DeBruijn Counting for Visualization Algorithms

We describe how to determine the number of cases that arise in visualization al- gorithms such as Marching Cubes by applying the deBruijn extension of Pólya counting. This technique constructs a polynomial, using the cycle index, encoding the case counts that arise when a discrete function (or “color”) is evaluated at each vertex of a polytope. The technique can serve as a valuable aid in debugging visualization algorithms that extend Marching Cubes, Separating Surfaces, Interval Volumes, Sweeping Simplices, etc., to larger dimensions and to more colors.