# Poincaré Extension of Möbius Transformations

Vladimir V. Kisil On leave from Odessa University.Email:
<EMAIL_ADDRESS>Web:
http://www.maths.leeds.ac.uk/~kisilv/AMSMSC[2010]: Primary 30C20; Secondary
30C35. School of Mathematics, University of Leeds, Leeds LS2 9JT, England

###### Abstract

Given sphere preserving (Möbius) transformations in $n$-dimensional Euclidean
space one can use the Poincaré extension to obtain sphere preserving
transformations in a half-space of $n+1$ dimensions. The Poincaré extension is
usually provided either by an explicit formula or by some geometric
construction. We investigate its algebraic background and describe all
available options. The solution is given in terms of one-parameter subgroups
of Möbius transformations acting on triples of quadratic forms. To focus on
the concepts, this paper deals with the Möbius transformations of the real
line only.

###### keywords:

Möbius transformation, Poincaré extension, quadratic forms, Cayley–Klein
geometries.

_Dedicated to John Ryan on the occasion of his 60th birthday_

##  1 Introduction

It is known, that Möbius transformations on $\mathbb{R}^{n}{}$ can be expanded
to the “upper” half-space in $\mathbb{R}^{n+1}{}$ using the Poincaré extension
[1, § 3.3] [2, § 5.2]. An explicit formula is usually presented without a
discussion of its origin. In particular, one may get an impression that the
solution is unique. This paper considers various aspects of such extension and
describes different possible realisations. Our consideration is restricted to
the case of extension from the real line to the upper half-plane. However, we
made an effort to present it in a way, which allows numerous further
generalisations.

##  2 Geometric construction

We start from the geometric procedure in the standard situation. The group
$\mathrm{SL}_{2}(\mathbb{R}{})$ consists of real $2\times 2$ matrices with the
unit determinant. $\mathrm{SL}_{2}(\mathbb{R}{})$ acts on the real line by
linear-fractional maps:

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}:x\mapsto\frac{ax+b}{cx+d},\quad\text{
where }x\in\mathbb{R}{}\text{ and }\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R}{}).$ (1)

A pair of (possibly equal) real numbers $x$ and $y$ uniquely determines a
semicircle ${C_{xy}}$ in the upper half-plane with the diameter $[x,y]$. For a
linear-fractional transformation $M$ (1), the images $M(x)$ and $M(y)$ define
the semicircle with the diameter $[M(x),M(y)]$, thus, we can define the action
of $M$ on semicircles: $M({C_{xy}})={C_{M(x)M(y)}}$. Geometrically, the
Poincaré extension is based on the following lemma, see Fig. 1(a) and more
general Lem. 18 below:

######  Lemma 1.

If a pencil of semicircles in the upper half-plane has a common point, then
the images of these semicircles under a transformation (1) have a common point
as well.

Elementary geometry of right triangles tells that a pair of intersecting
intervals $[x,y]$, $[x^{\prime},y^{\prime}]$, where
$x<x^{\prime}<y<y^{\prime}$, defines the point

$\left(\frac{xy-x^{\prime}y^{\prime}}{x+y-x^{\prime}-y^{\prime}}\,,\,\frac{\sqrt{(x-y^{\prime})(x-x^{\prime})(x^{\prime}-y)(y-y^{\prime})}}{x+y-x^{\prime}-y^{\prime}}\right)\in\mathbb{R}^{2}_{+}{}.$
(2)

An alternative demonstration uses three observations:

  1. 1.

the scaling $x\mapsto ax$, $a>0$ on the real line produces the scaling
$(u,v)\mapsto(au,av)$ on pairs (2);

  2. 2.

the horizontal shift $x\mapsto x+b$ on the real line produces the horizontal
shift $(u,v)\mapsto(u+b,v)$ on pairs (2);

  3. 3.

for the special case $y=-x^{-1}$ and $y^{\prime}=-{x^{\prime}}^{-1}$ the pair
(2) is $(0,1)$.

Finally, expression (2), as well as (3)–(4) below, can be calculated by the
specialised CAS for Möbius invariant geometries [3, 4].

(a)  (b)
(c)  (d)
(e)  (f)

Figure 1: Poincaré extensions: first column presents points defined by the
intersecting intervals $[x,y]$ and $[x^{\prime},y^{\prime}]$, the second
column—by disjoint intervals. Each row uses the same type of conic
sections—circles, parabolas and hyperbolas respectively. Pictures are produced
by the software [4].

