Seminar on B-Spline over triangular domain

1. Seminar on B-Spline over triangular domain Reporter: Gang Xu Institute of Computer Images and Graphics, Math Dept. ZJU October 26

2. Outline 1.Introduction 2.Mathmatic Preliminaries 3.B-Patches and Simplex Splines 4.DMS-Splines and its application 5.G-Patches 6.Future Work

3. Introduction Ramshaw’s “juiciest” challenge(1987) “Find a natural way to construct a triangular patch surface that builds in the appropriate continuity conditions, similar to what is done with the B-Spline.”

4. Introduction Bézier Curves Triangular Bézier patches B-Spline Curves What?

5. Desirable Control Scheme Attributes • Piecewise polynomial of a fixed degree • Individual piecewise polynomials are associated to regions of the domain • Control Points and Interactivity • Local Control • Automatic Continuity Maintenance • Simplifies to Univariate Splines • Numerical Stability

6. A Bleak Property Triangular Bézier Patch Continuity Constraints For a surface consisting of degree n≥1 triangular Bézier patches the highest degree of continuity possible, while still providing local flexibility, is -continuity, where

7. Examples • Cubic • Quartic The examples are from (Zhang et.al, 2005)

8. Current Situation A lot of work focus on this problem Each has its own specialized use, but Inevitably each has its own fundamental limits None is the true generalization of the B-spline!

9. Mathematical Preliminaries • BarycentricCoordinates on line

10. Mathematical Preliminaries • Barycentric coordinates on plane

11. Blossom • Fundmental Idea Represent the univariate complex function by the multivariate simple function • Related to Polar Forms

12. Blossom Principle • Symmetric • Multi-affine • Diagonal

13. Some Terminologies Multi-affine blossom of F Blossom value Blossom argument

14. Some Examples

15. Blossom form of CAGD • Most of curves and surfaces in CAGD have a blossom form • Bézier Curves • B-Spline Curves • Tensor product surfaces • Triangular Bézier patches • C- Bézier curves and H- Bézier curves, • Also their tensor product surfaces

16. Blossom of Bézier curves

17. de Casteljau algorithm in Blossom

18. Example

19. Blossom of B-spline curves

20. Blossom of Triangular Bézier patches

21. Pyramid Algorithms of B-B Surface

22. Shortcomings of B-B Surfaces • Modeling sufficiently complex surfaces requires the surfaces to have an extremely high degree Divide the domain into small triangular regions, define a B-B surfaces for each region, as B-spline curves. How can we get it?

23. B-Patches • Motivation Bézier curves B-spline curves

24. B-Patches Triangular Bézier patches

25. B-Patch’s control net

26. de Boor style algorithm of B-Patches

27. Shortcomings of B-Patches In order to be continuity, the knots along the shared domain edge must be Collinear. (Seidel,1991) Not extend well to a network of patches!

28. Example It is useless for surface modeling!

29. Simplex Splines The major problem with B-Patches is that the underlying basis functions don’t automatically provide the required degrees of continuity The simplex splines overcome it!

30. Simplex Splines Piecewise polynomial functions defined using a set of points in .The set of these points is called knot set (knot clouds). The simplex splines defined using knots has degree The simplex splines has overall continuity provided that the knot set does not contain a collinear subset of three knots. The simplex spline does not have control points.

31. Half-open Convex Hull x belongs to [v) if and only if there exists a small triangle that lies entirely within the [v] x belongs to exactly one triangle

32. Examples

33. Definition of simplex splines A degree simplex spline with knots is defined recursively as follows are barycentric coordinates with respect to

34. Examples 1

35. Examples 2

36. Examples 3

37. Examples 4

38. Shortcomings of Simplex Splines The choice of the knots to place in W during each recursive evaluation can effect the results of the computation if not chosen carefully. Plagued with numerical stability issues Computationally expensive Have no control points It is useless for surface modeling!

39. DMS-Splines Motivation nice labelling of control points B-Patches DMS-Splines Smooth basis functions Simplex splines Take the advantage of them!

40. The inventor Dahmen, Micchelli, Seidel, 1992 TVCG, IJSM,CAGD, TVC,GMOD,CGF

41. Definition of DMS-splines Triangulate the domain A knot cloud is arranged with each corner of the domain. For a degree n triangular domain, n knots are pulled out. quadratic

42. Definition of DMS-splines For a domain region control points Similar with simplex splines, define a set To be normalized, define

43. Definition of DMS-splines

44. Examples 1

45. Examples 2

46. Examples 3

47. Properties of DMS-splines Convex hull property Local control Smoothness Parametric affine invariance

48. Continuity Control by Placing Knots Make several knots collinear to decrease continuity discontinuity Four knots collinear quadratic Three knots collinear

49. Examples

50. Application(1) Filling Holes