On the Height of a Homotopy

Given 2 homotopic curves in a topological space, there are several ways to measure similarity between the curves, including Hausdorff distance and Frechet dis- tance. In this paper, we examine a different measure of similarity which considers the family of curves repre- sented in the homotopy between the curves, and mea- sures the longest such curve, known as the height of the homotopy. In other words, if we have two homotopic curves on a surface and view a homotopy as a way to morph one curve into the other, we wish to find the longest intermediate curve along the morphing. In this paper, our model assumes we are given a pair of disjoint embedded homotopic curves (where the endpoints remained fixed over the course of the homo- topy) in an edge-weighted planar triangulation satisfy- ing the triangle inequality. We prove that among min- imal height homotopies between the two curves, there exists an embedded isotopy; in other words, the homo- topy with minimum height never makes a "backwards" move and results in disjoint simple intermediate curves.