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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.
Keywords
Steiner problem, length minimization, regular tetrahedron,
piecewise-linear surface
Mathematical Subject Classification 2010
Primary: 05C05
Secondary: 51M15
Milestones
Received: 14 January 2011
Revised: 22 March 2011
Accepted: 24 March 2011
Published: 21 March 2012
Communicated by Frank Morgan