cartesian/polar conversion

I have a small problem which I suspect can be solved in an easy and
elegant way, and it is simply lack of sleep/stupidity preventing me
from finding it.

I have a number of line drawings comprised of cartesian points. I need
to project them onto a sphere (they are drawings of constellations, so
I need to project them onto an imaginary celestial sphere) and convert
the points into sphercal co-ordinates.
I *can* map the points to a sphere, that's no problem. But obviously,
when I do this they become distorted. What I want to do is find out
what the points should be to get an undistorted view - i.e the mapping
that will produce a projection so the points match the cartesian
version. Obviously, I realise this depends on the projection used etc,
but basically, a small amount of distortion is fine. I am sure there
must be a simple way to achieve this, but I can't seem to manage it.


Nick Veitch