This standard approach can be widened as follows. The above semicircle can be
equivalently described through the unique circle passing $x$ and $y$ and
orthogonal to the real axis. Similarly, an interval $[x,y]$ uniquely defines a
right-angle hyperbola in $\mathbb{R}^{2}{}$ orthogonal to the real line and
passing (actually, having her vertices at) $(x,0)$ and $(y,0)$. An
intersection with the second such hyperbola having vertices $(x^{\prime},0)$
and $(y^{\prime},0)$ defines a point with coordinates, see Fig. 1(f):

$\left(\frac{xy-x^{\prime}y^{\prime}}{x+y-x^{\prime}-y^{\prime}}\,,\,\frac{\sqrt{(x-y^{\prime})(x-x^{\prime})(x^{\prime}-y)(y^{\prime}-y)}}{x+y-x^{\prime}-y^{\prime}}\right),$
(3)

where $x<y<x^{\prime}<y^{\prime}$. Note, the opposite sign of the product
under the square roots in (2) and (3).

If we wish to consider the third type of conic sections—parabolas—we cannot
use the unaltered procedure: there is no a non-degenerate parabola orthogonal
to the real line and intersecting the real line in two points. We may recall,
that a circle (or hyperbola) is orthogonal to the real line if its centre
belongs to the real line. Analogously, a parabola is _focally orthogonal_ (see
[5, § 6.6] for a general consideration) to the real line if its focus belongs
to the real line. Then, an interval $[x,y]$ uniquely defines a downward-opened
parabola with the real roots $x$ and $y$ and focally orthogonal to the real
line. Two such parabolas defined by intervals $[x,y]$ and
$[x^{\prime},y^{\prime}]$ have a common point, see Fig. 1(c):

$\left(\frac{xy^{\prime}-yx^{\prime}+D}{x-y-x^{\prime}+y^{\prime}}\,,\,\frac{(x^{\prime}-x)(y^{\prime}-y)(y-x+y^{\prime}-x^{\prime})+(x+y-x^{\prime}-y^{\prime})D}{(x-y-x^{\prime}+y^{\prime})^{2}}\right),$
(4)

where $D=\pm\sqrt{(x-x^{\prime})(y-y^{\prime})(y-x)(y^{\prime}-x^{\prime})}$.
For pencils of such hyperbolas and parabolas respective variants of Lem. 1
hold.

Focally orthogonal parabolas make the angle $45^{\circ}$ with the real line.
This suggests to replace orthogonal circles and hyperbolas by conic sections
with a fixed angle to the real line, see Fig. 1(b)–(e). Of course, to be
consistent this procedure requires a suitable modification of Lem. 1, we will
obtain it as a byproduct of our study, see Lem. 18. However, the respective
alterations of the above formulae (2)–(4) become more complicated in the
general case.

The considered geometric construction is elementary and visually appealing.
Now we turn to respective algebraic consideration.

##  3 Möbius transformations and Cycles

The group $\mathrm{SL}_{2}(\mathbb{R}{})$ acts on $\mathbb{R}^{2}{}$ by matrix
multiplication on column vectors:

$\mathcal{L}_{g}:\ \begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}\mapsto\begin{pmatrix}ax_{1}+bx_{2}\\\
cx_{1}+dx_{2}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}x_{1}\\\ x_{2}\end{pmatrix}\,,\quad\text{ where
}g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R}{}).$
(5)

A linear action respects the equivalence relation $\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}\sim\begin{pmatrix}\lambda x_{1}\\\ \lambda
x_{2}\end{pmatrix}$, $\lambda\neq 0$ on $\mathbb{R}^{2}{}$. The collection of
all cosets for non-zero vectors in $\mathbb{R}^{2}{}$ is the _projective line_
$P\mathbb{R}^{1}{}$. Explicitly, a non-zero vector $\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}\in\mathbb{R}^{2}{}$ corresponds to the point with
homogeneous coordinates $[x_{1}:x_{2}]\in P\mathbb{R}^{1}{}$. If $x_{2}\neq 0$
then this point is represented by $[\frac{x_{1}}{x_{2}}:1]$ as well. The
embedding $\mathbb{R}{}\rightarrow P\mathbb{R}^{1}{}$ defined by
$x\mapsto[x:1]$, $x\in\mathbb{R}{}$ covers the all but one of points in
$P\mathbb{R}^{1}{}$. The exceptional point $[1:0]$ is naturally identified
with the infinity.

