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 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.

13. Topological Change of ICs

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 R3 {x, y, z} Morse theory of height function in augmented parametric space R5 { s1,s2,ŝ1,ŝ2,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 2nd order shape computation at critical points of a 2-manifold in a parametric 5-space.

23. Thank you!