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# Constrained Multi-Material Marching Squares

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In ordinary marching squares, we're trying to find isolines on a height map for some particular height. It's a delightfully simple algorithm because we can use a look-up table to determine the structure of edges and vertices within each grid cell based on whether each corner is above or below the desired height.

In multi-material marching squares, each point on the grid has some proportion of several materials and we're trying to draw the boundaries between the areas where each material is dominant. This is less simple, since there are more than two options for each corner of each cell; at worst each corner could have a distinct dominant material. Even so, it's not too hard to approach this problem with a look-up table based on the corners of each cell.

Finally, we have constrained multi-material marching squares, which is much like other constrained triangulation problems. In addition to the multi-material grid, we now have pre-defined boundary edges in some of the grid cells, and the multi-material marching squares must respect those pre-defined edges as if they accurately represent the boundary between two materials. I'm finding it hard to wrap my head around this problem. It seems that a look-up table will be of no use because the pre-defined edges create too many possibilities, even if those edges are restricted to the kinds of edges that marching square would naturally produce, but doing this without a look-up table also seems daunting.

Motivation: In principle the goal seems quite simple. Take a 2D grid and use it to define terrain as a height map and as a material map that will form the foundation for a procedurally constructed mesh. Aside from the usual hills and valleys of a plain height map, the multi-material aspect of the grid allows us to define swamp, forest, desert regions on the map and apply particular procedural meshing for each. In addition to that, we want vertical cliffs that get their own special meshing and define the region boundaries. The cliffs are the constraints of constrained multi-material marching squares because when there is a cliff running through a grid cell, that should always act as the boundary if the material at the top of the cliff is different from the material at the bottom, even if marching squares would have naturally put the boundary somewhere else.

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