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December 2008 14 Textures and Maps T exture mapping is an essential technique in computer graphics for adding fine, detailed material properties to 3D models. In order to wrap a 2D image around an object, artists need to find a way to flatten, or "unwrap" the surfaces, and map them into the two-dimensional texture space—which is done by introducing the least amount of distortion possible. However, to texture even the most ba- sic object, the sphere, with a single image is no trivial task. e perfect solution for this problem has been sought for hundreds of years now. And as it turns out, such a solution simply does not exist. Cartographers, or mapmakers, have been trying to repre- sent the round surface of the earth, only on a flat surface, for ages. ey face the same problem as young children trying to peel an orange and then flatten the peel without any tearing: impossible. It was in the early 1800s when Carl Friedrich Gauss, a German mathematician, proved that curved surfaces couldn't be represented on a flat surface without distortion. To understand the true nature of this distortion effect, which often causes trouble for texture artists, it is a good idea to take a look at some basic con- cepts of cartography. Creating maps is much more than simply measuring the world around us and printing the data. Charts need to be precise, easy to read, and communicate information effectively. World maps of different purposes, therefore, may be essentially different. at's because the applied projection—a mathematical formula mapping the sphere-like body of the earth to a two-dimensional plane—dis- plays some important properties at the expense of others. us, a perfect world map does not exist. Understanding Distortion To understand the inevitable distortion of maps or, conversely, of the textures on 3D models in computer graphics, we must consid- er the different metric properties that they do, or do not, preserve. Projections that preserve area, called equal-area maps, do not dis- tort the relative size of regions at the expense of not preserving their shape. Using such a chart, a person can determine the relative size of landmasses, for instance. Because most of the maps used in school do not have this property, children as well as adults often overestimate the sizes of Australasia and Antarctica. In the case of texture mapping, the equal-area property would translate to constant pixel density over the surface. An equal-area UV map would contain an image that is wrapped around the sur- face without scaling some parts up and some others down, so a uniform pattern on the model would keep its constant density when rendered. Shape-preserving, or conformal, mapping does not distort shapes locally, so angles at any point are correct. As a result, small features, such as the shape of islands on a map or small texture details, do not look distorted; they are not squashed or stretched. However, these mappings do distort the sizes of areas. e maps in the most popular online mapping applications have this trait; otherwise, it would be extremely challenging for us- ers to relate different formations, such as streets, cities, or rivers, to their real-world counterparts. Wrapping a painted texture around SURFACING By GERGELEY VASS A former Maya TD and instructor, Gergely Vass eventually moved to the Image Science Team of Autodesk Media and Entertainment. Currently he is developing advanced postproduction tools for Colorfront in Hungary, one of Europe's lead- ing DI and post facilities. Vass can be reached at A cylindrical texture properly preserves area and shape in the middle region, but does not represent the top and the bottom parts of the image well. Image ©

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