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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
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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
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Deformation as Generalized Offset
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Crv-Crv IP Under Deformation
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Tangent Movement
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Evolution Vector Field
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Evolution Algorithm
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Surface Case
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A Local Basis
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Evolution Vector Field
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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.
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Topological Change of ICs
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2-Manifold in Parametric 5-space
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IC as Height Contour
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Critical Points of Height Function
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4 Generic Transition Events
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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 }
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Tangent Vector Fields
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Computing Tangent Vector Fields
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Computing Transition Events
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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.
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Thank you!