Hyperbolic Inversion (Inversion II)

The first time I taught hyperbolic geometry, I thought I could be done in a week. I also thought that it would help to do spherical and hyperbolic geometry side by side. To save time, I did it not side by side, but simultaneously, using the quadratic forms

For negative ε, the spheres become hyperboloids, but the formula for inversion I(x) = x/ x ‧ x still works, with all its suitably formulated properties. Hence one can use the stereographic projection both for spheres and the hyperboloid in Lorentz space to get all the models with its simultaneously.

Needless to say, this was a complete disaster. Let’s just study the 2-dimensional hyperbolic inversion. Circles in this geometry are the level sets of the quadratic form above, which are hyperbolas. Below you see concentric hyperbolas with growing radius. For each radius, you get two branches. The squared radius can be 0, in which case you get the two diagonals, and even negative, so that the hyperbolas open upward and downward.

These and their translates are the only hyperbolas we will consider, others belong to other quadratic forms. Let’s begin simple and invert a family of parallel horizontal lines at the blue hyperbola. Their images are hyperbolas with one branch intersecting the mirror hyperbola in the same points as the lines, and all touching at the origin.

In fact, all lines are mapped to hyperbolas. Below are to families of line segments and their image hyperbolas.

If you want to check this with a computation: A line through (a,b) making angle φ with the x-axis is mapped to a hyperbola with “center” p and squared radius

Then, certain hyperbolas (namely level sets of our quadratic form) are mapped to hyperbolas. Below, the purplish hyperbolas are concentric, and their greenish images pass through the same pair points. (Where do these intersections come from – – -?).

In formulas: A hyperbola with center (a,b) and radius r is mapped to another hyperbola with

Are circles mapped to circles?

Evidently not. Euclid isn’t here anymore, circles are not round, and we better don’t mention the rest.