Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given 2 homotopic curves in a topological space, there are several ways to measure similarity beten the curves, including Hausdorff distance and Frechet dis- tance. ,  examine a different measure of similarity which considers the family of curves repre- sented in the homotopy beten the curves, and mea- sures the longest such curve, known as the height of the homotopy. In other words, if  have two homotopic curves on a surface and view a homotopy as a way to morph one curve into the other,  wish to find the longest intermediate curve along the morphing. , our model assumes  are given a pair of disjoint embedded homotopic curves (where the endpoints remained fixed over the course of the homo- topy) in an edge-ighted planar triangulation satisfy- ing the triangle inequality.  prove that among min- imal height homotopies beten 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.
['Erin W. Chambers', 'David Letscher']
I am writing a research paper on a idea On the Height of a Homotopy
[{"from": "human", "value": "I am writing a research paper on a idea On the Height of a Homotopy"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given 2 homotopic curves in a topological space, there are several ways to measure similarity beten the curves, including Hausdorff distance and Frechet dis- tance. ,  examine a different measure of similarity which considers the family of curves repre- sented in the homotopy beten the curves, and mea- sures the longest such curve, known as the height of the homotopy. In other words, if  have two homotopic curves on a surface and view a homotopy as a way to morph one curve into the other,  wish to find the longest intermediate curve along the morphing. , our model assumes  are given a pair of disjoint embedded homotopic curves (where the endpoints remained fixed over the course of the homo- topy) in an edge-ighted planar triangulation satisfy- ing the triangle inequality.  prove that among min- imal height homotopies beten 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."}]