Relationships between areas in a triangulation of a square 公开
Andreae, Phillip Volkmar (2010)
Published
Abstract
For a triangulation of the unit square--that is, a tiling of the square by triangles--we consider the general problem of studying the relationships between the areas of these triangles. For a particular combinatorial arrangement of vertices and edges, by a dimension count we expect there to exist a relation that must be satisfied by the triangles, regardless of where the vertices are placed. By generalizing the notion of triangulating a square and applying some facts from algebraic geometry, we can prove that this relation is in fact a homogeneous polynomial equation that is an invariant of the combinatorial triangulation. Our focus is on calculating the degree of this polynomial for any arbitrary triangulation. We develop and implement an algorithm to compute this degree by inductively relating a triangulation to simpler "factor" triangulations and studying the relationship between the associated polynomials.
Table of Contents
1. Introduction - 1
2. Preliminaries - 5
3. A brief introduction to algebraic geometry - 14
4. The mystery polynomial - 20
5. The algorithm - 32
6. Conclusion - 50
About this Honors Thesis
| School | |
|---|---|
| Department | |
| Degree | |
| Submission | |
| Language |
|
| Research Field | |
| 关键词 | |
| Committee Chair / Thesis Advisor | |
| Committee Members |
Primary PDF
| Thumbnail | Title | Date Uploaded | Actions |
|---|---|---|---|
|
|
Relationships between areas in a triangulation of a square () | 2018-08-28 13:12:12 -0400 |
|
Supplemental Files
| Thumbnail | Title | Date Uploaded | Actions |
|---|