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hendrikboom at 05:35pm on 10/07/2008
hendrikboom at 05:35pm on 10/07/2008Take a tetrahedron. No, it doesn't have to be regular. Call it tetrahedron ABCD. Pick one of its edges, maybe the longest one. Let's say that's edge CD. Now pick a point E on CD. Maybe near the middle is a good choice. Now divide ABCD into two, and get get ABCE and ABED.
Keep doing this, and get more and more tetrahedra. If you'd like the tetrahedra to line up, you could have the convention that each edge is divided only once, no matter how many tetrahedrons is it on -- all those tetrahedra will have to agree on where they are divided. Or one could systematically, each time divide the longest edge in the whole shebang.
Now while creating this whole shebang of nested, more-or-less edge-matched tetrahedra, you could apply various attributes to the tetrahedron (such as what materials it's mostly made of), and modify them when progressing from a tetrahedron to its components. Or when dividing an edge in two, you could pick a point slightly off the edge, breaking it up into two edges that aren't extensions of each other.
Would this make a usable space division for, say, a mining video game? The nice thing about it is that the division of tetrahedrons in a volume can be compatible with a division into triangles of the surface. Triangle meshes are well-known in computer graphics. You could generate a whole approximate planet by starting with an icosahedron and dividing its volume into twenty tetrahedra that all meet at the centre.
In fact, this generalizes nicely to multidimensional hyperspace divisions, in case anyone should ever want to make that kind of a simulation game. Or physics simulation. Maybe evaluating some kind of Hamiltonian or Lagrangian and using that to affect subdivision statistics would yield something interesting. A multiscale differential equation solver on lumpy space, anyone?
Keep doing this, and get more and more tetrahedra. If you'd like the tetrahedra to line up, you could have the convention that each edge is divided only once, no matter how many tetrahedrons is it on -- all those tetrahedra will have to agree on where they are divided. Or one could systematically, each time divide the longest edge in the whole shebang.
Now while creating this whole shebang of nested, more-or-less edge-matched tetrahedra, you could apply various attributes to the tetrahedron (such as what materials it's mostly made of), and modify them when progressing from a tetrahedron to its components. Or when dividing an edge in two, you could pick a point slightly off the edge, breaking it up into two edges that aren't extensions of each other.
Would this make a usable space division for, say, a mining video game? The nice thing about it is that the division of tetrahedrons in a volume can be compatible with a division into triangles of the surface. Triangle meshes are well-known in computer graphics. You could generate a whole approximate planet by starting with an icosahedron and dividing its volume into twenty tetrahedra that all meet at the centre.
In fact, this generalizes nicely to multidimensional hyperspace divisions, in case anyone should ever want to make that kind of a simulation game. Or physics simulation. Maybe evaluating some kind of Hamiltonian or Lagrangian and using that to affect subdivision statistics would yield something interesting. A multiscale differential equation solver on lumpy space, anyone?
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We had planned on building the asteroid out of tetrahedra and having a triangular-tessellated surface. We weren't going to do it by subdividing tetrahedra though. The plan was to produce a surface by choosing semi-random radii from a center of mass for a covering set of points on a nominal 'sphere', building a triangular mesh from that, and then working inwards to divide into tetrahedra while maintaining the already-known center of mass.
I'd settled on this because NASA records asteroid shapes as a set of radii at latitude/longitude points which meant I could suck in actual asteroid data to work out the statistics by which I generated my fake ones.
The whole scheme never went past the planning stage.