inversion sampling

[Wesley Smith]

Hi,
I'm trying to figure out how to evenly sample an arc of a circle under
inversive transformations. See the attached diagram for a visual of
the setup.

Here's a step-by-step of what's going on:

Given line L and circle C1 intersecting L
- generate C2 by inverting into C1 via C1*L/C1
- take the intersection of C1 and L
- sample evenly from one intersection point to the other across the
arc of C2 that lies within C1

For the list part in the image, I only showed samples from one
intersection point to the center of C1 (C2 always passes through the
center of C1).

One approach to generating the samples is to use circular translation
with a dual line generated by C2 and some scaling factor. I don't
want to do that because I want to generalize the situation from lines
and circles to planes and spheres. So, what I'd like to figure out is
what sequence distances from C2 will get me equal sampling along the
arc. I know for instance that:

dist(Pa - center(C2)) = radius_sq(C2)/radius_sq(C1)*dist(Pc-center(C1))

What I don't know is how to space out the values from [0, radius(C2)]
such that I get the desired result. It seems to depend on the circle
C3, but I'm not sure how to incorporate that information into what I'm
trying to derive. Any ideas?

thanks,
wes