation - failures causes serious problems in graphics/CAD

Implicit Representation

2D Curves: f(x,y)=0 - f() is a ‘membership’ testing function (x,y) is on the curve if f(x,y)=0 - In general no analytic way to find points on the curve

ie to evaluate y value corresponding to given x - overcomes some of coordinate system dependent problems - represents all lines and circles

Example: 2D Line ax+by+c=0 - represent vertical line a=-1, b=0

Example: 2D Circle - represents the entire circle

0222 =−+ ryx

3D Surfaces: f(x,y,z)=0 - membership test is (x,y,z) on the curve or surface - general representation in 3-space - represents any plane or line in 3-space

Plane: ax+by+cz+d=0 - slopes a,b,c wrt x,y,z-axis

Sphere of radius r: 02222 =−++ rzyx

3D Curves: - not easily represented in 3D - intersection of two implicit surfaces f(x,y,z)=0 g(x,y,z)=0 point (x,y,z) on curve must be a member of both function - line represented by intersection of 2 planes

Algebraic surfaces - class of implicit surfaces where f(x,y,z) is a polynomial function of the three variables

is an nth order polynomial - quadric surfaces n=2 are of particular importance as they represent several common surfaces sphere, cone, torus, ellipsoid

Quadrics have the property that intersection with a line gives at most 2 intersection points (used for rendering)

0)(1

=+++∑=

ii

ii

n

i

ii zcybxad

Parametric Representation

Parametric curve in 3-space is a function of 1-parameter, u x=x(u) y=y(u) z=z(u) - general representation - same in 2-space and 3-space - u is independent variable and x,y,z dependent

Curve can be written as the set (locus) of points:

p(u) = [x(u) y(u) z(u)]

Derivative of curve:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

duudzduudyduudx

duudp

)(

)(

)(

)(

- partial derivative wrt each function - points in the direction tangent to the curve - magnitude is rate of change wrt to u (velocity)

Example: 3D Line

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−+−+

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

−+=

)()()(

)1(

)1()(

212

212

212

2

2

2

1

1

1

21

zzzyyyxxx

zyx

zyx

ppp

ααα

αα

ααα

Derivatives:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

==)()()(

)()(

21

21

21

zzyyxx

pddp α

αα

α

Parametric surface: x=x(u,v) y=y(u,v) z=z(u,v)

- 2 independent parameters u,v p(u,v) = [ x(u,v) y(u,v) z(u,v)]T

plane tangent therepresent sderivative Partial - d(v)p(v)

),(),(),(

uv)z(u,

uv)y(u,

uv)x(u,

up(u)

sDerivative Partial

v

u

u

u

u

p

vuzvuyvux

p

=∂

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂∂

∂∂

∂

==∂

∂

Parametric representation - widely used in computer graphics for curves and surfaces and CAD - general representation of many forms. - simple computation of derivatives - requires a parameterization (u,v) of the surface ie underlying 2D coordinate system for a 3D surface - representation depends on 2D coordinate system - polynomial parametric forms used to represent a wide variety of surfaces. How can we define the parametric form of a surface?

Parametric Polynomial Curves

∑∑∑

∑

===

=

===

+

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⎤

⎢⎢⎢

⎣

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=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

n

kzk

kn

kyk

kn

kxk

k

zk

yk

xk

k

k

n

kk

k

cuuzcuuycuux

ccc

c

c

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uzuyux

up

000

0

)()()(

:equations of setst independen 3 givingfreedom, of degrees 1)3(n has curve parametric The

zy,in x, tscoefficient independenh matrix witcolumn a is where

)(

:bygiven isn degree of curve parametric polynomialA )()()(

)( :as drepresente is space-3in curve parametricA

We can define the curves x(u), y(u), z(u) over a range of u

maxmin uuu ≤≤

- p(u) over this range gives a curve segment

Without loss of generality we can define the curve segment:

10 ≤≤ u

Parametric Polynomial Surfaces

Define a surface by n and m order polynomials in u and v

∑∑= =

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

n

k

m

jjk

jk cvuvuzvuyvux

vup0 0),(

),(),(

),(

Surface has 3(n+1)(m+1) degrees of freedom

Usually, n=m and we parameterize u,v over the range 0,1 giving a surface patch. The surface can be viewed in the limit as a collection of curves generated by holding u or v constant

∑∑

∑∑

= =

−

= =

−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

n

k

m

jjk

jk

v

v

v

v

n

k

m

jjk

jk

u

u

u

u

cvjuvuzvuyvux

vup

cvkuvuzvuyvux

vup

0 0

1

0 0

1

),(),(),(

),(

),(),(),(

),(

sDerivative Partial

∑∑

∑∑

= =

−−

= =

−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

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−=⎥⎥⎥

⎦

⎤

⎢⎢⎢

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n

k

m

jjk

jk

v

v

v

vuuv

n

k

m

jjk

jk

u

u

u

uu

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vup

0 0

11

0 0

2

),(),(),(

),(),(

)1(),(),(),(

),(

sDerivative PartialOrder 2nd

Design Criteria Representations of curves and surfaces in computer graphics and CAD should satisfy the following:

•  Local control of shape •  Smoothness and continuity •  Direct evaluation of derivatives •  Stability

(small change in parameters gives a small change in shape) •  Fast Rendering

Use representations based on low-order parametric polynomials to satisfy the above

Local Control of Shape

- ideally control local shape with a set of local parameters rather than global change in shape - easier to design desired shape Represent curve shape p(u) by a set of ‘control points’:

- local curve shape is based on the position of the control points Interpolating curve: passes through all control point

