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Counting cases in substitope algorithms

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Abstract

We describe how to count the cases that arise in a family of visualization techniques, including Marching Cubes, Sweeping Simplices, Contour Meshing, Interval Volumes, and Separating Surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes ( geometric substitutions of polytopes). We demonstrate the method using "GAP," a software system for computational group theory. The case-counts are organized into a table that provides a taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculations confirm previously reported case-counts for four dimensions that are too large to check by hand and predict the number of cases that will arise in substitope algorithms that have not yet been invented. We show how Polya theory produces a closed-form upper bound on the case counts.

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Original languageEnglish
Pages (from-to)371-384
Number of pages14
JournalIEEE Transactions on Visualization and Computer Graphics
Volume10
Issue number4
DOIs
Publication statusPublished - Jul 2004

    Research areas

  • Isosurface, Level set, Group action, Orbit, Geometric substitution, Marching Cubes, Separating surface, Polya counting, Substitope

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