Reading 7: Discrete Conformal Geometry
Preview of a first from the reading, showing some text and an illustration of two small triangulations with congruent triangles, illustrating the failure of naïve definitions of conformal equivalence.
Your next reading will take a deep dive into conformal geometry and the many ways to discretize and compute conformal maps. This subject makes some beautiful and unexpected connections to other areas of mathematics (such as circle packings, and hyperbolic geometry), and is in some sense one of the biggest “success stories” of DDG, since there is now a complete uniformization theorem that mirrors the one on the smooth side. You'll find out more about what this all means in the reading! The reading comes from the note, Crane, “Conformal Geometry of Simplicial Surfaces.” For your assignment you will need to read the Overview (1.1) and Preliminaries (1.2); you must also pick one of Part I, Part II, or Part III to read, each of which covers a different perspective on discrete conformal maps. If you read Part II or Part III, you may also be interested in seeing the beautiful connections to hyperbolic geometry in Part IV (but this part is not required for your reading assignment). Handin instructions can be found in the Readings section of the main page.