The linear action (5) induces a morphism of the projective line
$P\mathbb{R}^{1}{}$, which is called a Möbius transformation. Considered on
the real line within $P\mathbb{R}^{1}{}$, Möbius transformations takes
fraction-linear form:

$g:\ [x:1]\mapsto\left[\frac{ax+b}{cx+d}:1\right]\,,\quad\text{ where
}g=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R}{})\text{ and }cx+d\neq 0.$

This $\mathrm{SL}_{2}(\mathbb{R}{})$-action on $P\mathbb{R}^{1}{}$ is denoted
as $g:x\mapsto g\cdot x$. We note that the correspondence of column vectors
and row vectors $\mathbold{i}:\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}\mapsto(x_{2},-x_{1})$ intertwines the left multiplication
$\mathcal{L}_{g}$ (5) and the right multiplication $\mathcal{R}_{g^{-1}}$ by
the inverse matrix:

$\mathcal{R}_{g^{-1}}:\
(x_{2},-x_{1})\mapsto(cx_{1}+dx_{2},\,-ax_{1}-bx_{2})=(x_{2},-x_{1})\begin{pmatrix}d&-b\\\
-c&a\end{pmatrix}\,.$ (6)

We extended the map $\mathbold{i}$ to $2\times 2$-matrices by the rule:

$\mathbold{i}:\ \begin{pmatrix}x_{1}&y_{1}\\\
x_{2}&y_{2}\end{pmatrix}\mapsto\begin{pmatrix}y_{2}&-y_{1}\\\
x_{2}&-x_{1}\end{pmatrix}\,.$ (7)

Two columns $\begin{pmatrix}x\\\ 1\end{pmatrix}$ and $\begin{pmatrix}y\\\
1\end{pmatrix}$ form the $2\times 2$ matrix $M_{xy}=\begin{pmatrix}x&y\\\
1&1\end{pmatrix}$. For geometrical reasons appearing in Cor. 3, we call a
_cycle_ the $2\times 2$-matrix ${C_{xy}}$ defined by

${C_{xy}}=\frac{1}{2}M_{xy}\cdot\mathbold{i}(M_{xy})=\frac{1}{2}M_{yx}\cdot\mathbold{i}(M_{yx})=\begin{pmatrix}\frac{x+y}{2}&-xy\\\
1&-\frac{x+y}{2}\end{pmatrix}\,.$ (8)

We note that $\det{C_{xy}}=-(x-y)^{2}/4$, thus $\det{C_{xy}}=0$ if and only if
$x=y$. Also, we can consider the Möbius transformation produced by the
$2\times 2$-matrix ${C_{xy}}$ and calculate:

${C_{xy}}\begin{pmatrix}x\\\ 1\end{pmatrix}=\lambda\begin{pmatrix}x\\\
1\end{pmatrix}\quad\text{and }\quad{C_{xy}}\begin{pmatrix}y\\\
1\end{pmatrix}=-\lambda\begin{pmatrix}y\\\ 1\end{pmatrix}\quad\text{where
}\lambda=\frac{x-y}{2}.$ (9)

Thus, points $[x:1]$, $[y:1]\in P\mathbb{R}^{1}{}$ are fixed by ${C_{xy}}$.
Also, ${C_{xy}}$ swaps the interval $[x,y]$ and its complement.

Due to their structure, matrices ${C_{xy}}$ can be parametrised by points of
$\mathbb{R}^{3}{}$. Furthermore, the map from
$\mathbb{R}^{2}{}\rightarrow\mathbb{R}^{3}{}$ given by $(x,y)\mapsto{C_{xy}}$
naturally induces the projective map $(P\mathbb{R}^{1}{})^{2}\rightarrow
P\mathbb{R}^{2}{}$ due to the identity:

$\frac{1}{2}\begin{pmatrix}\lambda x&\mu y\\\
\lambda&\mu\end{pmatrix}\begin{pmatrix}\mu&-\mu y\\\ \lambda&-\lambda
x\end{pmatrix}=\lambda\mu\begin{pmatrix}\frac{x+y}{2}&-xy\\\
1&-\frac{x+y}{2}\end{pmatrix}=\lambda\mu{C_{xy}}\,.$

Conversely, a zero-trace matrix $\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}$ with
a non-positive determinant is projectively equivalent to a product ${C_{xy}}$
(8) with $x,y=\frac{a\pm\sqrt{a^{2}+bc}}{c}$. In particular, we can embed a
point $[x:1]\in P\mathbb{R}^{1}{}$ to $2\times 2$-matrix ${C_{xx}}$ with zero
determinant.

