Textures and maps.(Viewpoint: SURFACING)(digital mapping)

Texture 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 basic object, the sphere, with a single image is no trivial task. The 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.

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Cartographers, or mapmakers, have been trying to represent the round surface of the earth, only on a flat surface, for ages. They 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 concepts 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. That's because the applied projection--a mathematical formula mapping the sphere-like body of the earth to a two-dimensional plane--displays some important properties at the expense of others. Thus, 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 consider the different metric properties that they do, or do not, preserve. Projections that preserve area, called equal-area maps, do not distort 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 surface 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.