Linear Steiner Trees for Infinite Spirals

A full Steiner tree T for a given set of points P is defined to be linear if all Steiner points lie on one path called the trunk of T . A (nonfull) Steiner tree is linear if it is a degeneracy of a full linear Steiner tree. Suppose P is a simple polygonal line. Roughly speaking, T is similar to P if its trunk turns to the left or right when P does. P is a left-turn (or right-turn) polygonal spiral if it always turns to the left (or right) at its vertices. P is an infinite spiral if n tends to infinity. In this paper we first prove some results on nonminimal paths and the decomposition of Steiner minimal trees, and then, based on these results, we study the case in which an infinite spiral P has a Steiner minimal tree that is linear and similar to P itself.