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Marching Cubes with Multiple Materials

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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

Edited by Outliner

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That paper seems about right. Basically in the simple form of marching cubes you create a surface in every cube that contains a sign-change. To add materials, you need to also produce a surface in every cube that contains a material change.

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

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19 minutes ago, LifeIsGood said:

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

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.

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27 minutes ago, Outliner said:

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.

Their is a more straightforward way of dealing with this. For any one material, there are at most 2^8 possible configurations. And there are at most 8 possible materials that could intersect in any one cell.

Worst case is you have to run the exact same marching cubes algorithm 8 times (once for each material present in the current cube). The tables don't expand any, if you don't mind hard transitions between materials.

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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.

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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.

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18 hours ago, Outliner said:

but if we just do 8 separate marching cubes, then what would prevent the materials from clipping into each other?

Depends how watertight/deterministic your marching cubes is, I guess. I'll have to give this a try at some point.

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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.

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In the past, I never bothered with marching different meshes for different terrain materials. I just marches the terrain as a single mesh, then used vertex colors (generated after marching the surface, using various techniques) to blend between terrain textures in the shader. Something like this (very quick example):

With a tri-planar shader that displays different textures for the top surface than what it displays for the side surfaces, then you can just paint the v-colors (either procedurally, or by hand if that is your wish, in a post-process step) for different materials, and the shader will handle blending between the types and applying the tri-planar projection. A single color layer provides for 5 base terrain materials, if you count black(0,0,0,0) as one material, red(1,0,0,0), green(0,1,0,0), blue(0,0,1,0) and alpha(0,0,0,1) as the others. Provide another RGBA v-color layer and you can bump that to 9.

Doing it this way, you don't have to be content with sharp edges between terrain types, since the shader is content to smoothly blend between materials as needed, and you don't deal with the hassle of marching multiple terrain meshes.

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• Here is the original blog post.
Edit: Sorry, I can't get embedded LaTeX to display properly.
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This post is part of my Game Math Series.
Source files are on GitHub.
Shortcut to sterp implementation.
Shortcut to code used to generate animations in this post.
An Alternative to Slerp
Slerp, spherical linear interpolation, is an operation that interpolates from one orientation to another, using a rotational axis paired with the smallest angle possible.
Quick note: Jonathan Blow explains here how you should avoid using slerp, if normalized quaternion linear interpolation (nlerp) suffices. Long store short, nlerp is faster but does not maintain constant angular velocity, while slerp is slower but maintains constant angular velocity; use nlerp if you’re interpolating across small angles or you don’t care about constant angular velocity; use slerp if you’re interpolating across large angles and you care about constant angular velocity. But for the sake of using a more commonly known and used building block, the remaining post will only mention slerp. Replacing all following occurrences of slerp with nlerp would not change the validity of this post.
In general, slerp is considered superior over interpolating individual components of Euler angles, as the latter method usually yields orientational sways.
But, sometimes slerp might not be ideal. Look at the image below showing two different orientations of a rod. On the left is one orientation, and on the right is the resulting orientation of rotating around the axis shown as a cyan arrow, where the pivot is at one end of the rod.

If we slerp between the two orientations, this is what we get:

Mathematically, slerp takes the “shortest rotational path”. The quaternion representing the rod’s orientation travels along the shortest arc on a 4D hyper sphere. But, given the rod’s elongated appearance, the rod’s moving end seems to be deviating from the shortest arc on a 3D sphere.
My intended effect here is for the rod’s moving end to travel along the shortest arc in 3D, like this:

The difference is more obvious if we compare them side-by-side:

This is where swing-twist decomposition comes in.

