djones@united.berkeley.edu (David Jones) (09/09/88)
I have a problem that involves placing "evenly spaced" points on an ellipsoid. Can you help? Evenly spacing points on a circle is easy - we use regular polygons. We might try regular polyhedra for points on a sphere, but there are only 5 of these: they have 4, 6, 8, 12, and 20 faces, which are triangles, squares, triangles, pentagons, and triangles respectively. For my purposes, a semi-regular polyhedron may be sufficient. A semi-regular polyhedron can be created from a regular polyhedron in a straighforward manner (though I can't seem to describe it in a concise way). For example, subdividing the 20 vertex, 12 face (pentagons) regular polyhedron gives a familiar soccerball with 32 faces (12 pentagons and 20 hexagons). Problem #1: For which values of N can we find regular or semi-regular polyhedra with N vertices? This would allow me to place "evenly spaced" vertices on a sphere. Problem #2: Is there a similar method to place "evenly spaced" vertices on an ellipsoid? In particular, I'm interested in the special case of 2 2 2 X + 4Y + 4Z = 4 (ie one axis is twice as long as the others) Any solutions, or pointers to math texts or papers would be very much appreciated. Please reply directly to me, since I have limited access to Net News. If I get a great solution, I'll post it. thanks, David Jones djones@united.berkeley.edu