Interactive reciprocation and inversion of a circle.<br><br>I&#39;m trying to...
- 2014-10-06 20:37:18+0000 - Updated: 2014-10-06 20:37:18+0000
Interactive reciprocation and inversion of a circle.

I'm trying to understand how the hyperbola in hyperbolic space arises. Why wouldn't a parabola work, for example? And for projecting the hyperbola onto the Poincaré disk, is there a nice operation like there is in the spherical geometry case, where circle inversion takes us from the sphere to the plane and vice versa?

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Reshared by: Vijay Sharma
- 2014-10-07 08:19:45+0000 - Updated: 2014-10-07 09:30:20+0000
I found this: "hyperbolic space is a hyperboloid of two sheets that may be thought of as a sphere of squared radius -1 or of radius i = sqrt(-1)"
http://www.math.brown.edu/~rkenyon/papers/cannon.pdf
Now this makes more sense to me. A hyperboloid is a sphere with negative curvature. This is exactly what we want for exploring plane tessellations with different curvatures.

I read elsewhere that the intersection of two sphere is a circle if they intersect, and a circle with imaginary radius if they don't intersect. That means that two non-intersecting spheres intersect at a hyperboloid? Interesting.
- 2014-10-07 12:16:43+0000 - Updated: 2014-10-07 12:21:10+0000