>> Also, do any mathematicians know if the cause of the stretching artefacts are caused by the fact there is a singularity in the co-ordinate system?
>> And if so, is there any theoretical way around the problem?
About the best explanation I've heard is that the mapping function needs to be conformal in order to remove stretching. Z^2 is conformal, which means that if you make any tiny shape out of points in Z, then square each one, you end up with the same shape without any stretching, just scaled and rotated.
There doesn't seem to be much literature about 3d conformal mappings, but here are 3 ones that I think are conformal:
1. just scaling and rotating the 3d space, e.g. x,y,z = 2(x,y,z) or x,y,z = 2(-y, x, z)
2. A certain type of toroid may be conformal
3. A mapping that looks like a cardoid swept around its main axis is also probably conformal
The problem is that 1 is boring and 2 & 3 don't have an attractor at infinity... everything tends to sweep back to 0,0,0
I personally think there is a way around the problem, but the resulting bulb may not be spherical, since all spheres have a singularity.. see
http://en.wikipedia.org/wiki/Hairy_ball_theorem
December 02, 2009, 11:11:56 AM
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