Ridges and Umbilics of Polynomial Parametric Surfaces
Given a smooth surface, a blue (red) ridge is a curve such that at each of its point, the maximum (minimum) principal curvature has an extremum along its curvature line. As curves of extremal curvature, ridges are relevant in a number of applications including surface segmentation, analysis, registration, matching. In spite of these interests, given a smooth surface, no algorithm reporting a certified approximation of its ridges was known so far, even for restricted classes of generic surfaces.
This paper partly fills this gap by developing the first algorithm for polynomial parametric surfaces — a class of surfaces ubiquitous in CAGD. The algorithm consists of two stages. First, a polynomial bivariate implicit characterization of ridges P = 0 is computed using an implicitization theorem for ridges of a parametric surface. Second, the singular structure of P = 0 is exploited, and the approximation problem is reduced to solving zero dimensional systems using Rational Univariate Representations. An experimental section illustrates the efficiency of the algorithm on Bézier patches.
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- 1.D. Attali, J.-D. Boissonnat, and A. Lieutier. Complexity of the delaunay triangulation of points on surfaces the smooth case. In ACM SoCG, San Diego, 2003.Google Scholar
- 2.F. Cazals, J.-C. Faugère, M. Pouget, and F. Rouillier. Topologically certified approxima-tion of umbilics and ridges on polynomial parametric surface. Technical Report 5674, INRIA, 2005.Google Scholar
- 4.F. Cazals and M. Pouget. Topology driven algorithms for ridge extraction on meshes. Technical Report RR-5526, INRIA, 2005.Google Scholar
- 6.J.-C. Faugère. A new efficient algorithm for computing gröbner bases without reduction to zero f5 . In International Symposium on Symbolic and Algebraic Computation Symposium - ISSAC 2002, Villeneuve d’Ascq, France, Jul 2002.Google Scholar
- 7.G. Gatellier, A. Labrouzy, B. Mourrain, and J.-P. Tècourt. Computing the topology of 3-dimensional algebraic curves. In Computational Methods for Algebraic Spline Surfaces, pages 27-44. Springer-Verlag, 2004.Google Scholar
- 8.J. Gravesen. Third order invariants of surfaces. In T. Dokken and B. Juttler, editors, Computational methods for algebraic spline surfaces. Springer, 2005.Google Scholar
- 9.L. Gonzalez-Vega and I. Necula. Efficient topology determination of implicitly defined algebraic plane curves. Computer Aided Geometric Design, 19(9), 2002.Google Scholar
- 10.D. Hilbert and S. Cohn-Vossen. Geometry and the Imagination. Chelsea, 1952.Google Scholar
- 11.P. W. Hallinan, G. Gordon, A.L. Yuille, P. Giblin, and D. Mumford. Two-and Three-Dimensional Patterns of the Face. A.K.Peters, 1999.Google Scholar
- 12.J.J. Koenderink. Solid Shape. MIT, 1990.Google Scholar
- 13.R. Morris. Symmetry of Curves and the Geometry of Surfaces: two Explorations with the aid of Computer Graphics. Phd Thesis, 1990.Google Scholar
- 14.R. Morris. The sub-parabolic lines of a surface. In Glen Mullineux, editor, Mathematics of Surfaces VI, IMA new series 58, pages 79-102. Clarendon Press, Oxford, 1996.Google Scholar
- 16.X. Pennec, N. Ayache, and J.-P. Thirion. Landmark-based registration using features identified through differential geometry. In I. Bankman, editor, Handbook of Medical Imaging. Academic Press, 2000.Google Scholar
- 17.I. Porteous. The normal singularities of a submanifold. J. Diff. Geom., 5, 1971.Google Scholar
- 18.I. Porteous. Geometric Differentiation (2nd Edition). Cambridge University Press, 2001.Google Scholar
- 21.R. Seidel and N. Wolpert. On the exact computation of the topology of real algebraic curves. In SCG’05: Proceedings of the twenty-first annual symposium on Computational geometry, pages 107-115, New York, NY, USA, 2005. ACM Press.Google Scholar