The Steiner Problem on Narrow and Wide Cones

Given a surface and n fixed points on the surface, the Steiner problem asks to find the minimal length path network in the surface connecting the n fixed points. The solution to a Steiner problem is called a Steiner minimal tree. The Steiner problem in the plane has been well studied, but until recently few results have been found for non-planar surfaces. In this paper we examine the Steiner problem on both narrow and wide cones. We prove several important properties of Steiner minimal trees on cones and present an algorithm solving the three point problem. We discuss how the algorithm for the 3 point problem can be generalized to an algorithm for the n point problem. These results are preliminary to solving the Steiner problem on piecewise linear or polyhedral surfaces, since any piecewise linear surface may be viewed as a locally finite union of overlapping cones.