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Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces

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Geometric Modeling and Processing - GMP 2006 (GMP 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4077))

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Abstract

This paper applies singularity theory of mappings of surfaces to 3-space and the generic transitions occurring in their deformations to develop algorithms for continuously and robustly tracking the intersection curves of two deforming parametric spline surfaces, when the deformation is represented as a family of generalized offset surfaces. This paper presents the mathematical framework, and develops algorithms accordingly, to continuously and robustly track the intersection curves of two deforming parametric surfaces, with the deformation represented as generalized offset vector fields. The set of intersection curves of 2 deforming surfaces over all time is formulated as an implicit 2-manifold \(\mathcal{I}\) in the augmented (by time domain) parametric space \(\mathbb R^5\). Hyper-planes corresponding to some fixed time instants may touch \(\mathcal{I}\) at some isolated transition points, which delineate transition events, i.e., the topological changes to the intersection curves. These transition points are the 0-dimensional solution to a rational system of 5 constraints in 5 variables, and can be computed efficiently and robustly with a rational constraint solver using subdivision and hyper-tangent bounding cones. The actual transition events are computed by contouring the local osculating paraboloids. Away from any transition points, the intersection curves do not change topology and evolve according to a simple evolution vector field that is constructed in the euclidean space in which the surfaces are embedded.

This work is supported in part by NSF CCR-0310705, NSF CCR-0310546, and NSF DMS-0405947. All opinions, findings, conclusions or recommendations expressed in this document are those of the authors and do not necessarily reflect the views of the sponsoring agencies.

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Author information

Authors and Affiliations

  1. School of Computing, University of Utah, Salt Lake City, UT, 84112, USA

    Xianming Chen, Richard F. Riesenfeld & Elaine Cohen

  2. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    James Damon

Authors
  1. Xianming Chen
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  2. Richard F. Riesenfeld
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  3. Elaine Cohen
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  4. James Damon
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Editor information

Editors and Affiliations

  1. Seoul National University, Korea

    Myung-Soo Kim

  2. Carnegie Mellon University, Pittsburgh, PA, USA

    Kenji Shimada

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© 2006 Springer-Verlag Berlin Heidelberg

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Chen, X., Riesenfeld, R.F., Cohen, E., Damon, J. (2006). Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces. In: Kim, MS., Shimada, K. (eds) Geometric Modeling and Processing - GMP 2006. GMP 2006. Lecture Notes in Computer Science, vol 4077. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11802914_8

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  • DOI: https://doi.org/10.1007/11802914_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-36711-6

  • Online ISBN: 978-3-540-36865-6

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