1 Dr. Scott Schaefer Coons Patches and Gregory Patches

1

Dr. Scott Schaefer

Coons Patches and Gregory Patches

2/39

Patches With Arbitrary Boundaries

Given any 4 curves, f(s,0), f(s,1), f(0,t), f(1,t) that meet continuously at the corners, construct a smooth surface interpolating these curves

3/39

Patches With Arbitrary Boundaries

Given any 4 curves, f(s,0), f(s,1), f(0,t), f(1,t) that meet continuously at the corners, construct a smooth surface interpolating these curves

)0,(sf

)1,(sf

),0( tf

),1( tf

s

t

4/39

Coons Patches

Build a ruled surface between pairs of curves

)0,(sf

)1,(sf

),0( tf

),1( tf

s

t

5/39

Coons Patches


)0,(sf

)1,(sf

),0( tf

),1( tf

s

t

)1,()0,()1(),(1 sftsfttsf

),1(),0()1(),(2 tfstfstsf

6/39

Coons Patches


)0,(sf

)1,(sf

s

t

)1,()0,()1(),(1 sftsfttsf

7/39

Coons Patches


),0( tf

),1( tf

s

t

),1(),0()1(),(2 tfstfstsf

8/39

Coons Patches

“Correct” surface to make boundaries match

)0,(sf

)1,(sf

s

t

))1,1()0,1()1()1,0()1()0,0()1)(1((),(),( 21 ftsftsftsftstsftsf

),0( tf

),1( tf

9/39

Coons Patches

“Correct” surface to make boundaries match

)0,(sf

)1,(sf

s

t

t

t

ff

ffss

tf

tfss

t

tsfsf

1

)1,1()0,1(

)1,0()0,0(1

),1(

),0(1

1)1,()0,(

),0( tf

),1( tf

10/39

Properties of Coons Patches

Interpolate arbitrary boundaries Smoothness of surface equivalent to

minimum smoothness of boundary curves Don’t provide higher continuity across

boundaries

11/39

Hermite Coons Patches

Given any 4 curves, f(s,0), f(s,1), f(0,t), f(1,t) that meet continuously at the corners and cross-boundary derivatives along these edges

, construct a smooth surface interpolating these curves and derivatives

stf

stf

tsf

tsf

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

12/39


Use Hermite interpolation!!!

2

2

2

2

)23(

)1(

)1(

)21()1(

)(

tt

tt

tt

tt

tH

)()1,()0,(),( )1,()0,(1 tHsfsftsf t

sftsf

13/39



)(),1(),0(),( ),1(),0(2 sHtftftsf s

tfstf

2

2

2

2

)23(

)1(

)1(

)21()1(

)(

tt

tt

tt

tt

tH

14/39



)(

)1,1()0,1(

)1,0()0,0(

)(),(),(),(

)1,1()0,1(

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

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

)1,0()0,0(

21 tH

ff

ff

sHtsftsftsf

vf

vf

uf

vuf

vuf

uf

uf

vuf

vuf

uf

vf

vf

T

2

2

2

2

)23(

)1(

)1(

)21()1(

)(

tt

tt

tt

tt

tH

15/39



)(

)1,1()0,1(

)1,0()0,0(

)(),(),(),(

)1,1()0,1(

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

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

)1,0()0,0(

21 tH

ff

ff

sHtsftsftsf

vf

vf

uf

vuf

vuf

uf

uf

vuf

vuf

uf

vf

vf

T

2

2

2

2

)23(

)1(

)1(

)21()1(

)(

tt

tt

tt

tt

tH

Requires mixed partials

16/39

Problems With Bezier Patches

00p 10p 20p 30p

01p 11p 21p 31p

02p 12p 22p 32p

03p 13p 23p 33p

17/39


s

tsf

),(

18/39


t

tsf

),(

19/39


Derivatives along edges not independent!!!

Solution

20/39

Solution

21/39

22/39

Gregory Patches

00p 10p 20p 30p

01p tp11tp21

31p

02p sp12sp22 32p

03p 13p 23p 33p

tp12tp22

sp21sp11

23/39

Gregory Patch Evaluation

00p 10p 20p 30p

01p tp11tp21

31p

02p sp12sp22 32p

03p 13p 23p 33p

tp12tp22

sp21sp11

ts

psptp

ts

111111

ts

psptp

ts

1

)1( 212121

ts

psptp

ts

2

)1()1( 222222

ts

psptp

ts

1

)1( 121212

24/39

Gregory Patch Evaluation

00p 10p 20p 30p

01p tp11tp21

31p

02p sp12sp22 32p

03p 13p 23p 33p

tp12tp22

sp21sp11

Derivative along edge decoupled from adjacent edge at interior points

25/39

Gregory Patch Properties

Rational patches Independent control of derivatives along

edges except at end-points Don’t have to specify mixed partial

derivatives Interior derivatives more complicated due to

rational structure Special care must be taken at corners (poles

in rational functions)

26/39

Constructing Smooth Surfaces With Gregory Patches Assume a network of cubic curves forming

quad shapes with curves meeting with C1 continuity

Construct a C1 surface that interpolates these curves

27/39

Constructing Smooth Surfaces With Gregory Patches Need to specify interior points for cross-

boundary derivatives Gregory patches allow us to consider each

edge independently!!!

28/39




Fixed control points!!

29/39




0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u

30/39




ii

i vtBtv )()( 3

ii

i vtBtv ˆ)()(ˆ 3

ii

i utBtu )()( 2

0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u

31/39




)(ˆ)()( tvtutv

Derivatives must be linearly dependent!!! 0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u

32/39




33333

00000

ˆ

ˆ

vuv

vuv

By construction, property holds at end-points!!! 0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u

33/39




0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u)(ˆ)())1(()())1(( 3030 tvtutttvtt

Assume weights change linearly

34/39




0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u)(ˆ)())1(()())1(( 3030 tvtutttvtt

Assume weights change linearly

A quartic function. Not possible!!!

35/39




0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u)(ˆ)())1(()())1(( 3030 tvtutttvtt

Require v(t) to be quadratic

033 3210 vvvv

36/39




0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u323

2133322

3032

130011

0330

3003

ˆ

ˆ

uuvvv

uuvvv

033 3210 vvvv

37/39

Constructing Smooth Surfaces With Gregory Patches Problem: construction is not symmetric

is quadratic is cubic

0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u323

2133322

3032

130011

0330

3003

ˆ

ˆ

uuvvv

uuvvv

033 3210 vvvv

)(ˆ tv)(tv

38/39

Constructing Smooth Surfaces With Gregory Patches Solution: assume v(t) is linear and use

to find Same operation to find

0v 1v 2v 3v

0v̂ 1̂v 2v̂ 3v̂

0u 1u 2u323

2133322

3032

130011

0330

3003

ˆ

ˆ

uuvvv

uuvvv

033 3210 vvvv

332

031

2331

032

1 , vvvvvv 21 ˆ,ˆ vv

21, vv

39/39

Constructing Smooth Surfaces With Gregory Patches Advantages

Simple construction with finite set of (rational) polynomials

DisadvantagesNot very flexible since cross-boundary

derivatives are not full cubics

If cubic curves not available, can estimate tangent planes and build hermite curves

Documents

1 Dr. Scott Schaefer Coons Patches and Gregory Patches