Subdivision Analysis of the Trilinear Interpolant

Isosurfaces are fundamental volumetric visualization tools and are generated by approximating contours of trilinearly interpolated scalar fields. While a complete set of cases has recently been published by Nielson, the formal proof that these cases are the only ones possible and that they are topologically correct is difficult to follow. We present a more straightforward proof of the correctness and completeness of these cases based on a variation of the Dividing Cubes algorithm. Since this proof is based on topological arguments and a divide-and-conquer approach, this also sets the stage for developing tessellation cases for higher order interpolants and the quadrilinear interpolant in four dimensions. We also demonstrate that apart from degenerate cases, Nielson´s cases are, in fact, subsets of two basic configurations of the trilinear interpolant.