fundamentals of computer graphics lecture 3 parametric ... · cn curve continuity...

《Fundamentals of Computer Graphics》

Lecture 3 Parametric curve and surface

Yong-Jin Liu

liuyongjin@tsinghua.edu.cn

Smooth curve and surface

Design criteria of smooth curve and

surface

• Smooth and continuity

• Ability to control the local shape

• Stability

• Is it easy to draw

Smooth and continuity

• Regard the curve as the orbit of point changing with its

parameter u

• Regard the differential of one point as the its speed

along the orbit, the direction of speed is the tangent

direction of this point

2

0 1 2( )

( ) ( )

( )

n

nx u a a u a u a u

C u y u

z u

1

1 2( ) 2

( ) ( )

( )

n

nx u a a u na u

C u y u

z u

Cn metric of curve smooth

• Differentiability of function：if parametric curve has

continuous derivative until n order at any point in the

interval [a, b]，i.e. n order continuous differentiable,

then the curve is n order parametric continuous in the

interval [a, b].

• Polynomial curve has infinite order parametric

continuous C。

Polynomial curve of n order pn(x)=a0+a1x+a2x2+anx

n

Consider two curves with a common point

• Differentiability of function：if parametric curve has

continuous derivative until n order at any point in the

interval [a, b]，i.e. n order continuous differentiable,

then the curve is n order parametric continuous in the

interval [a, b].

C0 continuous C1 continuous

Cn curve continuity

• Differentiability of function：if combined parametric

curve has continuous derivative until n order at the

joint point, it is called Cn or n-order parametric

continuous.

• Is there any thing wrong？

• The same curve can have different parameters.

2

2

2

1( )

( ) cos( ) 1 or ( ) sin( ) 2

( )1

tx t

x u u t

y u u ty t

t

circle：

Different parameters will get different

derivative vector！

)2,0()1(

)1(2,

)1(

4))(),((

)1,0())cos(),sin(())(),((

22

2

22

t

t

t

ttytx

uuuyux

C(u), u [a, b]

C(t), t [0, 1], ab

aut

( ) ( ) 1 ( ) 1( ) ( )

dC u dC t dt dC tC u C t

du dt du b a dt b a

Two continuity definitions

• Differentiability of function：if combined parametric

curve has continuous derivative until n order at the

joint，i.e. n order continuous differentiable, such

smooth is called Cn or n order parametric continuous.

• Geometry continuity：if the direction of

derivative vector of combined curve at the joint is

equal, then it is n order geometry continuous，

denoted by Gn.

If require G0 or C0 continuous at the joint,

that is to say two curves are continuous at the

joint.

If require G1 continuous at the joint，that is

to say two curves are G0 continuous at the

joint, and have the common derivative vector

P’(1)=aQ’(0).

when a＝1，G1 continuous is also C1 continuous

If require G2 continuous at the joint，that is

to say two curves are G1 continuous at the

joint, and have common curvature vector.

This equation can be wrote as :

a, b is arbitrary constant.

When a=1，b=0，G2 continuous is also C2 continuous.

3 3

(1) (1) (0) (0)Curvature formula

(1) (0)

P P Q Q

P Q

2(0) (1) (1)Q a P bP

)(tP

)(tQ

)0(P

)1(P

)0(Q

)1(Q

Figure 3.1.7 continuous of two curves

Arc length parameterization of curve

• The same curve can have different parameter

• If u(s) is a monotonic function，then

C(s)=C(u(s)) is a new parameterization about s

of the curve

• Compute the length L(u) of curve C(u)

• If there exists parameter s，

so that L(s)=s, s 0? s = 0

Parameter s

Length L(s)

Curve C(s)

0

1

Let | ( ) |

| ( ) | ( )

( ) ( )

u

dsC u

du

s C u du L u

u s L s

Arc length parameterization of curve

( ) ( ( )) 1( ) ( )

| ( ) |

dC s dC u s duv s C u

ds du ds C u

| ( ) | 1v s

Design criteria of smooth curve and

surface

• Smooth and continuity

• Ability to control the local shape

• Stability

• Is it easy to draw

Power based representation of polynomial curve

•

•

•

0

( ) ( ( ), ( ), ( ))n

i

i

i

C u x u y u z u u

a

( , , )i i i ix y za

0 1

1

( )T

n i i

n

uC u u

u

a a a a

n order polynomial curve Cn(u)=a0+a1u+a2u

2++anun

1 order polynomial curve:

a1u+a0

a0

a1+a0

n order polynomial curve Cn(u)=a0+a1u+a2u

2++anun

See from the aspect of human

computer interaction…

2 order polynomial curve:

a2u2+a1u+a0

a0

a1

a2+a1+a0

2a2+a1

Parabola, ellipse, hyperbola？

n order polynomial curve Cn(u)=a0+a1u+a2u

2++anun

3 order polynomial curve:

a3u3+a2u

2+a1u+a0

n order polynomial curve Cn(u)=a0+a1u+a2u

2++anun

Power based representation of polynomial curve

•

• The representation and control ability is very bad

Bernstein based representation of polynomial

curve

n order polynomial curve Cn(u)=a0+a1u+a2u

2++anun

,

0

,

( ) ( )

