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Analysis of Subdivision Surfaces at Extraordinary Vertices

Explore structure of subdivision surfaces, smoothness at extraordinary vertices, eigenvalues/vectors, and matrix analysis in C1 smoothness. Learn to evaluate surfaces with extraordinary vertices effortlessly.

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Analysis of Subdivision Surfaces at Extraordinary Vertices

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

  2. Structure of Subdivision Surfaces

  3. Structure of Subdivision Surfaces

  4. Structure of Subdivision Surfaces

  5. Structure of Subdivision Surfaces

  6. Structure of Subdivision Surfaces

  7. Structure of Subdivision Surfaces

  8. Structure of Subdivision Surfaces

  9. Structure of Subdivision Surfaces

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

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

  12. Subdivision Matrices • Encode local subdivision rules around extraordinary vertex

  13. Subdivision Matrix Example

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

  15. 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 C1 at the extraordinary vertex • A sufficient condition for C1 smoothness is that the characteristic map must be regular and injective

  16. 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. The Characteristic Map

  18. The Characteristic Map

  19. The Characteristic Map

  20. The Characteristic Map

  21. 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. Structure of Subdivision Matrices

  23. Structure of Subdivision Matrices Circulant matrix

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

  25. 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. 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. 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. Block-Circulant Matrices • Matrix composed of circulant matrices

  29. Block-Circulant Matrices • Matrix composed of circulant matrices

  30. Block-Circulant Matrices • Matrix composed of circulant matrices

  31. Eigenvalues/vectors ofBlock-Circulant Matrices • Find eigenvalues/vectors of block matrix eigenvectors eigenvalues inverse of eigenvectors

  32. Eigenvalues/vectors ofBlock-Circulant Matrices • Find eigenvalues/vectors of block matrix • Eigenvalues of block matrix are eigenvalues of expanded matrix evaluated at eigenvectors eigenvalues inverse of eigenvectors

  33. Eigenvalues/vectors ofBlock-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 eigenvectors eigenvalues inverse of eigenvectors

  34. Eigenvalues/vectors ofBlock-Circulant Matrices

  35. Eigenvalues/vectors ofBlock-Circulant Matrices

  36. Example: Loop Subdivision

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

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

  39. Example: Loop Subdivision

  40. Example: Loop Subdivision

  41. Example: Loop Subdivision

  42. Example: Loop Subdivision • Subdominant eigenvalue is • Corresponding eigenvector is

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

  44. Example:Loop Subdivision

  45. S Application: Exact Evaluation

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

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

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

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

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

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