kmmcs, jan. 2006, spline methods in cagd, [email protected] spline methods in cagd byung-gook lee...

KMMCS, Jan. 2006, Spline Methods in CAGD, [email protected]

Spline Methods in CAGD

byung-gook leeDongseo Univ.

[email protected] http://kowon.dongseo.ac.kr/~lbg/


Contents

• Affine combination• Lagrange• Bezier• Spline• B-spline• E-spline

• Box-spline• Refinement relation• Cross-sectional volumes • Subdivision schemes

Reference“Spline Methods Draft”

Tom Lyche and Knut Morken“Subdivision Methods for

Geometric Design:A Constructive Approach”

Joe Warren and Henrik Weimer





http://www.uio.no/studier/emner/matnat/ifi/INF-MAT5340/v04/litteratur.xmlhttp://www.subdivision.org/subdivision/index.jsp

http://www.microsoft.com/typography/ClearTypeInfo.mspxhttp://www.cgal.org/


Affine combination

• Linear combinations

• Affine(Barycentric) combinations

• Convex combinations

• Barycentric coordinates


Affine combination

Euclidean coordinate system

Coordinate-free system


Polynomial interpolation


Polynomial interpolation

• Lagrange polynomials


Examples of cubic interpolation


Bezier


Representation Bezier


Bezier

• Paul de Faget de Casteljau, Citroen, 1959• Pierre Bezier, Renault, UNISUF system, 1962• A.R. Forrest, Cambridge, 1968


Properties of Bezier

• Affine invariance• Convex hull property• Endpoint interpolation• Symmetry• Linear precision• Pseudo-local control• Variation Diminishing Property


Spline curves

• J. Ferguson , Boeing Co., 1963• C. de Boor, W. Gordon, General Motors, 1963

• to interpolate given data • piecewise polynomial curves with certain

differentiability constraints • not to design free form curves


Piecewise cubic hermite interpolation


Cubic spline interpolation


Linear B-spline


Quadratic B-spline


B-spline

• Recurrence Relation

• Bernstein polynomial


B-spline


B-spline

• Smoothness=Degree-Multiplicity


Representation B-spline


Cubic B-spline


B-spline space


B-spline

• C. de Boor, 1972• W. Gordon, Richard F. Riesenfeld, 1974

• Larry L. Schumaker• Tom Lyche• Nira Dyn


B-spline basis functions

B0(x) =

(1; 0 · x < 1;0; otherwise

B1(x) =

8><

>:

x; 0 · x < 1;2¡ x; 1 · x < 2;0; otherwise

B2(x) =

8>>><

>>>:

12x2; 0 · x < 1;¡ 3

2 + 3x ¡ x2; 1 · x < 2;12(¡ 3+x)2; 2 · x < 3;0; otherwise

B(x : 0;1)

B(x : 0;1;2)

B(x : 0;1;2;3)


Refinement relation for B-splineB0(x) = B0(2x) +B0(2x ¡ 1)

B1(x) =12B1(2x) + B1(2x ¡ 1) +

12B1(2x ¡ 2)

B2(x) =14B2(2x) +

34B2(2x ¡ 1) +

34B2(2x ¡ 2) +

14B2(2x ¡ 3)

Bn(x) =12n

n+1X

i=0

µn +1

i

¶Bn(2x ¡ i)


Repeated integration for B-spline

B0(x) =

(1; 0 · x < 1;0; otherwise

Bn(x) =

Z 1

0Bn¡ 1(x ¡ t)dt

= B0(x) Bn¡ 1(x) p(x) q(x) =

Z 1

¡ 1p(t)q(x ¡ t)dt


E-splines

½®(t) = 1+(t)e®t

½¡!®(t) =P N d

m=1

P n( m )n=1 cm;n

tn ¡ 1+

(n¡ 1)!e®(m ) (t)

½¡!®(t) = (½®1 ½®2

¢¢¢ ½®n)(t) where ¡!®= (®1; : : :;®N ¡ 1;®N )

¡!®= (®n(1)

(1) ;®n( 2)

(2) ; : : : ;®n(N d )

(N d ) ) whereP N d

m=1 n(m) = N


E-splines

¯®(t) = ½®(t) ¡ e®½®(t ¡ 1) =

(e®t; 0 · t < 1;0; otherwise

¯¡!®(t) = (¯®1 ¯®2

¢¢¢ ¯®n)(t)


Truncated powers for B-spline

B1(x) = c1(x) ¡ 2c1(x ¡ 1) +c1(x ¡ 2)

cn(x) =

(1n!x

n; x ¸ 0;0; otherwise

B2(x) = c2(x) ¡ 3c2(x ¡ 1) +3c2(x ¡ 2) ¡ c2(x ¡ 3)

Bn(x) =n+1X

i=0

(¡ 1)i

µn +1

i

¶cn(x ¡ i)


