The plane "sphere in the bag" on a spherical projection

Question

I am looking for a mathematical transformation to convert points on a two-dimensional plane [0,1]x[0,1]to units of units.

The most common projection is to display latitude and longitude by interpreting uand vas angles for spherical coordinates (mapping uto [0,2PI]and vto [-PI/2, PI/2])

This gives a strong distortion at the poles of the sphere. You might think of this transformation as wrapping a sphere in paper, curling paper at both ends. This will give distortion at these two ends.

The transformation I'm looking for can be simpler in placing the sphere in the middle of the paper and putting all the sides around the sphere and merging them in one place - so you get a small paper bag with your sphere in it. This reduces the minimum distortion at the bottom of the “bag” and the maximum distortion from above - and, when viewed from below, the distortion is equal in all directions.

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

Answer 1

: (x, y) → (r, alpha), r [0,1], O (0,5,0,5) P (x, y) , . r [-PI/2, PI/2] alpha [0,2PI].

Answer 2

, , . , , .

, , Dymaxion, , .

. .

Answer 3

, . . .

Answer 4

x-y 0 1 (.. ), 0 pi/2. (1,1) , (2). , (1,1) , .