Interactive reciprocation and inversion

Interactive reciprocation and inversion of a circle.<br><br>I&#39;m trying to...
Tim HuttonTim Hutton - 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?

For full functionality of this site it is necessary to enable ...

Shared with: Public
Reshared by: Vijay Sharma
Tim Hutton - 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)"
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.
Tim Hutton - 2014-10-07 12:16:43+0000 - Updated: 2014-10-07 12:21:10+0000
Further reading:
The change in sign of the squared radius is expressed neatly by the metric signature of the space (x,y,z) changing from (+,+,+) for the sphere to (+,+,-) for the hyperboloid. Is that right?
Tim Hutton - 2014-10-07 14:11:41+0000 - Updated: 2014-10-07 14:17:45+0000
Yes, it seems right. As hoped, the formula for stereographic projection is exactly as normal, but the distance function now depends on the metric signature. I've put a demo and explanation here:
Tim Hutton - 2014-10-07 14:17:26+0000
So to answer my own questions: the original demo in this post, about reciprocation, was a complete red herring. Under the correct metric signature the hyperbola is very naturally the inverted form of the circle. And yes, there is a nice operation that does stereographic projection: it's the usual stereographic projection, only with the appropriate distance function for the metric signature. See the previous comment for the formulae.

This post was originally on Google+