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Thaw: a tool for approximating cut loci on a triangulation of a surface. (English) Zbl 1093.53003

The numerical computation of the cut locus on smooth surfaces in \(\mathbb R^3\) is highly sensitive to the discretization of the surface as a simplicial mesh. The authors’ algorithm lets concentric wavefronts on the polyhedral surface evolve and smoothes any intermediate branching of the fronts especially introduced at positively curved vertices of the simplicial mesh. Compared to exact geodesic algorithms this method smoothes branchings related to the surface discretization, and focusses on an approximation of the cut locus of the underlying smooth surface. The evolution algorithm is similar to the discrete geodesic flow in [K. Polthier and M. Schmies, Hege, Hans-Christian (ed.) et al., Mathematical visualization. Algorithms, applications, and numerics. International workshop Visualization and mathematics, Berlin, Germany, 1997. Berlin: Springer, 135–150 (1998; Zbl 0940.68153)], but here applied to extract the cut locus. The implementation of the algorithm in the Thaw software is applied to Zoll’s surface of revolution as a test case, and to numerically study the conjecture that the cut locus of a general ellipsoid is part of a curvature line.

MSC:

53A05 Surfaces in Euclidean and related spaces
53-04 Software, source code, etc. for problems pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
68Q25 Analysis of algorithms and problem complexity

Citations:

Zbl 0940.68153

Software:

LaTeX; Thaw; Loki
PDFBibTeX XMLCite
Full Text: DOI Euclid EuDML

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