# Interactive reciprocation and inversion

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?

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

*This post was originally on Google+*

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.

http://en.wikipedia.org/wiki/Minkowski_space

The change in sign of the squared radius is expressed neatly by the

metric signatureof the space (x,y,z) changing from (+,+,+) for the sphere to (+,+,-) for the hyperboloid. Is that right?