Swing-Twist Decomposition
Swing-Twist decomposition is an operation that splits a rotation into two concatenated rotations, swing and twist. Given a twist axis, we would like to separate out the portion of a rotation that contributes to the twist around this axis, and what’s left behind is the remaining swing portion.
There are multiple ways to derive the formulas, but this particular one by Michaele Norel seems to be the most elegant and efficient, and it’s the only one I’ve come across that does not involve any use of trigonometry functions. I will first show the formulas now and then paraphrase his proof later:
Given a rotation represented by a quaternion R = [W_R, vec{V_R}] and a twist axis vec{V_T}, combine the scalar part from R the projection of vec{V_R} onto vec{V_T} to form a new quaternion: T = [W_R, proj_{vec{V_T}}(vec{V_R})]. We want to decompose R into a swing component and a twist component. Let the S denote the swing component, so we can write R = ST. The swing component is then calculated by multiplying R with the inverse (conjugate) of T: S= R T^{-1} Beware that S and T are not yet normalized at this point. It's a good idea to normalize them before use, as unit quaternions are just cuter. Below is my code implementation of swing-twist decomposition. Note that it also takes care of the singularity that occurs when the rotation to be decomposed represents a 180-degree rotation. public static void DecomposeSwingTwist ( Quaternion q, Vector3 twistAxis, out Quaternion swing, out Quaternion twist ) { Vector3 r = new Vector3(q.x, q.y, q.z); // singularity: rotation by 180 degree if (r.sqrMagnitude < MathUtil.Epsilon) { Vector3 rotatedTwistAxis = q * twistAxis; Vector3 swingAxis = Vector3.Cross(twistAxis, rotatedTwistAxis); if (swingAxis.sqrMagnitude > MathUtil.Epsilon) { float swingAngle = Vector3.Angle(twistAxis, rotatedTwistAxis); swing = Quaternion.AngleAxis(swingAngle, swingAxis); } else { // more singularity: // rotation axis parallel to twist axis swing = Quaternion.identity; // no swing } // always twist 180 degree on singularity twist = Quaternion.AngleAxis(180.0f, twistAxis); return; } // meat of swing-twist decomposition Vector3 p = Vector3.Project(r, twistAxis); twist = new Quaternion(p.x, p.y, p.z, q.w); twist = Normalize(twist); swing = q * Quaternion.Inverse(twist); } Now that we have the means to decompose a rotation into swing and twist components, we need a way to use them to interpolate the rod’s orientation, replacing slerp.
Swing-Twist Interpolation
Replacing slerp with the swing and twist components is actually pretty straightforward. Let the Q_0 and Q_1 denote the quaternions representing the rod's two orientations we are interpolating between. Given the interpolation parameter t, we use it to find "fractions" of swing and twist components and combine them together. Such fractiona can be obtained by performing slerp from the identity quaternion, Q_I, to the individual components. So we replace: Slerp(Q_0, Q_1, t) with: Slerp(Q_I, S, t) Slerp(Q_I, T, t) From the rod example, we choose the twist axis to align with the rod's longest side. Let's look at the effect of the individual components Slerp(Q_I, S, t) and Slerp(Q_I, T, t) as t varies over time below, swing on left and twist on right:
And as we concatenate these two components together, we get a swing-twist interpolation that rotates the rod such that its moving end travels in the shortest arc in 3D. Again, here is a side-by-side comparison of slerp (left) and swing-twist interpolation (right):

I decided to name my swing-twist interpolation function sterp. I think it’s cool because it sounds like it belongs to the function family of lerp and slerp. Here’s to hoping that this name catches on.
And here’s my code implementation:
public static Quaternion Sterp ( Quaternion a, Quaternion b, Vector3 twistAxis, float t ) { Quaternion deltaRotation = b * Quaternion.Inverse(a); Quaternion swingFull; Quaternion twistFull; QuaternionUtil.DecomposeSwingTwist ( deltaRotation, twistAxis, out swingFull, out twistFull ); Quaternion swing = Quaternion.Slerp(Quaternion.identity, swingFull, t); Quaternion twist = Quaternion.Slerp(Quaternion.identity, twistFull, t); return twist * swing; } Proof
Lastly, let’s look at the proof for the swing-twist decomposition formulas. All that needs to be proven is that the swing component S does not contribute to any rotation around the twist axis, i.e. the rotational axis of S is orthogonal to the twist axis. Let vec{V_{R_para}} denote the parallel component of vec{V_R} to vec{V_T}, which can be obtained by projecting vec{V_R} onto vec{V_T}: vec{V_{R_para}} = proj_{vec{V_T}}(vec{V_R}) Let vec{V_{R_perp}} denote the orthogonal component of vec{V_R} to vec{V_T}: vec{V_{R_perp}} = vec{V_R} - vec{V_{R_para}} So the scalar-vector form of T becomes: T = [W_R, proj_{vec{V_T}}(vec{V_R})] = [W_R, vec{V_{R_para}}] Using the quaternion multiplication formula, here is the scalar-vector form of the swing quaternion: S = R T^{-1} = [W_R, vec{V_R}] [W_R, -vec{V_{R_para}}] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_R} + W_R (-vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R (vec{V_R} -vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}}] Take notice of the vector part of the result: vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}} This is a vector parallel to the rotational axis of S. Both vec{V_R} X(-vec{V_{R_para}}) and vec{V_{R_perp}} are orthogonal to the twist axis vec{V_T}, so we have shown that the rotational axis of S is orthogonal to the twist axis. Hence, we have proven that the formulas for S and T are valid for swing-twist decomposition. Conclusion
That’s all.
Given a twist axis, I have shown how to decompose a rotation into a swing component and a twist component.
Such decomposition can be used for swing-twist interpolation, an alternative to slerp that interpolates between two orientations, which can be useful if you’d like some point on a rotating object to travel along the shortest arc.
I like to call such interpolation sterp.
Sterp is merely an alternative to slerp, not a replacement. Also, slerp is definitely more efficient than sterp. Most of the time slerp should work just fine, but if you find unwanted orientational sway on an object’s moving end, you might want to give sterp a try.

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