## Broad Area Colloquium For AI-Geometry-Graphics-Robotics-Vision (CS 528)

### Three Points Make a Triangle; Or a Circle...

Peter Schroeder, Caltech
May 9, 2005, 4:15PM
TCSeq 201
http://graphics.stanford.edu/ba-colloquium/

#### Abstract

Many algorithms in geometric modeling are based on concepts from differential geometry. Classical notions such as the metric or the curvature of a surface were formulated in the continuous (smooth) setting and a rich mathematical apparatus exists to help us understand what is possible. When it comes to the computational realm the picture is far murkier. Simply discretizing continuous notions to transfer them to the setting of meshes, for example, often leads to the loss of important structures. In numerical practice this typically results in performance problems and difficulties "making things work." A possible way out of this state of affairs is to reinvent differential geometry in the discrete setting from the ground up.

In my talk I will consider an example from texture mapping, or, more generally, finding a "nice" parameterization of a surface given as a triangle mesh. For example, we may be interested in finding parameterizations which preserve angles (are conformal). What is a good way to capture this in the discrete setting? This is where circles enter! (This notion is not unfamiliar, for example, in the case of Delaunay triangulations where the empty circumcircle property becomes the defining tool in the construction of particular triangulations.) So called "circle patterns" lead to a mathematically clean (and deep) way to capture discrete conformality, result in well defined numerical problems with efficient algorithms and unique solutions, and, this is where it gets good, give us tools to control the pesky area distortion issue in conformal mappings.

Joint work with Liliya Kharevych and Boris Springborn.

Caption for image: A discrete conformal map of the Max Planck dataset. Angles are well preserved and area experiences only low distortion. The map is globally continuous (seamless). You can have it all: angle preservation, low area distortion, and global continuity with the help of circle patterns with cone singularities (come to the talk to find out about the simple concepts and algorithms behind the fancy words!).