Sunday, August 7, 2016

Marching Squares, Surfaces of Marching Cubes, Diagonal Surfaces of Marching Cubes

Some say that Marching Squares have Ambiguous Cases, but they limit what has Ambiguous Cases. Here is a list of the regular cases:
Case 0: empty.
Case 1: lower left corner.
Case 2: lower right corner.
Case 3: lower half.
Case 4: upper right corner.
Case 5: backslash.
Case 6: right half.
Case 7: no upper left.
Case 8: upper left.
Case 9: left half.
Case 10: fore-slash.
Case 11: not upper right.
Case 12: upper half.
Case 13: not lower right.
Case 14: not lower left.
Case 15: full.

Here is a list of alternative cases. Basically, everything is not connected together. Case 5 and 10 are well known:
Case 3 Alternative
Case 5 Alternative
Case 6 Alternative
Case 7 Alternative
Case 9 Alternative
Case 10 Alternative
Case 11 Alternative
Case 12 Alternative
Case 13 Alternative
Case 14 Alternative
Case 15 Alternative

Here's another set of alternatives. Alternative 2 are connected vertically. Alternative 3 are connected horizontally. A and B distinguish for Case 15 Alternative 2 and Alternative 3:
Case 7 Alternative 2
Case 11 Alternative 2
Case 13 Alternative 2
Case 14 Alternative 2
Case 15 Alternative 2A
Case 15 Alternative 2B
Case 7 Alternative 3
Case 11 Alternative 3
Case 13 Alternative 3
Case 14 Alternative 3
Case 15 Alternative 3A
Case 15 Alternative 3B

Why talk about these additional alternative cases? Well, Marching Cubes indirectly mention these alternative cases. If we look at a Marching Cube, then there are six faces that are marching squares, and 6 diagonal surfaces of rectangle 1:SQ-RT(2) proportion that also follow marching squares (adjusted for the length increased). Marching Cubes only have ambiguous cases when at least one of the twelve Marching Squares has ambiguous cases.

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