The combination of (5)–(8) implies that the correspondence
$(x,y)\mapsto{C_{xy}}$ is $\mathrm{SL}_{2}(\mathbb{R}{})$-covariant in the
following sense:

$g{C_{xy}}g^{-1}={C_{x^{\prime}y^{\prime}}}\,,\quad\text{ where
}x^{\prime}=g\cdot x\text{ and }y^{\prime}=g\cdot y.$ (10)

To achieve a geometric interpretation of all matrices, we consider the
bilinear form
$Q:\mathbb{R}^{2}{}\times\mathbb{R}^{2}{}\rightarrow\mathbb{R}{}$ generated by
a $2\times 2$-matrix $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$:

$Q(x,y)=\begin{pmatrix}x_{1}&x_{2}\end{pmatrix}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}y_{1}\\\ y_{2}\end{pmatrix}\,,\quad\text{ where
}x=(x_{1},x_{2}),\ y=(y_{1},y_{2}).$ (11)

Due to linearity of $Q$, the null set

$\\{(x,y)\in\mathbb{R}^{2}{}\times\mathbb{R}^{2}{}\,\mid\,Q(x,y)=0\\}$ (12)

factors to a subset of $P\mathbb{R}^{1}{}\times P\mathbb{R}^{1}{}$.
Furthermore, for the matrices ${C_{xy}}$ (8), a direct calculation shows that:

######  Lemma 2.

The following identity holds:

${C_{xy}}(\mathbold{i}(x^{\prime}),y^{\prime})=\mathop{\mathrm{tr}}({C_{xy}}{C_{x^{\prime}y^{\prime}}})=\textstyle\frac{1}{2}(x+y)(x^{\prime}+y^{\prime})-(xy+x^{\prime}y^{\prime})\,.$
(13)

In particular, the above expression is a symmetric function of the pairs
$(x,y)$ and $(x^{\prime},y^{\prime})$.

The map $\mathbold{i}$ appearance in (13) is justified once more by the
following result.

######  Corollary 3.

The null set of the quadratic form
${C_{xy}}(x^{\prime})={C_{xy}}(\mathbold{i}(x^{\prime}),x^{\prime})$ consists
of two points $x$ and $y$.

Alternatively, the identities ${C_{xy}}(x)={C_{xy}}(y)=0$ follows from (9) and
the fact that $\mathbold{i}(z)$ is orthogonal to $z$ for all
$z\in\mathbb{R}^{2}{}$. Also, we note that:

$\mathbold{i}\left(\\!\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}\\!\right)\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}y_{1}\\\
y_{2}\end{pmatrix}=\begin{pmatrix}x_{1}&x_{2}\end{pmatrix}\begin{pmatrix}-c&-d\\\
a&b\end{pmatrix}\begin{pmatrix}y_{1}\\\ y_{2}\end{pmatrix}.$

Motivated by Lem. 2, we call
$\left\langle{C_{xy}},{C_{x^{\prime}y^{\prime}}}\right\rangle:=-\mathop{\mathrm{tr}}({C_{xy}}{C_{x^{\prime}y^{\prime}}})$
the _pairing_ of two cycles. It shall be noted that the pairing is _not_
positively defined, this follows from the explicit expression (13). The sign
is chosen in such way, that

$\left\langle{C_{xy}},{C_{xy}}\right\rangle=-2\det({C_{xy}})=\textstyle\frac{1}{2}(x-y)^{2}\geq
0.$

Also, an immediate consequence of Lem. 2 or identity (11) is

######  Corollary 4.

The pairing of cycles is invariant under the action (10) of
$\mathrm{SL}_{2}(\mathbb{R}{})$:

$\left\langle g\cdot{C_{xy}}\cdot g^{-1},g\cdot{C_{x^{\prime}y^{\prime}}}\cdot
g^{-1}\right\rangle=\left\langle{C_{xy}},{C_{x^{\prime}y^{\prime}}}\right\rangle.$

From (13), the null set (12) of the form $Q={C_{xy}}$ can be associated to the
family of cycles
$\\{{C_{x^{\prime}y^{\prime}}}\,\mid\,\left\langle{C_{xy}},{C_{x^{\prime}y^{\prime}}}\right\rangle=0,(x^{\prime},y^{\prime})\in\mathbb{R}^{2}{}\times\mathbb{R}^{2}{}\\}$
which we will call _orthogonal_ to ${C_{xy}}$.

