Display minimum distance and minimum-distance path between two points on a polyhedron

Asked Apr 18 '20 at 16:04

Active Apr 18 '20 at 16:18

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Given a genus-zero polyhedron (e.g., PolyhedronData["Dodecahedron"]) and a point $P$ anywhere on its surface, color all points on the polyhedron by the length of the minimum-length path from $P$.
For two specified points, $P_1$ and $P_2$, display that path on the polyhedron.

Optimizing over all possible paths between two points has proven both mathematically and Mathematically difficult--even for a cube! (I cannot find scholarly papers in the mathematical literature that solve this problem, in general.)

edited Apr 18 '20 at 16:18

J. M.'s missing motivation

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asked Apr 18 '20 at 16:04

David G. Stork

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Presumably, $P$ does not have to be a vertex? Is the polyhedron always genus 0, or can one expect holes (e.g. the Szilassi or Császár polyhedra)? – J. M.'s missing motivation Apr 18 '20 at 16:16
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I wonder if the methods in this question become applicable after a suitable triangularization of the given polyhedron? – J. M.'s missing motivation Apr 18 '20 at 16:22
Excellent suggestion! Before posting I searched but didn't find anything, including that valuable page. – David G. Stork Apr 18 '20 at 16:56
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also related: Minimum path on a rectangular prism – kglr Apr 18 '20 at 18:05

Display minimum distance and minimum-distance path between two points on a polyhedron

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