## Goal

I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.

## What I have tried

I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used to finding the corresponding transformation matrix, generalizing this approach.

But I also know that Möbius geometry can be described as an interpretation of $\mathbb{P}(\mathbb R^{3,1})$, as I'll describe in more detail below. My problem is, I don't know how to express a Möbius transformation defined by three points in that world. I know that apart from the points themselves, the circumcircle of the defining points has to map to the circumcircle of their images. But that's still only four constraints, while I believe I need five (since this is a three-dimensional projective space, similar to $\mathbb{RP}^3$).

The last constraint is probably that the light cone must be preserved, but that appears to be a non-linear condition, so I don't know how to incorporate that. Some knowledge about Lorentz transformations might help here, but I don't know much about those. In particular, I don't see how to characterize Lorentz transformations using *linear* conditions.

I also considered doing the $\mathbb{CP}^1$ approach to find a transformation (described as a matrix in $\mathbb C^{2\times 2}$), but trying to translate that into a $\mathbb R^{4\times 4}$ matrix for Möbius geometry I failed again: an intermediate representation had quadratic terms I couldn't get rid of.

## Question

How can I find a matrix in Möbius geometry, modeled as $\mathbb{P}(\mathbb R^{3,1})$, if my input is three points in the plane and their images?

## Möbius geometry

Here is my understanding of Möbius geometry in the plane. My primary source for this is currently section 3.2 of this paper (although I did change a sign in my vector below since theirs seemed to be incorrect). It suggests Introduction to Möbius Differential Geometry by Hertrich-Jeromin as well as Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. The former I find rather hard to read in a cookbook style, while the latter is not available from our library just now.

I imagine $\mathbb{P}(\mathbb R^{3,1})$ similar to $\mathbb{RP}^3$, i.e. projective three-space, but with the following bilinear form:

$$ \langle a,b\rangle = a_1b_1 + a_2b_2 + a_3b_3 - a_4b_4 $$

A circle with center $(x,y)$ and radius $r$ in this world is a vector

$$ \begin{pmatrix}2x \\ 2y \\ x^2+y^2-r^2-1 \\ x^2+y^2-r^2+1 \end{pmatrix} $$

or a multiple thereof. Two circles $a$, $b$ intersect orthogonally if and only if $\langle a,b\rangle = 0$ since

$$\langle a,b\rangle = 2\left(r_a^2 + r_b^2 - (x_a-x_b)^2 - (y_a-y_b)^2\right)$$

A circle of radius zero is a point. This kind of circle is the only one which is orthogonal to itself. Hence the light cone $\left\{p\in\mathbb{P}(\mathbb R^{3,1})\;\middle|\;\langle p,p\rangle = 0\right\}$ characterizes the set of all points. Since a Möbius transformation maps points to points, it has to preserve that light cone.