Solid shape
The extremal mesh and the understanding of 3D surfaces
International Journal of Computer Vision
Two- and three-dimensional patterns of the face
Two- and three-dimensional patterns of the face
Landmark-based registration using features identified through differential geometry
Handbook of medical imaging
The Sub-Parabolic Lines of a Surface
Proceedings of the 6th IMA Conference on the Mathematics of Surfaces
Complexity of the delaunay triangulation of points on surfaces the smooth case
Proceedings of the nineteenth annual symposium on Computational geometry
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On the complexity of real solving bivariate systems
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Ridge based curve and surface reconstruction
SGP '07 Proceedings of the fifth Eurographics symposium on Geometry processing
Fast, robust, and faithful methods for detecting crest lines on meshes
Computer Aided Geometric Design
An Intrinsic Framework for Analysis of Facial Surfaces
International Journal of Computer Vision
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
Tracing ridges on B-Spline surfaces
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
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Given a smooth surface, a blue (red) ridge is a curve such that at each of its points, the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately--in which case a local orientation procedure of principal directions is needed, but no method developed so far certifies the topology of the curves reported.On the way to developing certified algorithms independent from local orientation procedures, we make the following fundamental contributions. For any smooth parametric surface, we exhibit the implicit equation P = 0 of the singular curve P encoding all ridges and umbilics of the surface (blue and red), and show how to recover the colors from factors of P. Exploiting second order derivatives of the principal curvatures, we also derive a zero-dimensional system coding the so-called turning points, from which elliptic and hyperbolic ridge sections of the two colors can be derived. Both contributions exploit properties of the Weingarten map of the surface in the specific parametric setting and require computer algebra. Algorithms exploiting the structure of P for algebraic surfaces are developed in a companion paper.