Pre-Symmetry Sets of 3D shapes

Details Pre-Symmetry Sets of 3D shapes Pre-Symmetry Sets of 3D shapes

2005-05-05

arxiv.org/abs/math/0505088v1

Deep Structure, Singularities and Computer Vision. Lecture Notes in Computer Science 3753 (2005) 36-48.

Mathematics

Differential Geometry

ACM-class: I.2; I.5; I.4; J.2. Latex, 3 grouped figures. The final version will appear in the proceedings of the First Interna

Scientific paper

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinct points p and q in M, for which there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of this paper is to address problems related to the smoothness and the singularities of the pre-symmetry sets of 3D shapes. We show that the pre-symmetry set of a smooth surface in 3-space has locally the structure of the graph of a function from R^2 to R^2, in many cases of interest.

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