x-face marching cube ambiguity resolved

Built by Jarek Rossignac with Processing. Source code: xface pv2D

A trivial solution to the X-face ambiguity in marching cubes and other isosurface extraction algorithms
An x-face of a volumetric model has alternating green (larger than threshold v) and blue (smaller than v) values r0...r3 at its corners. The intersection of an x-face with a trianglulated approximation of the isosurface is two edges. There is an ambiguity as to which pair of edges should be used. An unambiguous solution is given by the topology of the isocurve of the bi-linear interpolation (orange curve). Hence, the correct topology choice is dictated by the value of (r0-v)*(r2-v)>(r1-v)*(r3-v). This test may be performed for consistently each face of each cell (cube) during the construction of an isosurface mesh.

E. V. Chernyaev. Marching Cubes 33: Construction of Topologically Correct Isosurfaces. Technical Report CERN CN 95-17, CERN, 1995.
A more complex test may establish whether the iso-surface of the field defined inside the cube as the tri-linear interpolation of the corner v alues makes one sheet per loop or connects two or more loops through tunels.

T. Lewiner, H. Lopes, A. Wilson Vieira1, G. Tavares. Efficient implementation of Marching Cubes’ cases with topological guarantees.

Note that the option of manking a dfferent choice (and hence one that is inconsistent with the topology of the tri-linear interpolation)

C. Andujar, P. Brunet, A. Chica, I. Navazo, J. Rossignac, A. Vinacua. Optimizing the topological and combinatorial complexity of Iso-Surfaces, Journal of Computer-Aided Design & Applications, 37(8):847-857, 2005.
may be exploited to minimize various measures of topological and combinatoric complexity. In that paper, we argue that in order to minimize the number of triangles in the iso-surface, each loop formed by the edges on the six faces of a cell is the boundary of a simply connected component of the iso-surface (hence, forbidding the creation of iso-surface tunnels in a cell). Combining this tunnel-free heuristic (which may be inconsistent with the topology of the iso-surface of the tri-linear field) and Chernyaev's rule for x-faces lead to a simple and unambiguous rule for the creation of iso-surface triangulations.