1. Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)

2. Jorg Peters, SurfLab Polynomial heritage of generalized spline subdivision Doo-Sabin Catmull-Clark

3. Jorg Peters, SurfLab Polynomial heritage of generalized spline subdivision Increasing regions are regular: points and faces have standard valence

4. Jorg Peters, SurfLab Polynomial heritage of generalized spline subdivision Doo-Sabin bi-2 B-spline Catmull-Clark bi-3 B-spline MidedgeZwart-Powell C^1 box-spline LoopC^2 box-spline box-spline = generalization of B-spline to shift-invariant partitions book: [de Boor, Hollig, Riemenschneider 94]

5. Jorg Peters, SurfLab Polynomial heritage of generalized spline subdivision Subdivision of the Zwart-Powell C^1 quadratic box-spline a cd Subdi vision basis function Subdivision Rule Subdi vision

6. Jorg Peters, SurfLab Polynomial heritage of generalized spline subdivision 2 steps4 steps Zwart-Powell subdivision = 2 steps of Midedge subdivision 2 1 1 regular: 4-valence, quadrilaterals Mid-edge Rule (“simplest rule”)

7. Jorg Peters, SurfLab Polynomial heritage of generalized spline subdivision Increasing regions are regular (polynomial) Union of surface layers at an extra- ordinary point

8. Jorg Peters, SurfLab Polynomial heritage of generalized spline subdivision Uses: Representation as Bezier patches Evaluation at non-binary points Fast moment computation

9. Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and Eigenvalues Jorg Peters SurfLab (Purdue,UFL)

10. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Challenge: Exponential increase in the number of facets! Volume Inertia Frame Center of mass

11. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces   V f     V SU dUnf dS nn/ f f dV   S dSnn/ f Theory: Gauss’ Divergence Theorem: The integral of the divergence over the volume equals the integral of the normal component over the surface S dV

12. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces  U n Theory: Change of variables The area of the surface element S equals the integral of the Jacobian |n| of the surface parametrization (x,y,z) over the domain U  S dS dU     V S   U nf dS nn/ f f dV

13. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces   VU uvvu dvdu ]yx-y[x z dV1 For example, f=[0,0,z] n = f n = z is piecewise polynomial in regular regions Volume = u vvu yxyx  u vvu yxyx     ppatch Up p u p v p v p u p dvdu )yxyx(z

14. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces “Volume” patch p =   Up p u p v p v p u p dvdu )yxyx(z Schema for bi-3 Bezier patch

15. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Volume patch p =   Up p u p v p v p u p dvdu )yxyx(z

16. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Volume patch p =   Up p u p v p v p u p dvdu )yxyx(z

17. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Volume patch p =   Up p u p v p v p u p dvdu )yxyx(z

18. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Work: at each subdivision steplinear for each extraordinary point add volume contribution of 3n patches Doo-Sabin

19. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces   Up p u p v p v p u p dvdu )yxyx(z m m i i WV    0 V i V i+1   i i V layerin p Volume m W

20. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Error estimation: bounding boxes

21. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Geometric decay of error volume 1, 1/8, 1/64,...

22. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Computing geometry given a fixed volume Bisection

23. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Higher moments and the inertia frame center of mass:  VVV dV zdV,y,dVx inertia tensor:  V dV,...xy..., eigenvector frame

24. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Higher moments and the inertia frame center of mass

25. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Physics-based animation Center of mass support

26. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Simple registration, comparison matching frames = computing a 3x3 matrix Q: IP Q = IS

27. Jorg Peters, SurfLab Moments of objects enclosed by generalized subdivision surfaces Solution: Moments efficiently and exactly computed via Gauss’ theorem and polynomial heritage Volume Inertia Frame Center of mass

28. Jorg Peters, SurfLab

29. Shape and eigenvalues Union of surface layers at an extra- ordinary point Control points transformed by the subdivision matrix

30. Jorg Peters, SurfLab Shape and eigenvalues i eigenvector expansion

31. Jorg Peters, SurfLab Shape and eigenvalues If all < 1, then collapse If some > 1, then unbounded growth Good sequence: 1,,, … where | | < 1 Eigenvectors of determine the tangent plane

32. Jorg Peters, SurfLab Shape and eigenvalues Fast contraction of 3-sided facets = (1+cos(2pi/ 3))/2 =.25 Slow contraction of large facets = (1+cos(2pi/16))/2 =.962... midedge subdivision

33. Jorg Peters, SurfLab Shape and eigenvalues adjust subdominant eigenvalues (modified midedge subdivision) =. 5

34. Jorg Peters, SurfLab Shape and eigenvalues

35. Jorg Peters, SurfLab Generalized spline subdivision Summary Polynomial Heritage regular regions Computing Moments Gauss’ theorem Shape and Eigenvalues subdominant values