Interesting Technical problem with distorting polar grids

[Runninghead]

Not expecting a quick solve for this one, but I hope it's of interest to some.

You may be familiar with map distortion techniques- make a double square image and it wraps nicely around a globe but you need to "fix the poles" where the distortion looks really bad.

In Photoshop you just use Filter > Distort > Polar Co-ordinates.

Here's a flat grid undistorted:
grid undistorted.jpg
You can tell which side will become which pole and where the least distorted "sweet spot" will be from the colours.
Now see the polar distorted version:
grid distorted.jpg

My question is this- how can I pre-distort the flat grid before running the Polar Co-ordinates filter so the squares of the grid are closer to being true squares in the final image?

Is it even possible?

I've tried some trickery with Filter > Distort > Spherize (Horizontal Only) but it seems like either it doesn't work or I'm not doing it right.

Any ideas?

[Redrobes]

Its impossible to distort a square onto a sphere without having points where it pinches up. There are always going to be some areas which are more distorted than others. I.e. you cant keep squares square when mapped onto a sphere. You can do it to some extent on a torus but I expect that wont help you. All of the map projections have some weaknesses as well as some benefits. You should check out map projections in wikipedia and look for the one which matches the closest to your own ideals. If you then want to warp an image you have using that projection then there are several apps about for it. GDAL is one which springs to mind and there is another with a name similar to GProjector but I forget exactly.

I think the question your asking is if I put some squarish shapes onto the sphere then what will they look like in the original image. Well you can just unproject that image back using the same apps.

Hope that helps.