Mapping problem ( sphere to rectangle )

This may be the wrong group for this problem,.. POLITE suggestions as to where I should go gratefully received :-)
Here is the problem :
If I have a sphere of radius R, with a band drawn on it of width W the edges of the band are circles.
If the sphere is centered on the origin, it's surface can be mapped to a square of 180 x 360 units with the unit being a degree of latitude or longitude. ( origin at centre ).
With a horizontal band the two edges are just horizontal lines at Y = ( + - ) SIN^-1 ( R / ( W / 2 )).
For a vertical band the plot of the edges is a pair of sort of " squared of " circles. Can someone please tell me how to calculate the shape of these "circles".
If I can plot a set of points around the shape, I can use AutoCAD to draw a spline curve to plot the shape with reasonable accuracy.
( the idea is to use the plot to print a series of colour bands on the " quarters " of a sphere )
Ideally I want a formula of the form Y = f ( X, (W/R)) with Y being latitude and X longitude and W/R the relative width of the strip on the sphere.
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Regards Jonathan

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Dear Jonathan Barnes:

You are attempting to create a printable surface that you can "roll up" into a sphere?
Sounds like what you asked for: http://www.vendian.org/mncharity/dir3/planet_globes / "Icosahedron" will get you other good hits, perhaps with algorithms.
You do understand that a flat surface cannot be "rolled up" into a true sphere, unless you make an infinte number or "points", right? It'll just be close...
David A. Smith
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Not quite rolling up.
I have software that wraps a rectangular bit map onto a virtual sphere using this mapping. I can export suitable bitmaps from AutoCAD if I can draw them.
In the real world it can be used like this :
Put a sphere on a stick ( Z axis ), with a rotary driver on the stick, Have a printing head on a rotary frame rotating it about the X axes.
trace the shape on a flat bed to get your shape on the sphere... direct control two stepper motors on the sphere and printer frame from linear motions of the plotter ( or programmed motions ).
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Regards Jonathan

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Dear Jonathan Barnes:
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If you still have difficulty getting the algorithm from the keyword above, let us know.
David A. Smith