Richard Riesenfeld University of Utah May 2008 Dynamic Geometric Computation of Interacting Models* * In collaboration with Xianming Chen¹, E Cohen¹, J Damon² _______________________________ 1. University of Utah 2. University of North Carolina May 2008 1 Dagstuhl
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Dynamic Geometric Computation of Interacting Models*
Dynamic Geometric Computation of Interacting Models*. Richard Riesenfeld University of Utah May 2008. * In collaboration with Xianming Chen ¹ , E Cohen ¹, J Damon ² _______________________________ 1. University of Utah 2. University of North Carolina. - PowerPoint PPT Presentation
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Richard RiesenfeldUniversity of Utah
May 2008
Dynamic Geometric Computation of Interacting Models*
* In collaboration with Xianming Chen¹, E Cohen¹, J Damon² _______________________________
1. University of Utah 2. University of North Carolina
May 2008 1Dagstuhl
Today: Intersection of Two Deforming Parametric Surfaces
May 2008 2Dagstuhl
Interactions are Complex
May 2008 Dagstuhl 3
Interactions are Complex
May 2008 Dagstuhl 4
Interactions are Complex
May 2008 Dagstuhl 5
Evolution
Overall Process
May 2008 Dagstuhl 6
Classification
Computation
Identification
Detection
Two Main Ideas
• Construct evolution vector field to follow the gradual change of intersection curve IC
• Use Singularity Theory and Shape Operator to compute topological change of IC
• Formulate locus of IC as 2-manifold in parametric 5-space
• Compute quadric approx at critical points of height function
May 2008 7Dagstuhl
Exchange Event
May 2008 Dagstuhl 8
Deformation as Generalized Offset
May 2008 9Dagstuhl
Curve /Curve IP Under Deformation
May 2008 10Dagstuhl
Tangent Movement
May 2008 Dagstuhl 11
Evolution Vector Field
May 2008 12Dagstuhl
Evolution Algorithm
May 2008 13Dagstuhl
Surface Case
May 2008 14Dagstuhl
Local Basis
May 2008 15Dagstuhl
Evolution Vector Field
May 2008 16Dagstuhl
Evolution Vector Field in Larger Context
• Well-defined actually in a neighborhood of any P in ³, 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.May 2008 17Dagstuhl
Topological Change of ICs
May 2008 18Dagstuhl
2-Manifold in Parametric 5-space
May 2008 19Dagstuhl
IC as Height Contour
May 2008 20Dagstuhl
Critical Points of Height Function
May 2008 21Dagstuhl
4 Generic Transition Events
May 2008 22Dagstuhl
Comment
May 2008 23Dagstuhl
Morse theory of height function in augmented parametric space
R5{ s1 , s2 , ŝ1
, ŝ2 , t }
Singularity theory of stable surface mapping in physical space
R3{x, y, z}
Tangent Vector Fields
May 2008 24Dagstuhl
Computing Tangent Vector Fields
May 2008 25Dagstuhl
Computing Transition Events
May 2008 26Dagstuhl
Future Directions
• Application uses• Real models• More complex interactions• More general situations• Better understanding of singularities
May 2008 Dagstuhl 27
Conclusion
• A general mathematical framework for dynamic geometric computation with B-splines– Evolve to neighboring solution by following
tangent– Identify transition points by solving a rational
system– Compute transition events by computing
2nd fundamental form on manifold
May 2008 Dagstuhl 28
Conclusion
General mathematical framework for dynamic geometric computation with B-splines– Encode all solutions as a manifold in product
space of curves/surfaces parametric space and deformation control space
– Construct families of tangent vectors on the manifold
May 2008 Dagstuhl 29
ReferencesTheoretically Based Algorithms for Robustly Tracking
Intersection Curves of Deforming Surfaces, Xianming Chen, R.F. Riesenfeld, Elaine Cohen, James N. Damon, CAD,Vol 39, No 5, May 2007,389-397
May 2008 Dagstuhl 30
Dagstuhl 31May 2008
vielen Dank für die Einladung
Dagstuhl 32May 2008
und auf Wiedersehen
May 2008 33Dagstuhl
Conclusion• Solve dynamic intersection curves of
2 deforming 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.