Smoothness and Continuity

‘Smoothness’ of a curve refers to the continuity of derivatives of the curve For a parametric curve all derivatives exist and are continuous

.....)1()()()(2

2

1

1

0∑∑∑=

−

=

−

=

−===n

k

kkuu

n

k

kk

n

ku

kk uckkupukcupucup

For two curves meeting at a join point the smoothness is defined by the highest derivative which is continuous

discontinuity in 1st derivative continuous in 1st derivative

space-3in continous is curve ly theconsequent andu spaceparameter in continous are sDerivative

)0()1(:

)0()1(:)0()1(:

continous isch whi parameters respect to with derivativeorder highest -

Continuity Parametric

2

1

0

uuuu

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n

qpCqpCqpC

C

=

==

direction same in the is curvature - )0()1(:

shape control toparameter free remaining use 3n rather tha derivateson sconstraint 2only requires -

direction same in the are curve to tangents- )0()1(:magnitudedifferent

but direction same in the are space-3in sderivative - Continuity Geometric

2

1

uuuu

uu

n

qpG

qpG

G

∝

∝

Parametric Cubic Polynomial Curves

What is the correct order of polynomial curve - low-order: less flexible - high-order: costly, bumpy (too many degrees of freedom)

Designing curve locally by a set of control points we can use a low-order curve to approximate complex shapes. Cubic polynomial curves are widely used:

c unknowns 12in equations 12

1

)(

3

2

3

2

1

0

3

0

33

2210

⎥⎥⎥

⎦

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⎦

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⎢⎢⎢⎢

⎣

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=

⎥⎥⎥⎥

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=

==+++= ∑=

kz

ky

kx

k

k

Tkk

ccc

c

uuu

u

cccc

c

cuucucucuccup

Want methods of deriving parameters c for a desired curve!

Cubic Polynomial Interpolation

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i

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=

=

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( ) ( )( ) ( )

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33

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33

31

22

31

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1

00

)1()(

)(

)0(:conditionsfour thegives This

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matricieselement 4x3 are ,1111

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1

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Blending Functions

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⎤

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==

===

)()()()(

)(

)(b functions blending ofset a toaccording p points control ofn combinatio theasion interpolat heconsider tcan We

3

2

1

0

ii

ubububub

uMub

pb(u)pMuc up(u) u

Tl

T l

TT

∑=

=+++=3

022221100

i

)(p (u)p (u)p (u)p (u))(

:pointeach of onscontributi individual the together blends''or weightsu which in polynomial a is)(bfunction blendingEach

iii pubbbbbup

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21

32

31

29

3

31

227

2

32

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1

32

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0

uabout symmetric arefunction Blending :Note))((

)1)(()1)((

)1)()(( :ioninterpolat cubicfor functions Blending

=−−=

−−−=−−=

−−−−=

uuu(u)buuu(u)buuu(u)b

uuu(u)b

All zeros of the blending functions are in the interval [0,1] - blending functions must vary substantially over [0,1]

ie rapid changes - results from requirement that curve pass through all control points

This results in limited usefulness of ‘interpolating’ cubic polynomial + polynomial is discontinuous between sections Apply same derivation process to polynomials with other constraints

Cubic Interpolation Patch

Extension of interpolating curve to a surface

[ ]

points ofset given a esinterpolath whicsurface theevaluate tofind want to which wetscoefficien 48 has - C

11

),(

:points 16by defined ispatch surface inginterpolat Bicubic

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i jij

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=

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=

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ijij

ij

p

p

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20

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∑∑= =

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Hermite Curves and Surfaces

Rather than interpolating points we interpolate between endpoints + tangents at end points

- ensures continuity between curve/surface segments

3211

3210300

32)1( )0( points-end at the tangent theConstrain

)1( )0( points-end intersects Curve

:as sconstraint define Curve a of Form Hermite

cccpcp

ccccppcpp

uu ++==

+++====

c

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

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=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

3210001011110001

)1(p(0)p

p(1)0)(p

q

:formmatrix In

u

u

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

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−−−

==

1122123301000001

matrix geometry' Hermite' Gives

:find toequations Solve

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qMuup HT=)(

:bygiven is polynomial Resulting

function blending inginterpolathan smoother t - [0,1]in szero' their of none have functions blendingfour The

232

132

)(

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23

23

23

23

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⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−+−

+−+−

==

=

uuuuuuu

uu

uMub

qubup

TH

T

curve continous C a givingbothfor [0,1]u with curve theofsection adjacent are q(u) and p(u) where

)0()1()0()1(

curveadjacent theof pointstart theas derivative same thehas curve one ofpoint end thesuch that -

derivates continuous with curve arepresent toused becan spolynomial Hermite

1

===

uu qpqp

This overcomes the problem with interpolating cubics where the end-points were only continuous in position

0100

11010100

3

0

3

0

ie points controlinput thefrom defined becan sDerivative

ion.interpolatdirect over the advantages haspatch surface Hermite Therefore,

patches.adjacent between derivative1st andposition in continous ish patch whic surface a gives Solving

)0,0()0,0()0,0()0,0( :define we(0,0)corner At

sderivative their and pointscorner 4 theeinterpolat todefined is

)()(),(

:patch surface Hermite

ppp

cpcpcpcp

qububvup

u

uvvu

ijji j

i

−=

====

=∑∑= =

Summary

Surface representation - explicit - implicit - parametric

parametric forms are widely used in computer graphics Parametric forms

- Interpolating curves and surfaces

Next lecture: other parametric forms of surfaces Hermite, Bezier, B-Spline, NURBS