Publication 2004 Issue No. 4 - July/August Abstract - Counting Cases in Substitope Algorithms
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Counting Cases in Substitope Algorithms
July/August 2004 (vol. 10 no. 4)
pp. 371-384
 ASCII Text x David C. Banks, Stephen A. Linton, Paul K. Stockmeyer, "Counting Cases in Substitope Algorithms," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 4, pp. 371-384, July/August, 2004.
 BibTex x @article{ 10.1109/TVCG.2004.6,author = {David C. Banks and Stephen A. Linton and Paul K. Stockmeyer},title = {Counting Cases in Substitope Algorithms},journal ={IEEE Transactions on Visualization and Computer Graphics},volume = {10},number = {4},issn = {1077-2626},year = {2004},pages = {371-384},doi = {http://doi.ieeecomputersociety.org/10.1109/TVCG.2004.6},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Visualization and Computer GraphicsTI - Counting Cases in Substitope AlgorithmsIS - 4SN - 1077-2626SP371EP384EPD - 371-384A1 - David C. Banks, A1 - Stephen A. Linton, A1 - Paul K. Stockmeyer, PY - 2004KW - IsosurfaceKW - level setKW - group actionKW - orbitKW - geometric substitutionKW - Marching CubesKW - separating surfaceKW - Pólya countingKW - substitope.VL - 10JA - IEEE Transactions on Visualization and Computer GraphicsER -

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 Pólya theory produces a closed-form upper bound on the case counts.

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Index Terms:
Isosurface, level set, group action, orbit, geometric substitution, Marching Cubes, separating surface, P&#243;lya counting, substitope.
Citation:
David C. Banks, Stephen A. Linton, Paul K. Stockmeyer, "Counting Cases in Substitope Algorithms," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 4, pp. 371-384, July-Aug. 2004, doi:10.1109/TVCG.2004.6