Dynamic Geometric Computation of Interacting Models*

1. Dynamic Geometric Computation of Interacting Models* Richard Riesenfeld University of Utah May 2008 • * In collaboration with • Xianming Chen¹, E Cohen¹, J Damon² • _______________________________ • 1. University of Utah • 2. University of North Carolina Dagstuhl

2. Today: Intersection of Two Deforming Parametric Surfaces Dagstuhl

3. Interactions are Complex Dagstuhl

4. Interactions are Complex Dagstuhl

5. Interactions are Complex Dagstuhl

6. Classification Computation Overall Process Detection Evolution Identification Dagstuhl

7. Two Main Ideas • Construct evolution vector field to follow the gradual change of intersection curve IC • Use Singularity Theory and Shape Operator to compute topological change of IC • Formulate locus of IC as 2-manifold in parametric 5-space • Compute quadric approx at critical points of height function Dagstuhl

8. Exchange Event Dagstuhl

9. Deformation as Generalized Offset Dagstuhl

10. Curve /Curve IPUnder Deformation Dagstuhl

11. Tangent Movement Dagstuhl

12. Evolution Vector Field Dagstuhl

13. Evolution Algorithm Dagstuhl

14. Surface Case Dagstuhl

15. Local Basis Dagstuhl

16. Evolution Vector Field Dagstuhl

17. Evolution Vector Field in Larger Context • Well-defined actually in a neighborhood of any P in ³, where two surfaces deform to P at t1and t2 • Vector field is on the tangent planes of level set surfaces defined by f=t1 - t2 • Locus of ICs is one of such level surfaces. Dagstuhl

18. Topological Change of ICs Dagstuhl

19. 2-Manifold in Parametric 5-space Dagstuhl

20. IC as Height Contour Dagstuhl

21. Critical Points of Height Function Dagstuhl

22. 4 Generic Transition Events Dagstuhl

23. Comment Morse theory of height function in augmented parametric space R5{ s1 , s2, ŝ1, ŝ2 , t} Singularity theory of stable surface mapping in physical space R3{x,y, z} Dagstuhl

24. Tangent Vector Fields Dagstuhl

25. Computing Tangent Vector Fields Dagstuhl

26. Computing Transition Events Dagstuhl

27. Future Directions • Application uses • Real models • More complex interactions • More general situations • Better understanding of singularities Dagstuhl

28. Conclusion • A general mathematical framework for dynamic geometric computation with B-splines • Evolve to neighboring solution by following tangent • Identify transition points by solving a rational system • Compute transition events by computing 2ndfundamental form on manifold Dagstuhl

29. Conclusion General mathematical framework for dynamic geometric computation with B-splines • Encode all solutions as a manifold in product space of curves/surfaces parametric space and deformation control space • Construct families of tangent vectors on the manifold Dagstuhl

30. References Theoretically Based Algorithms for Robustly Tracking Intersection Curves of Deforming Surfaces, Xianming Chen, R.F. Riesenfeld, Elaine Cohen, James N. Damon, CAD,Vol 39, No 5, May 2007,389-397  Dagstuhl

31. vielen Dank für die Einladung Dagstuhl

32. und auf Wiedersehen Dagstuhl

33. Dagstuhl

34. Conclusion • Solve dynamic intersection curves of 2 deforming B-splinesurfaces • Deformation represented as generalized offset surfaces • Implemented in B-splines, exploiting its symboliccomputation and subdivision-based 0-dimensional root finding. • Evolve ICs by following evolution vector field • Create, annihilate, merge or split IC by 2nd order shape computation at critical points of a 2-manifold in a parametric 5-space. Dagstuhl

35. Outline • Essential issues • General mathematical frame • Point-curve distance tracking • Surface-surface intersection tracking • Efficient NURBS symbolic computation Dagstuhl

36. Evolution • Identification • Detection • Classification • Computation Dagstuhl

37. Outline • Essential issues • General mathematical frame • Point-curve distance tracking • Surface-surface intersection tracking • Efficient NURBS symbolic computation Dagstuhl

38. Singularities of Differential Map • f :Rm→RnJacobian matrix singular • f :Rm→R f1 = f2 = … = fm = 0 • Hessian matrix H =( fij), nonsingular • Critical points classified by Morse index of H Dagstuhl

39. General Mathematical Frame -1 • Construct a manifold in the solution space Dagstuhl

40. General Mathematical Frame -2 • Construct dfamilies of tangent vector fields • Define projection map from the manifold to control space Dagstuhl

41. General Mathematical Frame -1 Construct a manifold in the solution space Dagstuhl

42. General Mathematical Frame -3 • Singularities of projection map • critical set in the solution space • transition set in the control space Dagstuhl

43. General Mathematical Frame -3 • Identify singularities • subdivision-based constraint solver • Robust guarantee for 0-dimensional solution • NURBS algebraic operation • Just for point-curve distance tracking • Robustness guarantee even though 1-dimensional Dagstuhl

44. General Mathematical Frame -4 • Evolution when away from transition set • d = 0 is simple • d > 0 needs extra effort • Heuristics from front propagation • Extra d constraints Dagstuhl

45. General Mathematical Frame -5 Transition when crossing transition set Restrict the projection to perturbation line Morse function Local 2nd order differential computation to catch global topology change Dagstuhl 45

46. General Mathematical Frame -5 • Transition when crossing transition set • Restrict the projection to perturbation line • Morse function • Local 2nd order differential computation to catch global topology change Dagstuhl

47. Outline • Essential issues • General mathematical frame • Point-curve distance tracking • Surface-surface intersection tracking • Efficient NURBS symbolic computation Dagstuhl

48. Critical Distance (CD) extremal and perpendicular extremal and perpendicular extremal and perpendicular Dagstuhl

49. Type Discriminant D Dagstuhl

50. Distance Tracking Problem • Given critical distances of P to the curve • If P is perturb on the plane by • Create any new CDs if any • Annihilate any old CDs if any • Evolve the rest of CDs • Distance tracking without global searching Dagstuhl