Overview

June 9- B-Spline Curves

June 16- NURBS Curves

June 30- B-Spline Surfaces

Curves

Surfaces

)()( uBu ii

i VQ

),(),( vuBvu rr

r VQ

???),( vuBr

)()(),( vBuBvuB jiij Tensor Product B-Spline

iu 1iu 2iu 3iu 4iu

iu 1iu2iu 3iu

4iu

jv1jv

2jv3jv

4jv

z

)(uBi

),( vuBij2C

Adding Knots

iu 1iu2iu 3iu

4iu

jv1jv

2jv3jv

4jv

z

5iu

iu 1iu2iu

3iu4iu

jv1jv

2jv3jv4jv

z

5iu

5jv

),( vuBij-Piecewise (16 parametric regions);

-Bicubic;

-C2;

-Local Support;

-Local supports cover the plane of the parameters in a regular fashion;

-Sum up to one;

Tensor Product B-Spline

Uniform cubic B-spline Curves

Uniform bicubic B-spline Surfaces

)()( uBu ii

i VQ ),(),( vuBvu iji j

ij VQ

iV ijV

)(uBi )()(),( vBuBvuB jiij

0u

Expression

Vertices

Basis Functions

Parameter Space

1u 2u Mu

0u 1u 2u 3u 4u 1Mu Mu

Mv

1Mv

0v

1v

3v

4v

)()(),(),(0

3

0

3,,, 11

vBuBvuvu sjrir s

sjrivvuuijjjii

VQQ

),( vuijQ

),()(0

3

0

3, vbub sr

r ssjri

V

10,10 vu

Local expression

u

Surfaces and Curves

),()()()( 03122130 vbvbvbvb WWWW

)()()(0

3

0

3,,, vbubv sr

r ssjriuji VQ

);()()()( 0,1,12,23,33 ubububub jijijiji VVVVW

);()()()( 01,11,121,231,32 ubububub jijijiji VVVVW

);()()()( 02,12,122,232,31 ubububub jijijiji VVVVW

).()()()( 03,13,123,233,30 ubububub jijijiji VVVVW

Continuity: C2

)()()()()( 04132231,1, vbvbvbvbvuji WWWWQ

).()()()( 01,11,121,23!,34 ubububub jijijiji VVVVW

ujv

1jv

iu 1iu

2jv

3,3,13,23,3

2,2,12,22,3

1,1,11,21,3

,,1,2,3

jijijiji

jijijiji

jijijiji

jijijiji

VVVV

VVVV

VVVV

VVVV

Surface Patch

3,13,3,13,23,3

2,12,2,12,22,3

1,11,1,11,21,3

,1,,1,2,3

1,11,1,11,21,3

jijijijiji

jijijijiji

jijijijiji

jijijijiji

jijijijiji

VVVVV

VVVVV

VVVVV

VVVVV

VVVVV

Four Patches

- C2

- Counting the surface patches…

- Convex Hull

- Rotation

- Scaling

- Translation

It requires 16 Control Vertices to define a patch.

11 nm Control Vertices

22 nm Patches

Properties: Uniform bicubic B-spline Surfaces

),( vuijQ )()(0

3

0

3, vbub sr

r ssjri

V

Boundary Conditions

3,3,13,23,3

2,2,12,22,3

1,1,11,21,3

,,1,2,3

jijijiji

jijijiji

jijijiji

jijijiji

VVVV

VVVV

VVVV

VVVV

3,3,3,13,23,33,3

3,3,3,13,23,33,3

2,2,2,12,22,32,3

1,1,1,11,21,31,3

,,,1,2,3,3

,,,1,2,3,3

jijijijijiji

jijijijijiji

jijijijijiji

jijijijijiji

jijijijijiji

jijijijijiji

VVVVVV

VVVVVV

VVVVVV

VVVVVV

VVVVVV

VVVVVV

3,3,3,3,13,23,33,33,3

3,3,3,3,13,23,33,33,3

3,3,3,3,13,23,33,33,3

2,2,2,2,12,22,32,32,3

1,1,1,1,11,21,31,31,3

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

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

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

jijijijijijijiji

jijijijijijijiji

jijijijijijijiji

jijijijijijijiji

jijijijijijijiji

jijijijijijijiji

jijijijijijijiji

jijijijijijijiji

VVVVVVVV

VVVVVVVV

VVVVVVVV

VVVVVVVV

VVVVVVVV

VVVVVVVV

VVVVVVVV

VVVVVVVV

Interpolation ?

“Closed” Surfaces

00V

10V

20V

30V

40V

50V

60V

0v1v2v

Generalization- Tensor Product Surfaces

)()(0 uBuB m

)(

)(0

vB

vB

m

mnm

n

VV

VV

0

000

),( vuQ

-Choice of basic functions;

-Given the vertices, we may compute the approximation surface;

-Given a set of points in the surface, we can compute the vertices of the interpolating surface.

Tensor Product Interpolants

)ˆ()ˆ(

)ˆ()ˆ(

)ˆ()ˆ(

)ˆ()ˆ(

0

000

0

000

0

000

mmm

m

mnm

n

mmm

m

vBvB

vBvB

uBuB

uBuB

VV

VV

)ˆ,ˆ()ˆ,ˆ(

)ˆ,ˆ()ˆ,ˆ(

0

000

mmm

m

vuvu

vuvu

QQ

QQ

Given

Wanted

SystemBVCQ

11 QCBV

2 steps:

;BDQ

.VCD

Solve (Schoenberg-Whitney)

(u-direction)

(v-direction)

Triangular Patch Surfaces

Barycentric Coordinates (r,s,t)

),,(),( ,,,,

,, tsrBvu nkji

nkji

kjikji

bQ

Control Vertices

kji ,,b

0,01,00,1),,( PPPP tsrtsr

Bernstein Polynomials

kjinkji tsr

kji

ntsrB

!!!

!),,(,,

Local Expression of a triangular Bezier Patch

P

0,1P0,0P

1,0P

Patch Domain

Parameter Space

Cubic Triangular Patch

Summary

Uniform bicubic B-Spline Functions

Generalization- Tensor Product Surfaces

Tensor Product Interpolants

Triangular Patch Surfaces