"sphere in a bag" plane to sphere projection

Question

I'm looking for the mathematical transformation to transform points on a 2D plane [0,1]x[0,1] onto the unitsphere.

The most common projection would latitude-longitude mapping by interpreting u and v as the angles for the spherical coordinates (map u to [0,2PI] and v to [-PI/2, PI/2])

This gives strong distortions on the poles of the sphere. One can think of this transform as like wrapping the sphere into a bonbon-paper twirling the paper at the both ends. This will give distortions at those two ends.

The transformation I'm looking for can be rater thoght of putting the sphere into the middle of a paper and putting all sides around the sphere and twirl them together on a single spot - so you get a little paper-bag with your sphere in it. This yelds minimal distortion on the bottom of the "bag" and maximum distoriton on the top - and if seen from below, the distortion is equal in all directions.

Can someone tell me how to calculate this kind of mapping?

Answer 1

For the mapping you describe, you can use polar coordinates: (x,y)-->(r,alpha), where r is in [0,1], representing the ratio between the distance from the center of the rectangle O(0.5,0.5) to the current point P(x,y), and the maximum length this segment could have at the current value of alpha. Then map r to [-PI/2, PI/2] and alpha to [0,2PI].