Topology is an area of Mathematics that deal with spatial properties that remain under the deformation of objects. The stretching of objects that do not tear or rip.
Getting the topology of a model correct is very important especially if the model is going to be animated. Efficient topology is crucial to being able to animate a model easily. It is important to be able to have smooth and correct topology with the least number of subdivisions possible in order to make the animation easier.
When creating topology for a model you have to remember to only use four sided shapes (quads) and more or any less and the shape becomes impossible to sub divide efficiently. When you divide a 3 or 5 sided shape in two it is impossible to do it and have the shapes equal this can cause real problems in the model and creates lumps and bumps. One way to fix this is to increase the number of subdivisions but this is a highly impractical solution. A model with a very high polygon count is harder to animate and would contain a lot more information than one with fewer polygons and more efficient subdivision.
When create facial topology the subdivision lines must follow the curves and muscular structure of the face as opposed to being simply a grid. This is extremely important when it comes to animating as the subdivided faces will stretch in a way that is unrealistic.
Poles are another element to consider in topology, poles are created when a vertex shares more than 4 edges. Poles should be avoided because they can create uneven shapes and bumps in the model especially when they occur on a raised surface.
Reference sources:
http://www.guerrillacg.org/home/3d-polygon-modeling/subdivision-topology-artifacts
http://www.subdivisionmodeling.com/forums/showthread.php?t=907
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