Question: How to map a 2-D surface on a sphere?

Actually, I could just cut to the chase here...

I need to print out a 2-D ring and then attach that to a 3-D sphere.  The problem is that the 2-D ring will buckle as I attempt to attach the outer circumference of it to the sphere.

I started wondering if I could use Maple to help me visualize other shapes (which must be closed) and show me how much they would buckle as I mapped them onto a sphere.  I realize that I could probably use a CAD program to do this, but don't believe that the CAD programs available to me will allow me to describe the 2-D shapes parametrically (e.g. sinewave for 1/4 cycle, cosine, etc...)

Anyhow... if anybody has already solved this problem, I would appreciate pointers or tips to the solution.

Failing that, if anybody would care to point a complete Maple beginner at how to get started drawing 2-D shapes and then folding them around a 3-D sphere, that should be enough to get me started.

(There also exists the possibility that what I want to do is physically impossible, in which case I would simply like to know how "close" I can get to the ideal).

Thanks for any help.

--wpd