What can we do with inversive geometry?

Question

I've recently been introduced to inversive geometry. This seems like it would be a very pretty area of study. Many sources that I have found seem a little old, however. I have two related questions:

1. The wiki page (linked above) suggests that there are various problems in geometry which are known to be solvable using inversive geometry. What are examples of these? I have found one very exciting one explained by Numberphile here. Are there more?

2. My looking around in various places suggests that inversive geometry is not such an active area of research at the moment. Is this true? Has it been superseded by other geometries, i.e. projective geometry?

3. Are there any authoritative books on inversive geometry, or any in particular which are known to be a good introduction to the subject?

Answer 1

In my opinion, it has been superseded.

Recall the idea of the complex plane: that we can view the complex numbers as being arranged in a Euclidean plane, similar to how the real numbers are often thought of as a number line.

The complex projective numbers are an incredibly important extension of the complex plane, formed by adding a single "point at infinity", usually called $\infty$, or sometimes $\omega$. (this has nothing to do with the first countable ordinal number) This is similar to the construction of the inversive plane.

Just like addition, multiplication, and conjugation of complex numbers has geometric interpretations, the reciprocal also has a geometric interpretation:

• Inversion in the unit circle about the origin is the same as the conjugate reciprocal $z \mapsto \bar{z}^{-1} $

Of central importance to the geometry of the complex projective numbers is the linear fractional transformations, also known as Möbius transformations. (aside: why does Wikipedia have two separate pages?!?!)

Linear fractional transformations, together with complex conjugation, subsume the basic geometric operations you'd do in inversive geometry: translations, reflections, rotations, and inversions. Conversely, every linear fractional transformation can be built out of these transformations.

It is in this form — via the study of linear fractional transformations of complex numbers — that one most typically encounters the ideas and methods of inversive geometry.

Answer 2

Inversive geometry is part of a wider field of research. For a general outlook: https://archive.org/details/treatiseonthecir033247mbp/page/n6.

Fashions come and go. Today there is a generalization trend, as in https://www.researchgate.net/publication/305313075_A_generalization_of_the_spherical_inversion, or "An Introduction to Inversion in an Ellipse" (https://arxiv.org/pdf/1309.6378.pdf). Projective geometry deals with a topic linked to inversion. However, circles are replaced by conics, and the inversion operation itself should be replaced by reciprocation.

A preview of the book suggestion on Wikipedia -- D.E. Blair, Inversion Theory and Conformal Mappings (2000) -- is available online.(https://books.google.it/books/about/Inversion_Theory_and_Conformal_Mapping.html?id=xcuFAwAAQBAJ&printsec=frontcover&source=kp_read_button&redir_esc=y#v=onepage&q&f=false)