Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
Improving the Robustness and Accuracy of the Marching Cubes Algorithm for Isosurfacing
IEEE Transactions on Visualization and Computer Graphics
The asymptotic decider: resolving the ambiguity in marching cubes
VIS '91 Proceedings of the 2nd conference on Visualization '91
IEEE Transactions on Visualization and Computer Graphics
On accuracy of marching isosurfacing methods
SPBG'08 Proceedings of the Fifth Eurographics / IEEE VGTC conference on Point-Based Graphics
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Isosurfaces, one of the most fundamental volumetric visualization tools, are commonly rendered using the wellknown 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.