Mapping a cube's net to a sphere and circle with minimal distortion?

I’m making a set of basic geometric solids (cube, cylinder, sphere, possibly various other shapes) that all have the same template for their UV mapping, so any texture that works with one works with another. This template is the net of a cube; a row of four 256x256 squares, with two more above and below the third in the line.

How would I go about figuring out how to distort each texture the least? I’ve cut up my sphere into what seems to be the most like the cube’s net and mapped it to a grid for the sides and a set of squares for the poles. This results in minimal distortion at the sides, so they’re fine, but at the poles, a square becomes a circle and a circle becomes a mess, as does a triangle. I want textures to look good on both cubes and spheres. How would I go about figuring where to put the points, if I want it to be as not-distorted as possible while still having the whole square texture?

Solutions I’ve come up with for each ring of points (shapes are outside-in, shapes are full size then 3/4 then 1/2 then 1/4 unless otherwise noted, meaning sorta equidistant):

  • To heck with it, square square square square centerpoint.
  • Square, square-with-bent-edges, square with even more bent edges, circle, centerpoint.
  • Double number of corners every time (square, octagon, 16-sided shape, 32 sided shape, centerpoint.)
  • Square, circle, circle, circle, centerpoint (meaning anything between the square and the circle are distorted)
  • Above, but with the largest circle being 256 in diameter instead of 3/4 of 256 and the circles inside scaling to match

Which would be best? Which would look nicest? I’m going to try square then circles with the “normal” spacing (1/8th of the space avaliable between each layer at their closest)

Please clearly show what you are talking about so we don’t have to try and decipher what is in your mind.

Show us something, supply demos etc etc