Mathematics of Beautiful Graphics

Photorealistic computer-generated images are everywhere — from blockbuster films to video games to architectural visualization. Modern software tends to hide the mathematics behind them, but it's beautiful stuff: differential geometry, measure theory, and statistics all play essential roles. In this talk, we'll see how simple questions about light and color naturally lead to an infinite-dimensional problem with a curious recursive structure — to compute anything, you seemingly need to have already computed everything. The way out is to recast the problem as integration over a space of light paths. Different visual phenomena — mirrors, glass, fog — correspond to different measures on this space. Understanding this structure puts realistic rendering within reach of anyone willing to do the analysis. It also offers mathematicians a source of compelling high-dimensional problems and a new way of seeing the world — from the translucence of a jade sculpture to the glow of a sunset, the geometry of path space is all around us.