[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

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

  • Conference paper
Geometric Modeling and Processing - GMP 2006 (GMP 2006)

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

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Mather, J.: Stability of C  ∞ –mappings, I: The Division Theorem. Ann. of Math. 89, 89–104 (1969); II. Infinitesimal stability implies stability. Ann. of Math. 89, 254–291 (1969); III. Finitely determined map germs. Inst. Hautes Etudes Sci. Publ. Math. 36, 127–156 (1968); IV. Classification of stable germs by \(\mathbb R\)–algebras. Inst. Hautes Etudes Sci. Publ. Math. 37, 223–248 (1969); V. Transversality. Adv. in Math. 37, 301–336 (1970); VI. The nice dimensions. In: Liverpool Singularities Symposium I. Springer Lecture Notes in Math. vol. 192, pp. 207–253 (1970)

    Google Scholar 

  2. Abdel-Malek, K., Yeh, H.: Determining intersection curves between surfaces of two solids. Computer-Aided Design 28(6-7), 539–549 (1996)

    Article  Google Scholar 

  3. Bajaj, C.L., Hoffmann, C.M., Lynch, R.E., Hopcroft, J.E.H.: Tracing surface intersections. Computer Aided Geometric Design 5(4), 285–307 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barnhill, R.E., Farin, G., Jordan, M., Piper, B.R.: Surface/surface intersection. Computer Aided Geometric Design 4(1-2), 3–16 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barnhill, R.E., Kersey, S.N.: A marching method for parametric surface/surface intersection. Computer Aided Geometric Design 7(1-4), 257–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Damon, J.: On the Smoothness and Geometry of Boundaries Associated to Skeletal Structures I: Sufficient Conditions for Smoothness. Annales Inst. Fourier 53, 1941–1985 (2003)

    MathSciNet  Google Scholar 

  7. Damon, J.: On the Smoothness and Geometry of Boundaries Associated to Skeletal Structures II: Geometry in the Blum Case. Compositio Mathematica 140(6), 1657–1674 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Damon, J.: Determining the Geometry of Boundaries of Objects from Medial Data. Int. Jour. Comp. Vision 63(1), 45–64 (2005)

    Article  Google Scholar 

  9. Elber, G., Cohen, E.: Error bounded variable distance offset operator for free form curves and surfaces. Int. J. Comput. Geometry Appl. 1(1), 67–78 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Elber, G., Kim, M.-S.: Geometric constraint solver using multivariate rational spline functions. In: Symposium on Solid Modeling and Applications, pp. 1–10 (2001)

    Google Scholar 

  11. Elber, G., Lee, I.-K., Kim, M.-S.: Comparing offset curve approximation methods. IEEE Computer Graphics and Applications 17, 62–71 (1997)

    Article  Google Scholar 

  12. Farouki, R.T., Neff, C.A.: Analytic properties of plane offset curves. Computer Aided Geometric Design 7(1-4), 83–99 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Goldman, R.: Curvature formulas for implicit curves and surfaces. cagd 22(7), 632–658 (2005)

    MATH  Google Scholar 

  14. Hamann, B.: Visualization and Modeling Contours of Trivariate Functions, Ph.D. thesis, Arizona State Univeristy (1991)

    Google Scholar 

  15. Hohmeyer, M.E.: A surface intersection algorithm based on loop detection. In: Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications, May 1991, pp. 197–207. ACM Press, New York (1991)

    Chapter  Google Scholar 

  16. Hu, C.Y., Maekawa, T., Patrikalakis, N.M., Ye, X.: Robust Interval Algorithm for Surface Intersections. Computer Aided Design 29(9), 617–627 (1997)

    Article  MATH  Google Scholar 

  17. Jun, C.-S., Kim, D.-S., Kim, D.-S., Lee, H.-C., Hwang, J., Chang, T.-C.: Surface slicing algorithm based on topology transition. Computer-Aided Design 33(11), 825–838 (2001)

    Article  Google Scholar 

  18. Kimmel, R., Bruckstein, A.M.: Shape offsets via level sets. Computer-Aided Design 25(3), 154–162 (1993)

    Article  MATH  Google Scholar 

  19. Koenderink, J.J.: Solid Shape. MIT Press, Cambridge (1990)

    Google Scholar 

  20. Kriezis, G.A., Patrikalakis, N.M., Wolter, F.E.: Topological and differential-equation methods for surface intersections. Computer-Aided Design 24(1), 41–55 (1992)

