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

  • Xianming Chen
  • Richard F. Riesenfeld
  • Elaine Cohen
  • James Damon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4077)


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.


Transition Point Tangent Plane Geometric Design Intersection Curve Surface Intersection 
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  1. 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. 2.
    Abdel-Malek, K., Yeh, H.: Determining intersection curves between surfaces of two solids. Computer-Aided Design 28(6-7), 539–549 (1996)CrossRefGoogle Scholar
  3. 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)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barnhill, R.E., Farin, G., Jordan, M., Piper, B.R.: Surface/surface intersection. Computer Aided Geometric Design 4(1-2), 3–16 (1987)MATHCrossRefMathSciNetGoogle Scholar
  5. 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)MATHCrossRefMathSciNetGoogle Scholar
  6. 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)MathSciNetGoogle Scholar
  7. 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)MATHMathSciNetGoogle Scholar
  8. 8.
    Damon, J.: Determining the Geometry of Boundaries of Objects from Medial Data. Int. Jour. Comp. Vision 63(1), 45–64 (2005)CrossRefGoogle Scholar
  9. 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)MATHCrossRefMathSciNetGoogle Scholar
  10. 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. 11.
    Elber, G., Lee, I.-K., Kim, M.-S.: Comparing offset curve approximation methods. IEEE Computer Graphics and Applications 17, 62–71 (1997)CrossRefGoogle Scholar
  12. 12.
    Farouki, R.T., Neff, C.A.: Analytic properties of plane offset curves. Computer Aided Geometric Design 7(1-4), 83–99 (1990)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Goldman, R.: Curvature formulas for implicit curves and surfaces. cagd 22(7), 632–658 (2005)MATHGoogle Scholar
  14. 14.
    Hamann, B.: Visualization and Modeling Contours of Trivariate Functions, Ph.D. thesis, Arizona State Univeristy (1991)Google Scholar
  15. 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)CrossRefGoogle Scholar
  16. 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)MATHCrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 18.
    Kimmel, R., Bruckstein, A.M.: Shape offsets via level sets. Computer-Aided Design 25(3), 154–162 (1993)MATHCrossRefGoogle Scholar
  19. 19.
    Koenderink, J.J.: Solid Shape. MIT Press, Cambridge (1990)Google Scholar
  20. 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)MATHCrossRefGoogle Scholar
  21. 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)CrossRefGoogle Scholar
  22. 22.
    Lang, S.: Undergraduate Analysis, 2nd edn. Springer, Heidelberg (1997)Google Scholar
  23. 23.
    Maekawa, T.: An overview of offset curves and surfaces. Computer-Aided Design 31(3), 165–173 (1999)MATHCrossRefGoogle Scholar
  24. 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)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Markot, R.P., Magedson, R.L.: Solutions of tangential surface and curve intersections. Computer-Aided Design 21(7), 421–427 (1989)CrossRefGoogle Scholar
  26. 26.
    O’Neill, B.: Elementary Differential Geometry, 2nd edn. Academic Press, London (1997)MATHGoogle Scholar
  27. 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)MATHCrossRefGoogle Scholar
  28. 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. 29.
    Pham, B.: Offset curves and surfaces: a brief survey. Computer-Aided Design 24(4), 223–229 (1992)CrossRefGoogle Scholar
  30. 30.
    Preparata, F.P., Shamos, M.I.: Computational geometry: an introduction. Springer, Heidelberg (1985)Google Scholar
  31. 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)MATHCrossRefGoogle Scholar
  32. 32.
    Sederberg, T.W., Meyers, R.J.: Loop detection in surface patch intersections. Computer Aided Geometric Design 5(2), 161–171 (1988)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Sherbrooke, E.C., Patrikalakis, N.M.: Computation of the solutions of nonlinear polynomial systems. Computer Aided Geometric Design 10(5), 379–405 (1993)MATHCrossRefMathSciNetGoogle Scholar
  34. 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)MATHCrossRefGoogle Scholar
  35. 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. 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)CrossRefGoogle Scholar
  37. 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)CrossRefGoogle Scholar
  38. 38.
    Xu, G., Bajaj, C.L.: Curvature Computations of 2-Manifolds in ℝk Google Scholar
  39. 39.
    Ye, X., Maekawa, T.: Differential Geometry of Intersection Curves of Two Surfaces. Computer Aided Geometric Design 16(8), 767–788 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Xianming Chen
    • 1
  • Richard F. Riesenfeld
    • 1
  • Elaine Cohen
    • 1
  • James Damon
    • 2
  1. 1.School of ComputingUniversity of UtahSalt Lake CityUSA
  2. 2.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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