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Critical points with discrete Morse theory

Published:31 July 2015Publication History

ABSTRACT

In this work, we present some of the unexpected observations resulted from our recent research. We, recently, needed to identify a small number of important critical points, i.e. minimum, maximum and saddle points, on a given manifold mesh surface. All critical points on a manifold triangular mesh can be identified using discrete Gaussian curvature, which is given as ki = 2π − Σj θi,j where ki is vertex defect (the discrete Gaussian curvature) of the vertex i and θi,j is the corner of the vertex in the triangle j. A very useful property coming with vertex defect is the discrete version of Gauss-Bonnet theorem: the sum of all vertex defects is always constant as Σi ki = 2π(2−2g) where g is the genus of the mesh. Any vertex with a non-zero vertex defect is really an critical point of the surface. However, identification of interesting critical points is hard with vertex defect alone. As it can be seen in Figure 1(a), even we ignore vertex defects that are small, too many vertices are still chosen and this information is not really useful to make any conclusion of the shape of the surface.

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  1. Critical points with discrete Morse theory

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      • Published in

        cover image ACM Conferences
        SIGGRAPH '15: ACM SIGGRAPH 2015 Posters
        July 2015
        95 pages
        ISBN:9781450336321
        DOI:10.1145/2787626

        Copyright © 2015 Owner/Author

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 31 July 2015

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