Finding the Centroid of a Spherical Crescent

[Dehstil]

Hello,

What I really want to do is deal with spherical "crescents" and incomplete annuli and see how well they are approximated by spherical caps, but here is my question:

How would you go about finding the centroid of a spherical crescent (one spherical cap minus the other) in the case when they are partially intersecting?

After some digging, I've managed to find the area but not much else for this case:
Page 10 on: ati.amd.com/developer/siggraph06/Oat-AmbientApetureLighting.pdf[/URL]
Page 2: [url]www.cse.ust.hk/~psander/docs/aperture.pdf[/url]
Page 12: [url]www3.interscience.wiley.com/cgi-bin/fulltext/121601807/PDFSTART[/url]

I've attempted some geometric approaches but have not gotten very far. Perhaps knowing how to integrate over a spherical cap or the intersection of two spherical caps would be useful in a calculus-based "center of mass" approach.

 

[flatmaster]

I'm pretty sure you want cylindrical coordinates with the center of both spheres on the z axis. Integral over theta is easy via symetry. You would have two remaining integrals over z and roh. You need to write equations for the surface of each sphere in terms of z and roh.