Is there a name for the map projection that makes a circle out of a sphere, where, in polar coordinates, the angle is the bearing angle of the point and the radius is the distance (on the surface of the sphere) between the point and the point defined to be the center?
This would be equivalent to making a hole inside the sphere (opposite to where you want your map projection's center to be) and then stretching out the sphere into a circle, preserving distance in the radial direction.