!( ) (1 )

!( )!

n

i i n

i

i n i

i n

C u B u

nB u u u

i n i

pBeizer curve

Bernstien polynomial

1

u

u2

a0 a1

a2

2

210)( uauaauf

B0,2

B1,2

B2,2

p0 p1

p2

2,222,112,00)()( BpBpBpuCuf

Bernstein based representation of polynomial curve

• have clear geometry meaning

, ,

0 0

( ) ( ) ( )n n

i i n i n i

i i

C u B u B u

p p

Bernstein based representation of polynomial curve

• Have clear geometry meaning

• Interpolation of endpoints

, ,

0 0

( ) ( ) ( ) , [0,1]n n

i i n i n i

i i

C u B u B u u

p p

,

!( ) (1 ) , [0, 1]

!( )!

i n i

i n

nB u u u u

i n i

, 0

0

,

0

(0) (0)

(1) (1)

n

i i n

i

n

i i n n

i

C B

C B

p p

p p

Bernstein based representation of polynomial curve

• Have clear geometry meaning

• Why there exists control polygon effect when use

Bernstein based representation？

, 0,1 0 1,1 1

0

,

( ) ( ) ( ) ( )

!( ) (1 ) , [0, 1]

!( )!

n

i n i

i

i n i

i n

C u B u B u B u

nB u u u u

i n i

p p p

1 order Bezier curve

u=0 u=1

, 0,2 0 1,2 1 2,2 2

0

,

( ) ( ) ( ) ( ) ( )

!( ) (1 ) , [0, 1]

!( )!

n

i n i

i

i n i

i n

C u B u B u B u B u

nB u u u u

i n i

p p p p

2 order Bezier curve

u=0 u=1

, 0,3 0 1,3 1 2,3 2 3,3 3

0

,

( ) ( ) ( ) ( ) ( ) ( )

!( ) (1 ) , [0, 1]

!( )!

n

i n i

i

i n i

i n

C u B u B u B u B u B u

nB u u u u

i n i

p p p p p

3 order Bezier curve

u=0 u=1

9

,

0

,

( ) ( )

!( ) (1 ) , [0, 1]

!( )!

i n i

i

i n i

i n

C u B u

nB u u u u

i n i

p

9 order Bezier curve

u=0 u=1 u=0.5

Properties of Bezier curve and Bernstein base

1. Bi,n(u) 0, for all i, n and 0 u 1

2. Partition of unity ni=0Bi,n(u) = 1, 0 u 1

3. B0,n(0) = Bn,n(1) = 1

4. Bi,n(u) has and only has a maximum value

in the interval [0,1] , it is at u=i/n

5. Base set {Bi,n(u)} is symmetry about u=1/2

,

!( ) (1 ) , [0, 1]

!( )!

i n i

i n

nB u u u u

i n i

Computation method of Bezier curve

• Method 1：

• Method 2：geometry drawing

9

,

0

,

( ) ( )

!( ) (1 ) , [0, 1]

!( )!

i n i

i

i n i

i n

C u B u

nB u u u u

i n i

p

Computation method of Bezier curve

• Method 2 (geometry drawing, ruler drawing)：

• example 2 order Bezier curve

2

,2

0

2 2

0 1 2

0 1 1 2

( ) ( )

(1 ) 2 (1 )

(1 )((1 ) ) ((1 ) )

i i

i

C u B u

u u u u

u u u u u u

P

P P P

P P P P

2

,2

0

2 2

0 1 2

0 1 1 2

( ) ( )

(1 ) 2 (1 )

(1 )((1 ) ) ((1 ) )

i i

i

C u B u

u u u u

u u u u u u

P

P P P

P P P P

Is this property general？

2 order Bezier curve

,

!( ) (1 ) , [0, 1]

!( )!

i n i

i n

nB u u u u

i n i

Properties of Bezier curve and Bernstein base

6. recursive definition：

Bi,n(u) = (1u)Bi,n1(u) +uBi1,n1(u)

with Bi,n(u) 0, if i < 0 or i > n

7. derivative：

B’i,n(u) = dBi,n(u) /du = n(Bi1,n1(u) Bi,n1(u) )

with B1,n1(u) Bn,n1(u) 0

Denote n order Bezier curve as

Cn(P0,,Pn), then we have

Cn(P0,,Pn)=(1u)Cn1(P0,,Pn-1)+uCn1(P1,,Pn)

Geometry drawing algorithm

n order Bezier curve has

denote Pi as P0,i

Cn(P0,,Pn)=(1u)Cn1(P0,,Pn-1)+uCn1(P1,,Pn)

Pk,i(u)=(1u)Pk1,i(u)+uPk1,i+1(u)

k = 1,,n

i = 0,,nk deCasteljau algorithm

Smooth curve and surface in 3D space

1D to 2D

• Regard the curve as the orbit of point

changing with its parameter u

2

0 1 2( )

( ) ( )

( )

n

nx u a a u a u a u

C u y u

z u

Tensor product surface

• Tensor product surfaces

• Point S(u0, v0)

• Isoparametric lines C(u) = S(u, v0) and

C(v) = S(u0, v)

