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 t1 and 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.

Topological Change of ICs

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

R3 {x, y, z}

Morse theory of height

function in augmented

parametric space R5

{ s1 , s2 , ŝ1, ŝ2 , 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 2nd order shape computation at critical points of a 2-manifold in a parametric 5-space.

Thank you!