##  4 Extending cycles

Since bilinear forms with matrices ${C_{xy}}$ have numerous geometric
connections with $P\mathbb{R}^{1}{}$, we are looking for a similar
interpretation of the generic matrices. The previous discussion identified the
key ingredient of the recipe: $\mathrm{SL}_{2}(\mathbb{R}{})$-invariant
pairing (13) of two forms. Keeping in mind the structure of ${C_{xy}}$, we
will parameterise111Further justification of this parametrisation will be
obtained from the equation of a quadratic curve (18). a generic $2\times 2$
matrix as $\begin{pmatrix}l+n&-m\\\ k&-l+n\end{pmatrix}$ and consider the
corresponding four dimensional vector $(n,l,k,m)$. Then, the similarity with
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R}{})$:

$\begin{pmatrix}{l}^{\prime}+{n}^{\prime}&-{m}^{\prime}\\\
{k}^{\prime}&-{l}^{\prime}+{n}^{\prime}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}l+n&-m\\\
k&-l+n\end{pmatrix}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}^{-1},$ (14)

corresponds to the linear transformation of $\mathbb{R}^{4}{}$, cf. [5, Ex.
4.15]:

$\begin{pmatrix}{n}^{\prime}\\\ {l}^{\prime}\\\ {k}^{\prime}\\\
{m}^{\prime}\end{pmatrix}=\begin{pmatrix}1&0&0&0\\\ 0&cb+ad&bd&ca\\\
0&2cd&d^{2}&c^{2}\\\ 0&2ab&b^{2}&a^{2}\end{pmatrix}\begin{pmatrix}n\\\ l\\\
k\\\ m\end{pmatrix}.$ (15)

In particular, this action commutes with the scaling of the first component:

$\lambda:(n,l,k,m)\mapsto(\lambda n,l,k,m).$ (16)

This expression is helpful in proving the following statement.

######  Lemma 5.

Any $\mathrm{SL}_{2}(\mathbb{R}{})$-invariant (in the sense of the action
(15)) pairing in $\mathbb{R}^{4}{}$ is isomorphic to

$2\tau\tilde{n}n-2\tilde{l}l+\tilde{k}m+\tilde{m}k=\begin{pmatrix}\tilde{n}&\tilde{l}&\tilde{k}&\tilde{m}\end{pmatrix}\begin{pmatrix}2\tau&0&0&0\\\
0&-2&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{pmatrix}\begin{pmatrix}n\\\ l\\\ k\\\
m\end{pmatrix},$

where $\tau=-1$, $0$ or $1$ and $(n,l,k,m)$,
$(\tilde{n},\tilde{l},\tilde{k},\tilde{m})\in\mathbb{R}^{4}{}$ .

###### Proof.

Let $T$ be $4\times 4$ a matrix from (15), if a
$\mathrm{SL}_{2}(\mathbb{R}{})$-invariant pairing is defined by a $4\times 4$
matrix $J=(j_{fg})$, then $T^{\prime}JT=J$, where $T^{\prime}$ is transpose of
$T$. The equivalent identity $T^{\prime}J=JT^{-1}$ produces a system of
homogeneous linear equations which has the generic solution:

$\displaystyle j_{12}$ $\displaystyle=j_{13}=j_{14}=j_{21}=j_{31}=j_{41}=0,$
$\displaystyle j_{22}$ $\displaystyle=\frac{(d-a)j_{42}-2bj_{44}}{c}-2j_{43},$
$\displaystyle j_{23}$ $\displaystyle=-\frac{b}{c}j_{42},$ $\displaystyle
j_{24}$ $\displaystyle=-\frac{cj_{42}+2(a-d)j_{44}}{c},$ $\displaystyle
j_{34}$ $\displaystyle=\frac{c(a-d)j_{42}+(a-d)^{2}j_{44}}{c^{2}}+j_{43},$
$\displaystyle j_{33}$ $\displaystyle=\frac{b^{2}}{c^{2}}j_{44},$
$\displaystyle j_{32}$
$\displaystyle=\frac{b}{c}\frac{cj_{42}+2(a-d)j_{44}}{c},$

with four free variables $j_{11}$, $j_{42}$, $j_{43}$ and $j_{44}$. Since a
solution shall not depend on $a$, $b$, $c$, $d$, we have to put
$j_{42}=j_{44}=0$. Then by the homogeneity of the identity
$T^{\prime}J=JT^{-1}$, we can scale $j_{43}$ to $1$. Thereafter, an
independent (sign-preserving) scaling (16) of $n$ leaves only three non-
isomorphic values $-1$, $0$, $1$ of $j_{11}$. ∎

The appearance of the three essential different cases $\tau=-1$, $0$ or $1$ in
Lem. 5 is a manifestation of the common division of mathematical objects into
elliptic, parabolic and hyperbolic cases [6] [5, Ch. 1]. Thus, we will use
letters “e”, “p”, “h” to encode the corresponding three values of $\tau$.

Now we may describe all $\mathrm{SL}_{2}(\mathbb{R}{})$-invariant pairings of
bilinear forms.

######  Corollary 6.

Any $\mathrm{SL}_{2}(\mathbb{R}{})$-invariant (in the sense of the similarity
(14)) pairing between two bilinear forms $Q=\begin{pmatrix}l+n&-m\\\
k&-l+n\end{pmatrix}$ and
$\tilde{Q}=\begin{pmatrix}\tilde{l}+\tilde{n}&-\tilde{m}\\\
\tilde{k}&-\tilde{l}+\tilde{n}\end{pmatrix}$ is isomorphic to:

$\displaystyle\left\langle Q,\tilde{Q}\right\rangle_{\tau}$
$\displaystyle=-\mathop{\mathrm{tr}}({Q}_{\tau}\tilde{Q})$ (17)
$\displaystyle=2\tau\tilde{n}n-2\tilde{l}l+\tilde{k}m+\tilde{m}k,\quad\text{
where }{Q}_{\tau}=\begin{pmatrix}{l}-\tau{n}&-{m}\\\
{k}&-{l}-\tau{n},\end{pmatrix}$

and $\tau=-1$, $0$ or $1$.

Note that we can explicitly write $Q_{\tau}$ for $Q=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$ as follows:

$Q_{e}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix},\quad
Q_{p}=\begin{pmatrix}\frac{1}{2}(a-d)&b\\\
c&-\frac{1}{2}(a-d)\end{pmatrix},\quad Q_{h}=\begin{pmatrix}-d&b\\\
c&-a\end{pmatrix}.$

In particular, $Q_{h}=-Q^{-1}$ and $Q_{p}=\frac{1}{2}(Q_{e}+Q_{h})$.
Furthermore, $Q_{p}$ has the same structure as ${C_{xy}}$. Now, we are ready
to extend the projective line $P\mathbb{R}^{1}{}$ to two dimensions using the
analogy with properties of cycles ${C_{xy}}$.

######  Definition 7.

  1. 1.

Two bilinear forms $Q$ and $\tilde{Q}$ are $\tau$-_orthogonal_ if
$\left\langle Q,\tilde{Q}\right\rangle_{\tau}=0$.

  2. 2.

A form is $\tau$-_isotropic_ if it is $\tau$-orthogonal to itself.

If a form $Q=\begin{pmatrix}l+n&-m\\\ k&-l+n\end{pmatrix}$ has $k\neq 0$ then
we can scale it to obtain $k=1$, this form of $Q$ is called _normalised_. A
normalised $\tau$-isotropic form is completely determined by its diagonal
values: $\begin{pmatrix}u+v&-u^{2}+\tau v^{2}\\\ 1&-u+v\end{pmatrix}$. Thus,
the set of such forms is in a one-to-one correspondence with points of
$\mathbb{R}^{2}{}$. Finally, a form $Q=\begin{pmatrix}l+n&-m\\\
k&-l+n\end{pmatrix}$ is e-orthogonal to the $\tau$-isotropic form
$\begin{pmatrix}u+v&-u^{2}+\tau v^{2}\\\ 1&-u+v\end{pmatrix}$ if:

$k(u^{2}-\tau v^{2})-2lu-2nv+m=0,$ (18)

that is the point $(u,v)\in\mathbb{R}^{2}{}$ belongs to the quadratic curve
with coefficients $(k,l,n,m)$.

##  5 Homogeneous spaces of cycles

Obviously, the group $\mathrm{SL}_{2}(\mathbb{R}{})$ acts on
$P\mathbb{R}^{1}{}$ transitively, in fact it is even $3$-transitive in the
following sense. We say that a triple $\\{x_{1},x_{2},x_{3}\\}\subset
P\mathbb{R}^{1}{}$ of distinct points is _positively oriented_ if

$\text{ either }\quad x_{1}<x_{2}<x_{3},\quad\text{ or }\quad
x_{3}<x_{1}<x_{2},$

where we agree that the ideal point $\infty\in P\mathbb{R}^{1}{}$ is greater
than any $x\in\mathbb{R}{}$. Equivalently, a triple $\\{x_{1},x_{2},x_{3}\\}$
of reals is positively oriented if:

$(x_{1}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1})>0.$

Also, a triple of distinct points, which is not positively oriented, is
_negatively oriented_. A simple calculation based on the resolvent-type
identity:

$\frac{ax+b}{cx+d}-\frac{ay+b}{cy+d}=\frac{(x-y)(ad-bc)}{(cx+b)(cy+d)}$

shows that both the positive and negative orientations of triples are
$\mathrm{SL}_{2}(\mathbb{R}{})$-invariant. On the other hand, the reflection
$x\mapsto-x$ swaps orientations of triples. Note, that the reflection is a
Moebius transformation associated to the cycle

${C_{0\infty}}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\quad\text{ with
}\det{C_{0\infty}}=-1.$ (19)

A significant amount of information about Möbius transformations follows from
the fact, that any continuous one-parametric subgroup of
$\mathrm{SL}_{2}(\mathbb{R}{})$ is conjugated to one of the three following
subgroups222A reader may know that $A$, $N$ and $K$ are factors in the Iwasawa
decomposition $\mathrm{SL}_{2}(\mathbb{R}{})=ANK$ (cf. Cor. 10, however this
important result does not play any rôle in our consideration.:

$A=\left\\{\begin{pmatrix}e^{-t}&0\\\ 0&e^{t}\end{pmatrix}\right\\},\quad
N=\left\\{\begin{pmatrix}1&t\\\ 0&1\end{pmatrix}\right\\},\quad
K=\left\\{\begin{pmatrix}\cos t&-\sin t\\\ \sin t&\cos
t\end{pmatrix}\right\\},$ (20)

where $t\in\mathbb{R}{}$. Also, it is useful to introduce subgroups
$A\\!^{\prime}$ and $N^{\prime}$ conjugated to $A$ and $N$ respectively:

$A\\!^{\prime}=\left\\{\begin{pmatrix}\cosh t&\sinh t\\\ \sinh t&\cosh
t\end{pmatrix}\,\mid\,t\in\mathbb{R}{}\right\\},\qquad
N^{\prime}=\left\\{\begin{pmatrix}1&0\\\
t&1\end{pmatrix}\,\mid\,t\in\mathbb{R}{}\right\\}.$ (21)

Thereafter, all three one-parameter subgroups $A\\!^{\prime}$, $N^{\prime}$
and $K$ consist of all matrices with the universal structure

$\begin{pmatrix}a&\tau b\\\ b&a\end{pmatrix}\quad\text{where
}\tau=1,\,0,\,-1\text{ for }A\\!^{\prime},\,N^{\prime}\,\text{ and }K\text{
respectively}.$ (22)

We use the notation $H_{\tau}$ for these subgroups. Again, any continuous one-
dimensional subgroup of $\mathrm{SL}_{2}(\mathbb{R}{})$ is conjugated to
$H_{\tau}$ for an appropriate $\tau$.

We note, that matrices from $A$, $N$ and $K$ with $t\neq 0$ have two, one and
none different real eigenvalues respectively. Eigenvectors in
$\mathbb{R}^{2}{}$ correspond to fixed points of Möbius transformations on
$P\mathbb{R}^{1}{}$. Clearly, the number of eigenvectors (and thus fixed
points) is limited by the dimensionality of the space, that is two. For this
reason, if $g_{1}$ and $g_{2}$ take equal values on three different points of
$P\mathbb{R}^{1}{}$, then $g_{1}=g_{2}$.

Also, eigenvectors provide an effective classification tool:
$g\in\mathrm{SL}_{2}(\mathbb{R}{})$ belongs to a one-dimensional continuous
subgroup conjugated to $A$, $N$ or $K$ if and only if the characteristic
polynomial $\det(g-\lambda I)$ has two, one and none different real root(s)
respectively. We will illustrate an application of fixed points techniques
through the following well-known result, which will be used later.

######  Lemma 8.

Let $\\{x_{1},x_{2},x_{3}\\}$ and $\\{y_{1},y_{2},y_{3}\\}$ be positively
oriented triples of points in $\dot{\mathbb{R}}$. Then, there is a unique
(computable!) Möbius map $\phi\in\mathrm{SL}_{2}(\mathbb{R}{})$ with
$\phi(x_{j})=y_{j}$ for $j=1$, $2$, $3$.

