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Unformatted text preview: Marching Cubes without Skinny Triangles Carlos A. Dietrich, Carlos E. Scheidegger, Jo˜ao L. D. Comba, Luciana P. Nedel and Cl´audio T. Silva, Senior Member, IEEE *† Abstract Most computational codes that use irregular grids de- pend on the triangle quality of the single worst triangle in the grid: skinny triangles can lead to bad performance and numerical instabilities. Marching Cubes is the standard iso- surface grid generation algorithm, and while most triangles it generates are good, it almost always generates some bad triangles. Here we show how simple changes to Marching Cubes can lead to a drastically reduced number of degener- ate triangles, making it a more practical choice for isosur- face grid generation, reducing or eliminating the need and costs of post-processing. 1. Introduction Marching Cubes  is currently the most popular algo- rithm for isosurface extraction. It is elegant, simple, fast, and robust. While the output mesh Marching Cubes gen- erates is adequate for visualization purposes, it is far from being suitable for use in numerical simulations. This def- ficiency arises from the degenerate triangles that MC typi- cally generates, and that, for example, a single badly-shaped triangle can lead to ill-conditioning of an entire finite ele- ment simulation . The current practice is to solve this problem by post-processing [1, 14], but here we present a simpler alternative. We first elucidate the causes of bad tri- angles in Marching Cubes, and then mitigate the problem with small specific changes. Our discussion of Marching Cubes is based on the notion of Edge Groups, recently introduced by Dietrich et al. . Each MC case generates up to 5 triangles, which are di- rectly encoded in a fixed table. More importantly, each tri- angle is created using vertices placed along the edges of a fixed cube, and so there’s only a limited number of ways a triangle is generated. We then identify equivalent triples of * Carlos A. Dietrich, Jo˜ao L. D. Comba and Luciana P. Nedel are with the Instituto de Inform´atica, UFRGS, Brazil, E-mail: [cadiet- rich,comba,nedel]@inf.ufrgs.br. † Carlos E. Scheidegger and Cl´audio T. Silva are with the Scientific Computing and Imaging Institute, University of Utah, USA, E-mail: [cscheid,csilva]@sci.utah.edu. edges under the cube’s symmetries, and arrive at 8 differ- ent edge groups, illustrated in Figure 1. Surprisingly, a single edge group is responsible for most degenerate triangles in MC. Some cases in the Marching Cubes table admit different triangulations, which use differ- ent edge groups. By systematically analyzing each case in the Marching Cubes table, we generate a table that leads to improved triangle qualities, building on our previous work [3,4]....
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- Spring '08