I was trying to nail down what sort of world I wanted my current game project to present to players. In my previous game engine project, Revolude, the terrain was a simple heightmap and a static ROAM type algorithm was used to perform a simple polygon reduction before dumping each terrain chunk to a display list consisting of triangle strips and fans.

Revolude, with some pseudo-trees and buildings, and some missiles hitting the ground. |

I was just about to opt for creating a much more involved version of this same type of terrain when I had a stroke of ingenuity. The algorithm is logically very simple, and produces the equivalent output of what I would like to call the dual of a cubrille mesh. In other words, it takes a Minecraft-esque type mesh and converts every face into a single point, and each point into a triangle. The dual of a cube is an octahedron, and the algorithm I wanted to create essentially produced an octahedral mesh from a voxel volume.

A cube, and its 'dual' - an octahedron. Notice how each of the six cube corners becomes a single triangle, and each cube face is collapsed into its center point. |

From an outside standpoint, this really seems very simple to do. One approach could be to first generate a simple boxy cube mesh, and then try to brute force convert it into its dual, an octahedral mesh. This became an appealing option after only a while, but I only approach problems with the intent of devising a novel solution, so that was out of the question.

Ideally, I was hoping to create an algorithm in some way similar to marching cubes - where each cube was inspected along with its neighbors to produce geometry. Something that achieved the desired output through such an immediate method was the goal. This approach, however, became much more involved than I had initially believed it could be.

The end product works in multiple passes, on the volume as a whole. The first pass examines 8 voxels at a time, determining whether vertices should be placed, and where. The second pass connects vertices to form edges. Edges can only exist between vertices where the dot product of their normals is greater than or equal to zero. So, vertices must either be facing in the same direction, or up to 90 degrees apart. The third pass involves detecting triangles formed by edges. This is done by searching for 'complement' edges for each vertex, where two edges connected to the vertex are connected to a third edge.

The last operation involves detecting all the left-over quads formed by edges, by doing a similar search to the triangle detecting pass. Instead of looking at the edges connected to a point, we look at the edges connected to an edge, and see if there is an opposite edge on the ends of its connected edges.

These operations, naively implemented, can be extremely slow. Fortunately, it is not too expensive to store extra connectivity information which is used to quickly search edges, vertices, and triangles for related geometry. The end product can then be reduced to simple vertex information and vertex indices for triangles.

Here is a quick video of the algorithm at work with an evolving volume. Forgive the choppiness of the video, my little netbook isn't very awesome. The slowest part of this demo is generating the volume itself. The next step will be to utilize a compressed data structure as the data source, as opposed to raw volume data.

The end product works in multiple passes, on the volume as a whole. The first pass examines 8 voxels at a time, determining whether vertices should be placed, and where. The second pass connects vertices to form edges. Edges can only exist between vertices where the dot product of their normals is greater than or equal to zero. So, vertices must either be facing in the same direction, or up to 90 degrees apart. The third pass involves detecting triangles formed by edges. This is done by searching for 'complement' edges for each vertex, where two edges connected to the vertex are connected to a third edge.

The last operation involves detecting all the left-over quads formed by edges, by doing a similar search to the triangle detecting pass. Instead of looking at the edges connected to a point, we look at the edges connected to an edge, and see if there is an opposite edge on the ends of its connected edges.

These operations, naively implemented, can be extremely slow. Fortunately, it is not too expensive to store extra connectivity information which is used to quickly search edges, vertices, and triangles for related geometry. The end product can then be reduced to simple vertex information and vertex indices for triangles.

Here is a quick video of the algorithm at work with an evolving volume. Forgive the choppiness of the video, my little netbook isn't very awesome. The slowest part of this demo is generating the volume itself. The next step will be to utilize a compressed data structure as the data source, as opposed to raw volume data.

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