Cross-sectional Volumes

cn(x) =1

pn + 1

voln

"(

f t0;t1; :: : ;tng2 (R +)n+1jnX

i=0

ti = x

) #


B-spline basis as Cross-sectional Volumes

Bn(x) =1

pn + 1

voln

"(

f t0;t1; : : : ;tng2 Hn+1jnX

i=0

ti = x

) #


Cross-sectional Volumes for subcubes

Bn(x) =1

pn + 1

voln

"(

f t0;t1; : : : ;tng2 Hn+1jnX

i=0

ti = x

) #


Bivariate Tensor Product B-spline

Bo.wrl

B1.wrl

B2.wrl

B12.wrl

B01.wrl

Bn;m(x;y) = Bn(x)Bm(y)


Box-spline as Cross-sectional Volumes

X= f fai ;bi g2 z2ji = 0;1;:: : ;ng

Bn(x) =1

pn + 1

voln

"(

f t0;t1; : : : ;tng2 Hn+1jnX

i=0

ti = x

) #

BP

n (x;y) = voln

"(

f t0;t1; : : :; tng2 Hn+1jnX

i=0

fai ;bi gti = fx;yg

) #


Bivariate Box spline over triangular grid


Shadows of boxes

Bo.wrl


Refinement Relation for B-spline

Sn(x) =

1

2(1+x)S

n¡ 1(x)

S0(x) = 1+x

S1(x) =

1

2(1+ 2x +x2

)

S2(x) =

1

4(1+3x + 3x2 +x3

)


Refinement Relation for Box-spline

BP

(x;y) =

(1; 0 · x;y < 1;0; otherwise

X= f (1;0);(0;1)g

S ~P (x;y) =12(1+xayb)S

P(x;y)

~X=

X[ f (a;b)g

SP

(x;y) =12(1+x)(1+ y)

BP

(x;y) =

BP

(2x;2y) + BP

(2x ¡ 1;2y) + BP

(2x;2y ¡ 1) +BP

(2x ¡ 1;2y ¡ 1)



~X= f (1;0);(0;1);(1;1)g

S ~P (x;y) =12(1+x)(1+y)(1+xy)

=£

1 x x2¤1

2

2

41 1 01 2 10 1 1

3

5

2

41yy2

3

5


Subdivision for Box-spline

·12 00 1

2

¸

£12

12

¤

£1

¤

·1212

¸

~X= f (1;0);(0;1);(1;1)g



~X= f(1;0);(0;1);(1;1);(¡ 1;1)g

S ~P (x;y) =14(1+x)(1+ y)(1+xy)(1+x¡ 1y)

=£

x¡ 1 1 x x2¤1

4

2

664

0 1 1 01 2 2 11 2 2 10 1 1 0

3

775

2

664

1yy2

y3

3

775


Subdivision for Box-spline

~X= f(1;0);(0;1);(1;1);(¡ 1;1)g

·14 012

14

¸

·0 1

414

12

¸

·14

12

0 14

¸

·12

14

14 0

¸



~X= f(1;0);(1;0);(0;1);(0;1)g

=£

1 x x2¤1

4

2

41 2 12 4 21 2 1

3

5

2

41yy2

3

5

S ~P (x;y) =14(1+ x)2(1+ y)2



~X= f (1;0);(1;0);(1;0);(0;1);(0;1);(0;1)g

=£

1 x x2 x3¤ 1

16

2

664

1 3 3 13 9 9 13 9 9 11 3 3 1

3

775

2

664

1yy2

y3

3

775

S ~P (x;y) =116

(1+x)3(1+ y)3



~X= f (1;0);(1;0);(0;1);(0;1);(1;1);(1;1)g

S ~P (x;y) =116

(1+x)2(1+y)2(1+xy)2

=£

1 x x2 x3 x4¤ 1

16

2

66664

1 2 1 0 02 6 6 2 01 6 10 6 10 2 6 6 20 0 1 2 1

3

77775

2

66664

1yy2

y3

y4

3

77775


Condition number


Condition number of B-spline basis

Tom Lyche and Karl Scherer, On the p-norm condition number of the multivariate triangular Bernstein basis, Journal of Computational and Applied Mathematics 119(2000) 259-273


Stability


Blossom


B-spline problems

• Degree Elevation• Degree Reduction• Knot Insertion• Knot Deletion

Gerald Farin, Curves and Surfaces for Computer Aided Geometric Design, 4 th ed, Academic Press (1996)Ronald N. Goldman, Tom Lyche, editors, Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces, SIAM (1993)


Bezier Degree Reduction


Bezier Degree Reduction

• Least square method• Legendre-Bernstein basis transformations

Rida T. Farouki, Legendre-Bernstein basis transformations, Journal of Computational and Applied Mathematics 119(2000) 145-160.Byung-Gook Lee, Yunbeom Park and Jaechil Yoo, Application of Legendre-Bernstein basis transformations to degree elevation and degree reduction, Computer Aided Geometric Design 19(2002) 709-718.


Bezier Degree Reduction with constrained


Contents

• Affine combination• Bezier curves• Spline curves• B-spline curves• Condition number• L1-norm spline• Quasi-interpolant• Box-spline• Cross-sectional volumes• Refinement relation• Subdivision









kmmcs, jan. 2006, spline methods in cagd, [email protected] spline methods in cagd byung-gook lee...

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