Subdivision Analysis of the Trilinear Interpolant

13 years 2 months ago

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— Isosurfaces are fundamental volumetric visualization tools, and are generated by the well-known Marching Cubes cases that approximate 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 for 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.

Hamish Carr, Nelson Max

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