    Article  MATH  Google Scholar 

  21. Kumar, G.V.V.R., Shastry, K.G., Prakash, B.G.: Computing offsets of trimmed NURBS surfaces. Computer-Aided Design 35(5), 411–420 (2003)

    Article  Google Scholar 

  22. Lang, S.: Undergraduate Analysis, 2nd edn. Springer, Heidelberg (1997)

    Google Scholar 

  23. Maekawa, T.: An overview of offset curves and surfaces. Computer-Aided Design 31(3), 165–173 (1999)

    Article  MATH  Google Scholar 

  24. Maekawa, T., Patrikalakis, N.M.: Computation of singularities and intersections of offsets of planar curves. Computer Aided Geometric Design 10(5), 407–429 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  25. Markot, R.P., Magedson, R.L.: Solutions of tangential surface and curve intersections. Computer-Aided Design 21(7), 421–427 (1989)

    Article  Google Scholar 

  26. O’Neill, B.: Elementary Differential Geometry, 2nd edn. Academic Press, London (1997)

    MATH  Google Scholar 

  27. Ouyang, Y., Tang, M., Lin, J., Dong, J.: Intersection of two offset parametric surfaces based on topology analysis. Journal of Zhejiang Univ. SCI 5(3), 259–268 (2004)

    Article  MATH  Google Scholar 

  28. Patrikalakis, N.M., Maekawa, T., Ko, K.H., Mukundan, H.: Surface to Surface Intersections. Computer-Aided Design and Applications 1(1-4), 449–458 (2004)

    Google Scholar 

  29. Pham, B.: Offset curves and surfaces: a brief survey. Computer-Aided Design 24(4), 223–229 (1992)

    Article  Google Scholar 

  30. Preparata, F.P., Shamos, M.I.: Computational geometry: an introduction. Springer, Heidelberg (1985)

    Google Scholar 

  31. Sederberg, T.W., Christiansen, H.N., Katz, S.: Improved test for closed loops in surface intersections. Computer-Aided Design 21(8), 505–508 (1989)

    Article  MATH  Google Scholar 

  32. Sederberg, T.W., Meyers, R.J.: Loop detection in surface patch intersections. Computer Aided Geometric Design 5(2), 161–171 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sherbrooke, E.C., Patrikalakis, N.M.: Computation of the solutions of nonlinear polynomial systems. Computer Aided Geometric Design 10(5), 379–405 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  34. Smith, T.S., Farouki, R.T., al Kandari, M., Pottmann, H.: Optimal slicing of free-form surfaces. Computer Aided Geometric Design 19(1), 43–64 (2002)

    Article  MATH  Google Scholar 

  35. Soldea, O., Elber, G., Rivlin, E.: Global Curvature Analysis and Segmentation of Volumetric Data Sets using Trivariate B-spline Functions. In: Geometric Modeling and Processing 2004, April 2004, pp. 217–226 (2004)

    Google Scholar 

  36. Thirion, J.-P., Gourdon, A.: Computing the Differential Characteristics of Isointensity Surfaces. Journal of Computer Vision and Image Understanding 61(2), 190–202 (1995)

    Article  Google Scholar 

  37. Wallner, J., Sakkalis, T., Maekawa, T., Pottmann, H., Yu, G.: Self-Intersections of Offset Curves and Surfaces. International Journal of Shape Modelling 7(1), 1–21 (2001)

    Article  Google Scholar 

  38. Xu, G., Bajaj, C.L.: Curvature Computations of 2-Manifolds in ℝk

    Google Scholar 

  39. Ye, X., Maekawa, T.: Differential Geometry of Intersection Curves of Two Surfaces. Computer Aided Geometric Design 16(8), 767–788 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard F. Riesenfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elaine Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. James Damon
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Seoul National University, Korea

    Myung-Soo Kim

  2. Carnegie Mellon University, Pittsburgh, PA, USA

    Kenji Shimada

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

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

Download citation

  • 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

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation