# Möbius transformation by 3 points in the Minkowski model

## 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.

• Although this question has been answered in terms of how to take the $\mathbb{CP}^1$ description and convert it to this world, I'm still interested in solutions which tackle the problem from a different angle, since there might be room for generalizations, e.g. defining a transformation in terms of circles not points, or generalizing to higher dimensions. – MvG Jan 30 '14 at 22:12
• The Poincaré and Beltrami-Klein models of the hyperbolic space ${\mathbb H}_{\mathbb R}^n$ (we discussed that, for $n=3$, the conformal and holomorphic structures on the absolute ${\mathbb S}^2$ are the same). The formula (I dislike as depending on coodinates) $z\mapsto\frac{2z}{|z|^2+1}$ establish an isometry (up to a scalar factor) between them. Another coincidence between models is ${\mathbb H}_{\mathbb R}^4={\mathbb H}_{\mathbb H}^1$. Hope arxiv.org/abs/math/0702714 arxiv.org/abs/0907.4469 arxiv.org/abs/0907.4469 arxiv.org/abs/1107.0346 may be helpful. – Sasha Anan'in Feb 4 '14 at 17:03

In fact, you are looking for an imbedding $\text{PGL}_2{\mathbb C}=\text{PSL}_2{\mathbb C}\hookrightarrow\text{PO}({\mathbb R};3,1)=\text{Isom}\,{\mathbb H}_{\mathbb R}^3$ that provides the subgroup of all orientation-preserving isometries of ${\mathbb H}_{\mathbb R}^3$.

Consider the $4$-dimensional ${\mathbb R}$-linear space $H$ formed all hermitian $2\times 2$-matrices. It is easy to verify that $q(h):=\det h\in{\mathbb R}$ defined for all $h\in H$ is a quadratic form of signature $(3,1)$.

The group $\text{SL}_2{\mathbb C}$ acts on $H$ by the rule $g\cdot h:=ghg^*$ (on the right-hand side, we use the matrix multiplication and the hermitian transpose). Obviously, this action preserves the quadratic form $q$ and, therefore, provides the desired imbedding.

The imbedding $g\mapsto i(g)$ is "quadratic" : the coefficients of $i(g)$ are quadratic homogeneous expressions in coefficients of $g$ (they also involve the complex conjugation). This "quadratic" feature is inescapable because, in a sense, an imbedding in question is unique. So, if you need, you can extract from it all particular formulae you may want to.

• Thanks a lot! I now see how I can represent my circles as $$h=\begin{pmatrix}x^2+y^2-r^2&x+iy\\x-iy&1\end{pmatrix}$$ and operate on them by $ghg^*$ as you described. If I had to, I could turn this into a $4\times4$ matrix on the original description, but for the moment I'm very happy with this representation. – MvG Jan 30 '14 at 22:03

It is fun to see in an invariant way where the embedding $\text{PGL}_2{\mathbb C}=\text{PSL}_2{\mathbb C}\hookrightarrow\text{PO}({\mathbb R};3,1)$ which Sasha speaks of comes from'. Take $V$ to be a two-dimensional complex vector space so that ${\mathbb C} {\mathbb P}^1$ is the quotient of $V \setminus 0$ by the group ${\mathbb C} \setminus 0$ of nonzero complex scalars. Now split the group up as ${\mathbb R}^+ \times S^1 =$ (scalings)x (rotations) and form the same quotient but in two steps: first by rotation, then by scaling.

Divide $V$ by the rotation group $S^1$ using invariant theory as follows.

A) LEMMA: The vector space $Q$ of real quadratic $S^1$-invariant functions on $V$ forms a 4-dimensional real vector space. Any $S^1$-invariant polynomial on $V$ is a function of these quadratic invariants. There is a single quadratic relation $R$ amongst the invariants.

STANDARD REALIZATION OF LEMMA. Choose linear coordinates $z_1, z_2$ for $V$. A basis for $Q$ is then $|z_1|^2, |z_2|^2, Re(z_1 \bar z_1), Im(z_1 \bar z_2)$. The relation $R$ comes from $|z_1|^2|z_2|^2 = |z_1 \bar z_2|^2$. EXERCISE: as a real quadratic form on $Q$ the relation $R$ has signature $3,1$.

B) Let $Q^*$ be the real dual vector space to $Q$. Let $\delta V \to Q^*$ be evaluation map $\delta(v) (P) = P(v)$, for $P \in Q$. It follows from the lemma that this map defines an isomorphism $V/S^1$ onto the image of $\delta$. The image of $\delta$ is the `positive half'' of the light cone in $Q^*$
relative to the dual quadratic form to the quadratic form $R$ on $Q$. (The origin is sent to the cone point.) In terms of the standard realization we can write
$\delta(v) = v v^*$ where $v= (z_1,z_2)$. We are then identifying $Q^*$ with the space of Hermitian matrices on $V$. The components of the rank 1 Hermitian matrix $v v^*$ are the basis of invariants. The map $\delta$ is a $Gl(V)=\text{GL}_2{\mathbb C}$ equivariant map (and $S^1$-invariant) map from $V$ to $Q^*$.

Next, divide by scalings $t \in {\mathbb R}^+$. Use $\delta(tv) = t^2 \delta (v)$. Thus dividing by scalings over on the image of $\delta$ side corresponds to dividing out the positive light cone by positive scalings. We get as a result a $\text{GL}_2{\mathbb C} : \text{O}({\mathbb R};3,1)$-equivariant embedding of ${\mathbb C} {\mathbb P}^1$ into real projective 3-space whose image is the projectivized light cone: a real projective conic.

Where are the circles of Mobius in this real conic? Three distinct points (or rays) on the light cone determine a unique 3-plane in $Q^*$. The intersection of this 3-plane with the light cone, viewed real-projectively is the corresponding circle connecting the three points.