,

0 0

,

( , ) ( ( , ), ( , ), ( , ))

( ) ( )

[ ( )] [ ][ ( )]

n m

i j i j

i j

T

i i j j

S u v x u v y u v z u v

f u g v

f u g v

b

b

Base function

Geometry coefficient

Tensor product Bezier surface

• Point S(u0, v0)

• Isoparametric lines C(u) = S(u, v0) and C(v) = S(u0, v)

, , ,

0 0

( , ) ( ) ( )n m

i n j m i j

i j

S u v B u B v

P

, , ,

0 0

( , ) ( ) ( )n m

i n j m i j

i j

S u v B u B v

P

2D Bernstein base

Computation method of Bezier curve

• Geometry drawing algorithm

• deCasteljau algorithm

Computation method of Bezier curve

• deCasteljau algorithm

, , ,

0 0

, , ,

0 0

,

0

( , ) ( ) ( )

( ) ( )

( )

n m

i n j m i j

i j

n m

i n j m i j

i j

n

i n i

i

S u v B u B v

B u B v

B u

P

P

Q u

v

Design criteria of smooth curve and

surface

• Smooth and continuity

• Ability to control the local shape

• Stability

• Is it easy to draw

• Is Bezier curve perfect?

Beizer curve

• The representation and control ability of local

shape is bad

• if interpolate many (n+1) discrete point s,

then require a high order (n) Bezier curve.

How to improve？

• Splicing many Bezier curve segments

n order polynomial curve Cn(u)=a0+a1u+a2u

2++anun

Design of B-spline curve

• Determine the order of continuity p

• Determine node vector U = {u0, u1, , um}；

• Determine a set of control points {P0, P1, ,

Pn}；

• The form of B-spline curve is

,

0

( ) ( )n

i p i

i

C u N u

P

B-spline surface

• Tensor product surface

, , ,

0 0

( , ) ( ) ( )n m

i p j q i j

i j

S u v N u N v

P

1

1 1

11

1

1 1

11

{0, ,0, , , ,1, ,1}

{0, ,0, , , ,1, ,1}

r

p r p

pp

s

q s q

qq

U u u

V u u

1 and 1r n p s m q

Ability of local shape change

Ability of local shape change

The reason is the local scope of basic function

, , ,

0 0

( , ) ( ) ( )n m

i p j q i j

i j

S u v N u N v

P

, , , 1 1( ) ( ) [ , ) [ , )i j i p j q i i p j j qN u N v u u v v P

Some extensions of NURBS curve and surface

• Free form deformation (FFD)

• Subdivision surface

Free deformation of parametric curve and

surface (FFD)

• 1D curve

• 2D surface

• 3D solid

,

0

( ) ( )n

i p i

i

C u N u

P

1

1 1

11

{0, ,0, , , ,1, ,1}

m

p m p

pp

U u u

Free deformation of parametric curve

and surface (FFD)

• 1D curve

• 2D surface

• 3D solid

, , ,

0 0

( , ) ( ) ( )n m

i p j q i j

i j

S u v N u N v

P

1

1 1

11

1

1 1

11

{0, ,0, , , ,1, ,1}

{0, ,0, , , ,1, ,1}

r

p r p

pp

s

q s q

qq

U u u

V u u

Free deformation of parametric curve and

surface (FFD)

• 1D curve

• 2D surface

• 3D solid

1

1 1

11

1

1 1

11

1

1 1

11

{0, ,0, , , ,1, ,1}

{0, ,0, , , ,1, ,1}

{0, ,0, , , ,1, ,1}

o

p r p

pp

s

q s q

qq

t

r t r

rr

U u u

V u u

W u u

, , , , ,

0 0 0

( , , ) ( ) ( ) ( )n m l

i p j q k r i j k

i j k

S u v w N u N v N v

P

Free deformation of parametric curve

and surface (FFD)

•1D curve

•2D surface

•3D solid

•Further expansion？

Free deformation of parametric curve

and surface (FFD)

•1D curve

•2D surface

•3D solid

•Further expansion？

Extended Free Form Deformation

Selective Course Project 6

Try to program the following shape

using GDI+:

Tips: The following code constructs a complicated region using two spline curves

using namespace Gdiplus; Graphics graphics( pDC->m_hDC ); Pen pen(Color::Blue, 3); Point point1( 50, 200); Point point2(100, 150); Point point3(160, 180); Point point4(200, 200); Point point5(230, 150); Point point6(220, 50); Point point7(190, 70); Point point8(130, 220); Point curvePoints[8] = {point1, point2, point3, point4, point5, point6, point7, point8}; Point* pcurvePoints = curvePoints; GraphicsPath path; path.AddClosedCurve(curvePoints, 8, 0.5); PathGradientBrush pthGrBrush(&path); pthGrBrush.SetCenterColor(Color(255, 0, 0, 255)); Color colors[] = {Color(0, 0, 0, 255)}; INT count = 1; pthGrBrush.SetSurroundColors(colors, &count); graphics.DrawClosedCurve(&pen, curvePoints, 8, 0.5); graphics.FillPath(&pthGrBrush, &path);