The cut locus from a point on the surface of a convex polyhedron is a tree containing a line segment beginning at every vertex. In the limit of infinitely small triangles, the cut locus from a point on a triangulation of a smooth surface therefore tends to become dense in the smooth surface, whereas the cut locus from the same point on the smooth surface is also a tree, but of finite length. We introduce a method for avoiding this problem. The method involves introducing a minimal angular resolution and discarding those points of the cut locus on the triangulation for which the angle measured between the shortest geodesic curves meeting at these points is smaller than the given angular resolution. We also describe software based upon this method that allows one to visualize the cut locus from a point on a surface of the form $(x/a)^{n}+(y/b)^{n}+(z/c)^{n}=1$, where $n$ is a positive even integer. We use the software to support a new conjecture that the cut locus of a general ellipsoid is a subarc of a curvature line of the ellipsoid.

References in zbMATH (referenced in 13 articles , 1 standard article )

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  1. Itoh, Jin-ichi; Kiyohara, Kazuyoshi: Cut loci and conjugate loci on Liouville surfaces (2011)
  2. He, Dong-qing; Lu, Zhe-an; Ren, Zhi-gang: Freeze-thaw cycling damage of layered hybrid fiber reinforced concrete (2010)
  3. Itoh, Jin-Ichi; Kiyohara, Kazuyoshi: The cut loci on ellipsoids and certain Liouville manifolds (2010)
  4. Duan, X.; Naterer, G. F.: Heat conduction with seasonal freezing and thawing in an active layer near a tower foundation (2009)
  5. Enomoto, Kazuyuki; Itoh, Jin-Ichi; Sinclair, Robert: The total absolute curvature of open curves in (E^3) (2008)
  6. Kesten, Harry; Sidoravicius, Vladas: Positive recurrence of a one-dimensional variant of diffusion limited aggregation (2008)
  7. Xie, ZhengHui; Song, LiYe; Feng, XiaoBing: A moving boundary problem derived from heat and water transfer processes in frozen and thawed soils and its numerical simulation (2008)
  8. Ahmed, Amal; Fluet, Matthew; Morrisett, Greg: (L^3): a linear language with locations (2007)
  9. Sinclair, Robert; Tanaka, Minoru: A bound on the number of endpoints of the cut locus (2006)
  10. Sinclair, Robert; Tanaka, Minoru: Jacobi’s last geometric statement extends to a wider class of Liouville surfaces (2006)
  11. Morrisett, Greg; Ahmed, Amal; Fluet, Matthew: (\textL^3): A linear language with locations (2005)
  12. Itoh, Jin-ichi; Sinclair, Robert: Thaw: a tool for approximating cut loci on a triangulation of a surface (2004)
  13. Sinclair, R.: On the last geometric statement of Jacobi (2003)