Hyperspheres made up of tetrahedrons are difficult...
So far I can either represent a hypersphere with 5 tetrahedron cells (4d version of face) or represent one with 600 tetrahedron cells. I do not know of any convex shape between the two with only tetrahedron shaped cells and all points equally distanced from the center. I tried making a "subdivide" function and using it on the 5 cell shape and then stretching points to be equally distanced from the center, but it becomes convex. Only way to fix it is to shorten 5 vertices until those cells form an octahedron. Pretty nonoptimal since that's 40 cells wasting rendering resources!
A problem is that traditional subdividing is impossible in 4D. A regular triangle can be cleanly subdivided into 4 identical smaller triangles, but a tetrahedron cannot be subdivided into identical tetrahedron. My solution was to divide it into 4 identical smaller tetrahedrons and 8 additional tetrahedrons of similar size but it was less than ideal with the above example. Similarly that makes LOD generation really really difficult.
On the positive side of things, I think I've figured out the best approach for having a terrain feature in my engine. Instead of modeling the terrain by hand, it'd be generated during runtime by a 3D texture acting as a heightmap. This would save a ton of memory, be easy to stream in chunks, and solve the LOD problem. Close terrain could be generated at max resolution while distant hills and mountains could be generated using a lower resolution version of the texture. It could use the texture for physics too which would save a lot of performance.
So far I can either represent a hypersphere with 5 tetrahedron cells (4d version of face) or represent one with 600 tetrahedron cells. I do not know of any convex shape between the two with only tetrahedron shaped cells and all points equally distanced from the center. I tried making a "subdivide" function and using it on the 5 cell shape and then stretching points to be equally distanced from the center, but it becomes convex. Only way to fix it is to shorten 5 vertices until those cells form an octahedron. Pretty nonoptimal since that's 40 cells wasting rendering resources!
A problem is that traditional subdividing is impossible in 4D. A regular triangle can be cleanly subdivided into 4 identical smaller triangles, but a tetrahedron cannot be subdivided into identical tetrahedron. My solution was to divide it into 4 identical smaller tetrahedrons and 8 additional tetrahedrons of similar size but it was less than ideal with the above example. Similarly that makes LOD generation really really difficult.
On the positive side of things, I think I've figured out the best approach for having a terrain feature in my engine. Instead of modeling the terrain by hand, it'd be generated during runtime by a 3D texture acting as a heightmap. This would save a ton of memory, be easy to stream in chunks, and solve the LOD problem. Close terrain could be generated at max resolution while distant hills and mountains could be generated using a lower resolution version of the texture. It could use the texture for physics too which would save a lot of performance.
