1. 1 Dr. Scott Schaefer Analysis of Subdivision Surfaces at Extraordinary Vertices

2. 2/50 Structure of Subdivision Surfaces

3. 3/50 Structure of Subdivision Surfaces

4. 4/50 Structure of Subdivision Surfaces

5. 5/50 Structure of Subdivision Surfaces

6. 6/50 Structure of Subdivision Surfaces

7. 7/50 Structure of Subdivision Surfaces

8. 8/50 Structure of Subdivision Surfaces

9. 9/50 Structure of Subdivision Surfaces

10. 10/50 Structure of Subdivision Surfaces If ordinary case is smooth, then obviously entire surface is smooth except possibly at extraordinary vertices

11. 11/50 Smoothness of Surfaces A surface is a C k manifold if locally the surface is the graph of a C k function Must develop a local parameterization around extraordinary vertices to analyze smoothness

12. 12/50 Subdivision Matrices Encode local subdivision rules around extraordinary vertex

13. 13/50 Subdivision Matrix Example

14. 14/50 Subdivision Matrix Example Repeated multiplication by S performs subdivision locally Only need to analyze S to determine smoothness of the subdivision surface

15. 15/50 Smoothness at Extraordinary Vertices Reif showed that it is necessary for the subdivision matrix S to have eigenvalues of the form where for the surface to be C 1 at the extraordinary vertex A sufficient condition for C 1 smoothness is that the characteristic map must be regular and injective

16. 16/50 The Characteristic Map Let the eigenvalues of S be of the form where. The eigenvectors associated with provide a local parameterization around the extraordinary vertex

17. 17/50 The Characteristic Map

18. 18/50 The Characteristic Map

19. 19/50 The Characteristic Map

20. 20/50 The Characteristic Map

21. 21/50 Analyzing Arbitrary Valence Matrices become very large, very quickly Must analyze every valence independently Need tools for somehow analyzing eigenvalues/vectors of arbitrary valence easily

22. 22/50 Structure of Subdivision Matrices

23. 23/50 Structure of Subdivision Matrices Circulant matrix

24. 24/50 Circulant Matrices Matrix whose rows are horizontal shifts of a single row

25. 25/50 Eigenvalues/vectors of Circulant Matrices Given an circulant matrix with rows associated with c(x), its eigenvalues are of the form and has eigenvectors where and

26. 26/50 Eigenvalues/vectors of Circulant Matrices Given an circulant matrix with rows associated with c(x), its eigenvalues are of the form and has eigenvectors where and

27. 27/50 Eigenvalues/vectors of Circulant Matrices Given an circulant matrix with rows associated with c(x), its eigenvalues are of the form and has eigenvectors where and

28. 28/50 Block-Circulant Matrices Matrix composed of circulant matrices

29. 29/50 Block-Circulant Matrices Matrix composed of circulant matrices

30. 30/50 Block-Circulant Matrices Matrix composed of circulant matrices

31. 31/50 Eigenvalues/vectors of Block-Circulant Matrices Find eigenvalues/vectors of block matrix eigenvectorseigenvaluesinverse of eigenvectors

32. 32/50 Eigenvalues/vectors of Block-Circulant Matrices Find eigenvalues/vectors of block matrix Eigenvalues of block matrix are eigenvalues of expanded matrix evaluated at eigenvectorseigenvaluesinverse of eigenvectors

33. 33/50 Eigenvalues/vectors of Block-Circulant Matrices Find eigenvalues/vectors of block matrix Eigenvalues of block matrix are eigenvalues of expanded matrix evaluated at Eigenvectors of block matrix are multiples of times eigenvectors of block matrix eigenvectorseigenvaluesinverse of eigenvectors

34. 34/50 Eigenvalues/vectors of Block-Circulant Matrices

35. 35/50 Eigenvalues/vectors of Block-Circulant Matrices

36. 36/50 Example: Loop Subdivision

37. 37/50 Example: Loop Subdivision Some parts of the matrix are not circulant

38. 38/50 Example: Loop Subdivision Eigenvectors/values for block-circulant portion are eigenvectors/values for entire matrix except at j=0

39. 39/50 Example: Loop Subdivision

40. 40/50 Example: Loop Subdivision

41. 41/50 Example: Loop Subdivision

42. 42/50 Example: Loop Subdivision Subdominant eigenvalue is Corresponding eigenvector is

43. 43/50 Example: Loop Subdivision Subdominant eigenvalue is Corresponding eigenvector is Plot real/imaginary parts to create char map

44. 44/50 Example:Loop Subdivision

45. 45/50 Application: Exact Evaluation S

46. 46/50 Application: Exact Evaluation Subdivide until x is in ordinary region

47. 47/50 Application: Exact Evaluation Subdivide until x is in ordinary region

48. 48/50 Application: Exact Evaluation Subdivide until x is in ordinary region

49. 49/50 Application: Exact Evaluation Subdivide until x is in ordinary region Extract B-spline control points and evaluate at x

50. 50/50 Application: Exact Evaluation Subdivide until x is in ordinary region Extract B-spline control points and evaluate at x