1. Tracking Intersection Curves of Two Deforming Parametric Surfaces Xianming Chen¹, Richard Riesenfeld¹ Elaine Cohen¹, James Damon² ¹School of Computing, University of Utah ²Department of Mathematics, UNC

3. Two Main Ideas Construct evolution vector field –To follow the gradual change of intersection curve (IC) Apply Morse theory and Shape Operator –To compute topological change of IC Formulate locus of IC as 2-manifold in parametric 5-space Compute quadric approximation at critical points of height function

4. Deformation as Generalized Offset

5. Crv-Crv IP Under Deformation

6. Tangent Movement

7. Evolution Vector Field

8. Evolution Algorithm

9. Surface Case

10. A Local Basis

11. Evolution Vector Field

12. Evolution Vector Field in Larger Context Well-defined actually in a neighborhood of any P in R³, where two surfaces deform to P at t 1 and t 2 Vector field is on the tangent planes of level set surfaces defined by f = t 1 - t 2 Locus of ICs is one of such level surfaces.

13. Topological Change of IC s

14. 2-Manifold in Parametric 5-space

15. IC as Height Contour

16. Critical Points of Height Function

17. 4 Generic Transition Events

18. A Comment Singularity theory of stable surface mapping in physical space R 3 {x, y, z } Morse theory of height function in augmented parametric space R 5 { s 1, s 2, ŝ 1, ŝ 2, t }

19. Tangent Vector Fields

20. Computing Tangent Vector Fields

21. Computing Transition Events

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

23. Thank you!