Observe that all parameter circles are tangential to the coordinate axes at 0. Since we use the inverse map to visualize what happens near infinity, we translate this tangential property of the circles back into the domain and say: “parallel lines are tangent to each other at infinity”.

The inverse function maps any straight line in the domain to a circle through 0.
Since we show domain and range we cannot use full lines. The stereo image shows
very well that all the (partial) circles lie on the unit sphere and the full
circles would go through the south pole (the stereographic image of 0).

One can see on the Riemann sphere that infinity is covered without any singular behavior. This
is an example of the fact that all rational functions are better visualized on the Riemann sphere
than in the Gaussian plane.

It is an important property of the inverse function that it maps lines and circles in the Gaussian
plane to circles on the Riemann sphere. This is no surprise since the inverse function as a map
from the Riemann sphere to the Riemann sphere is just a 180 degree rotation.
For computations use this representation of a circle: { z | (z-m)*conjugate(z-m) = r^2 }.