Consider how one makes terrain using marching cubes. By having a grid of floats we can represent a continuous field that marching cubes will interpolate and turn into a nice smooth isosurface for the player to walk around on. This is easy and excellent for creating mountains and valleys and so on, but what if we want more variety in our game? A game is not normally made of just grass and sky. Maybe some places should be sand, or water, or road. How could that be worked into the mesh that we're getting from marching cubes?

The obvious approach seems to be to have multiple fields, so each point on the grid has a certain level of sand, soil, rock, water, and so on. Then we modify the marching cubes algorithm to look for transitions between materials, so it puts a surface between areas of mostly one material and areas that are mostly other materials. We'd also want to keep track of when these surfaces touch the air, because that's the only time when we'd actually want to triangulate and render the surfaces.

Suddenly the delightfully simple marching cubes algorithm is looking a lot less obvious. Has anything like this ever been done? Does anyone have any tips? Is this the right approach?

Edit: Embarrassing mistake! I didn't think of phrasing the problem as "multiple materials" until I went to post this question, but now that I have I see there are plentiful google results for marching cubes with multiple materials. I'm still interested in any tips and advice, but now I have other resources to help with this problem.

From the Google results, this paper looks especially interesting: Automatic 3D Mesh Generation for A Domain with Multiple Materials

The paper on Dual Contouring also provides a solution to realize multiple materials.

19 minutes ago, LifeIsGood said:

The paper on Dual Contouring also provides a solution to realize multiple materials.

This paper? Dual Contouring of Hermite Data by Tao Ju et al.

If I have this right, then the essence of the problem is to convert the marching cubes algorithm from an algorithm where the corners of each cube can only be positive or negative, into an algorithm where the corners of each cube can take anywhere up to 8 distinct values. If I have this right, then while ordinary marching cubes deals with 2^8 possibilities, the multiple material marching cubes needs to deal with a far greater number of possibilities. It's not as many as 8^8, but it's still quite a few. Hopefully one of these papers will go into a way to enumerate the possibilities.

Supposing that we're dealing with sand, rock, and air, then each corner has three numbers, one for each material, and the material of the corner is whichever has the greatest number. When two adjacent corners have distinct materials, then the algorithm will put a vertex somewhere on the edge between the corners. Supposing that the one of the corners is sand and the other is rock, we can linearly interpolate the values of sand and rock between the two corners, and the position of the vertex should be the point at which we switch from sand having the greatest value to rock having the greatest value. It's possible that if we interpolated the air value we'd find that air has the greatest value somewhere along the edge, but it's probably fair to just ignore that.

I should focus on implementing a multiple material marching squares as a starting point, then move up to marching cubes once I'm confident I've got all the details worked out for the 2D version. Unfortunately it's not so easy to find resources for the 2D case.

2 minutes ago, swiftcoder said:

Worst case is you have to run the exact same marching cubes algorithm 8 times (once for each material present in the current cube).

Just running the marching cubes algorithm for each material doesn't seem like it would work. Hard transitions are exactly what I want, since that's easier to render than fading between materials, but if we just do 8 separate marching cubes, then what would prevent the materials from clipping into each other?

Suppose we have three materials on the corners of a cube, then it seems they ought to divide the cube among themselves around a point in the middle. Using three separate marching cube procedures the materials might end up leaving an empty space in the middle, or they might end up clipping into each other so some parts of the cube are claimed by more than one material, but it seems impossible for the cube to be properly partitioned among the materials. The geometry of the situation is more complicated than single-material marching cubes ever needs to deal with.

As it happens, I think I've found the actual number of possibilities for multi-material cubes. It's called the 8th Bell number, and it is 4140. It's roughly 16 times the 256 possibilities of single-material marching cubes, but still well within the range of a 16-bit integer.

Finding a good index for the lookup table is tricky. There are 4140 cases for a cube with various materials on its corners, but how should we number those cases? There are 28 comparisons we could do between the corners of a cube, and if we give each comparison a bit in the index, then we end up with an array of 268,435,456 elements, of which only 4140 will ever be used.

Here's a blog entry about generating lists of set partitions. It generates partitions in a particular order, but it's not clear how to reverse the process and generate a number when given a partition. Enumerating set partitions with Bell numbers and Stirling numbers of the second kind.

10 minutes ago, swiftcoder said:

Depends how watertight/deterministic your marching cubes is, I guess.

If I understand this correctly, then the issue isn't about floating point errors or deterministic behavior. Even with absolute precision there would still be no way to avoid clipping except by having large gaps between materials. For example:

On the left we have a Marching Squares square with three materials inserted with three separate single-material marching squares passes. Unfortunately there is a gap, so picture grass, sand, and water meeting at a point that includes a hole to infinity. If we want to fill that gap, we're forced to do something like shown on the right, where we increase the magnitude of one of the materials, but if we increase any of the materials even slightly it will not only start to fill the gap, but also start to overlap with the other materials.