(For more information, see: www.cs.caltech.edu The world around us is full of shapes: airplane wings and cell phones, brain tumors and rising loaves of bread, fossil records and freeform architectural surfaces. To a large extent, our ability to master these domains is limited by our capacity to design, process, and analyze geometry. But like much of mathematics, geometry makes liberal use of infinity -- a concept that is alien to machines with finite memory and limited precision. The driving force behind discrete differential geometry (DDG) is to develop a language that can be easily understood by a computer, yet still faithfully captures the way shape behaves in nature. A valuable consequence of constructing algorithmic descriptions is that real-world phenomena like "curvature" and "holonomy" (which traditionally demand expert terminology) can now be easily conveyed to anyone whose vocabulary includes words like "sum" and "triangle." In this talk I explore recent discoveries in the rapidly growing field of DDG, and demonstrate how a clear geometric perspective can lead to simpler, more efficient algorithms that are numerically robust and exhibit good scaling behavior. A somewhat remarkable fact is that a wide variety of seemingly dissimilar questions can be answered by computing solutions to a simple linear system known as a discrete Poisson equation. For instance: what's the shortest path from one point to another on a curved surface? How can you construct a flow with ...

• published: 03 Jun 2012
• views: 3402
• author: Keenan Crane