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December 2008 15 Viewpoint ■ ■ ■ ■ an object using shape-preserving mapping allows the paint strokes to keep their form but signifi cantly change their scale—and, thus, the texture resolution. ere is no projection that would map a curved surface to a plane, thereby pre- serving area and shape at the same time throughout the image. e various projec- tions developed by cartographers over cen- turies illustrate this perfectly: Maps can be constructed to preserve area, shape, direc- tion, distance, or scale, although not all of them simultaneously. For each specifi c application—marine navigation, aviation, or simply fi nding a restaurant—diff erent charts are designed. Travelers on transatlantic fl ights are often surprised to see that the aircraft's route is not straight on the real-time map displayed on the video screen in front of them. It's not the navigator's mistake, however; it is only the map that distorts distances sig- nifi cantly, making the shortest path between two points appear curved. So, it is not surprising that pilots prefer a diff erent kind of chart—for instance, a Lambert conformal conic projection—to plan their routes. Applying Tissot's Concept It is interesting to look at various map projections developed over the years. For instance, the French mathematician Nicolas Auguste Tissot developed a concept to illustrate and measure the deformation of diff erent projections. e basic idea of the Tissot's indicatrix, or ellipse of distortion, is that we draw circles of the same size onto a 3D surface and study their size and shape on the fi nal map. As a result, an equal-area projection will keep the sizes of the circles but will distort their shapes into ellipses. e circles will remain perfectly round on shape-preserving maps, while their size will vary. e smaller the circles, the less detail there is on the map. By painting identical brushstrokes on a surface with pre-assigned UV coordinates in a 3D application, we can also generate Tissot's indicatrix-like diagram. Looking at the resulting two- dimensional texture, we can see where the texture distorts shapes or has good/ poor pixel density. So the question is, are CG artists doomed to work with textures inevitably distorted or of poor quality at some regions? Not neces- sarily. As we peel the orange, we cannot change the scale or the shape of the skin, so we break it into pieces or introduce "cracks." By using multiple pieces of maps/textures or not constraining the shape to be rectan- gular, we can ease the distortion eff ect. e extreme example would be a map- ping that projects independent textures to every single face on the polygon model, completely eliminating distortions. e penalty of such an approach is the increased number of texture seams over the surface. It is extremely diffi cult to hide such seams by keeping perfect smoothness as we paint, edit, or simply fi lter our textures. So, it's not surprising, then, that cartographers also have created charts of the globe consisting of multiple, but connected, pieces of maps to minimize distortion—even though these charts are nonintuitive and hard to read. While textured spheres are used often in CG to represent environments or back- grounds, most of the surfaces we texture are complex and have some organic form. e standard workfl ow to create a UV map for such an object is to start with a standard projection and refi ne it on the vertex level to minimize distortion. us, we do not use a single mathematical formula to project the texture as cartographers do, but to store the texture space coordinates, the UV coordinates, ex- plicitly for each vertex. Current 3D packages have manual smooth- ing tools and automatic "relax" algorithms to minimize artifacts: An iterative optimization refi nes the UV coordinates to distribute extreme distortions across the whole mesh, so the end result is more uniform without overlapping or tangled parts. When choosing the adequate projection for primitive geometries or manually editing complex UV maps for organic objects, we should keep in mind the two essential properties used in cartogra- phy: shape and area preservation. If we are not planning to further edit or process the texture, equal-area mapping is ideal since it guar- antees constant pixel density over the complete surface. For example, cubic mapping should be chosen to store 360-degree panoramas or environment maps, rather than spherical or ball projection (like the mirror ball images taken on set). is way, we can avoid "singular" points having extremely poor texture quality. If we need to paint or retouch the texture, however, we should keep the number of texture seams minimal and aim for shape- preserving mapping. Spherical mapping could be a good choice in this case, since our texture is not cut into pieces and strong distortion occurs only at the two caps. By manually creating the UV map, we get the chance to balance the area and the shape-pre- serving aspect of our texture. ■ Examples of transferring curved surfaces onto fl at surfaces include an equal-area map (top) and a shape-preserving map (bottom).

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