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

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

Deformation as Generalized Offset

Crv-Crv IP Under Deformation

Tangent Movement

Evolution Vector Field

Evolution Algorithm

Surface Case

A Local Basis

Evolution Vector Field

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.

Topological Change of IC s

2-Manifold in Parametric 5-space

IC as Height Contour

Critical Points of Height Function

4 Generic Transition Events

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 }

Tangent Vector Fields

Computing Tangent Vector Fields

Computing Transition Events

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.

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