Consider the plight of Garak (the tailor on Startrek's Deep Space 9). Customers from all over the galaxy come to his shop looking for a custom made, well fitting suit, which doesn't cost too much (Ferengi rule of acquisition No. 3). Some of these customers have quite unusual body shapes, some are even of higher genus (and quite quarrelsome). The cost of a suit is related to the number of pieces that need to be cut and sewn together. The fit of a suit gets better and better as one uses more individual pieces for the pattern. You see his problem...

Let's abstract a bit. Consider a surface (2-manifold, possibly with boundary, arbitrary topology) given to you as a triangle mesh (Garak has a scanner in his shop). Abstractly we have a triangle graph with lengths on each edge (a discrete metric). We now seek a conformally equivalent discrete metric, a new assignment of lengths to all edges, such that the resulting metric (the layout of the mesh) is flat. If we allow a few vertices with non-zero curvature, so called cone singularities, we effectively have the setting of the tailor (think pleats and darts in sewing; a real tailor allows curves with non-zero curvature; we'll only allow individual vertices with non-zero curvature).

Finding such parameterizations is a first step in many digital geometry processing algorithms from texture mapping to meshing and I will discuss some recent work (joint with Ulrich Pinkall and Boris Springborn of TU Berlin) on an algorithm which addresses it in a very pretty way.

Bio: Peter Schröder is a professor of computer science and applied & computational mathematics at Caltech where he has been a member of the faculty since 1995. His main work is in the area of numerical algorithms for computer graphics applications and primarily in Digital Geometry Processing. Most recently he's been working on discrete analogs of classical differential geometry concepts (such as discrete conformal mappings) to produce efficient and accurate algorithms for computer graphics applications. His work has been recognized through a number of awards including a Packard Foundation Fellowship, the ACM/SIGGRAPH Computer Graphics Achievement Award, and, most recently, a Humboldt Foundation Forschungspreis.