The Steiner problem on the regular tetrahedron

Abstract

The Steiner problem involves finding a shortest path network connecting a specified set of points. In this paper, we examine the Steiner problem for three points on the surface of a regular tetrahedron. We prove several important properties about Steiner minimal trees on a regular tetrahedron. There are infinitely many ways to connect three points on a tetrahedron, so we present a way to eliminate all but a finite number of possible solutions. We provide an algorithm for finding a shortest network connecting three given points on a regular tetrahedron. The solution can be found by direct measurement of the remaining possible Steiner trees.

The 3-point Steiner problem on a cylinder

“…Analytic methods for finding the solution to Steiner problems on the hyperbolic plane and surfaces of revolution were given in [Halverson and March 2005] [Caffarelli et al 2012], respectively. Geometric methods for solving the two-and 3-point Steiner problems on the regular tetrahedron were provided in [Brune and Sipe 2009;Moon et al 2011]. A cutting algorithm to find the solution to 3-point Steiner problems on the cone, similar to the one in this paper, is given in [Lee et al 2011].…”

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“…[Caffarelli et al 2012], respectively. Geometric methods for solving the two-and 3-point Steiner problems on the regular tetrahedron were provided in [Brune and Sipe 2009;Moon et al 2011]. A cutting algorithm to find the solution to 3-point Steiner problems on the cone, similar to the one in this paper, is given in [Lee et al 2011].…”