###### Proof.

Often, the statement is quickly demonstrated through an explicit expression
for $\phi$, cf. [7, Thm. 13.2.1]. We will use properties of the subgroups $A$,
$N$ and $K$ to describe an algorithm to find such a map. First, we notice that
it is sufficient to show the Lemma for the particular case $y_{1}=0$,
$y_{2}=1$, $y_{3}=\infty$. The general case can be obtained from composition
of two such maps. Another useful observation is that the fixed point for $N$,
that is $\infty$, is also a fixed point of $A$.

Now, we will use subgroups $K$, $N$ and $A$ in order of increasing number of
their fixed points. First, for any $x_{3}$ the matrix
$g^{\prime}=\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos t\end{pmatrix}\in K$
such that $\cot t=x_{3}$ maps $x_{3}$ to $y_{3}=\infty$. Let
$x_{1}^{\prime}=g^{\prime}x_{1}$ and $x_{2}^{\prime}=g^{\prime}x_{2}$. Then
the matrix $g^{\prime\prime}=\begin{pmatrix}1&-x_{1}^{\prime}\\\
0&1\end{pmatrix}\in N$, fixes $\infty=g^{\prime}x_{3}$ and sends
$x^{\prime}_{1}$ to $y_{1}=0$. Let
$x^{\prime\prime}_{2}=g^{\prime\prime}x_{2}^{\prime}$, from positive
orientation of triples we have $0<x^{\prime\prime}_{2}<\infty$. Next, the
matrix $g^{\prime\prime\prime}=\begin{pmatrix}a^{-1}&0\\\ 0&a\end{pmatrix}\in
A$ with $a=\sqrt{x^{\prime\prime}_{2}}$ sends $x^{\prime\prime}_{2}$ to $1$
and fixes both $\infty=g^{\prime\prime}g^{\prime}x_{3}$ and
$0=g^{\prime\prime}g^{\prime}x_{1}$. Thus,
$g=g^{\prime\prime\prime}g^{\prime\prime}g^{\prime}$ makes the required
transformation $(x_{1},x_{2},x_{3})\mapsto(0,1,\infty)$. ∎

######  Corollary 9.

Let $\\{x_{1},x_{2},x_{3}\\}$ and $\\{y_{1},y_{2},y_{3}\\}$ be two triples
with the opposite orientations. Then, there is a unique Möbius map
$\phi\in\mathrm{SL}_{2}(\mathbb{R}{})$ with
$\phi\circ{C_{0\infty}}(x_{j})=y_{j}$ for $j=1$, $2$, $3$.

We will denote by $\phi_{XY}$ the unique map from Lem. 8 defined by triples
$X=\\{x_{1},x_{2},x_{3}\\}$ and $Y=\\{y_{1},y_{2},y_{3}\\}$.

Although we are not going to use it in this paper, we note that the following
important result [8, § III.1] is an immediate consequence of our _proof_ of
Lem. 8.

######  Corollary 10 (Iwasawa decomposition).

Any element of $g\in\mathrm{SL}_{2}(\mathbb{R}{})$ is a product
$g=g_{A}g_{N}g_{K}$, where $g_{A}$, $g_{N}$ and $g_{K}$ belong to subgroups
$A$, $N$, $K$ respectively and those factors are uniquely defined.

In particular, we note that it is not a coincidence that the subgroups appear
in the Iwasawa decomposition $\mathrm{SL}_{2}(\mathbb{R}{})=ANK$ in order of
decreasing number of their fixed points.

##  6 Triples of intervals

We change our point of view and instead of two ordered triples of points
consider three ordered pairs, that is—three intervals. For them we will need
the following definition.

######  Definition 11.

We say that a triple of intervals
$\\{[x_{1},y_{1}],[x_{2},y_{2}],[x_{3},y_{3}]\\}$ is _aligned_ if the triples
$X=\\{x_{1},x_{2},x_{3}\\}$ and $Y=\\{y_{1},y_{2},y_{3}\\}$ of their endpoints
have the same orientation.

Aligned triples determine certain one-parameter subgroups of Möbius
transformations as follows:

######  Proposition 12.

Let $\\{[x_{1},y_{1}],[x_{2},y_{2}],[x_{3},y_{3}]\\}$ be an aligned triple of
intervals.

  1. 1.

If $\phi_{XY}$ has at most one fixed point, then there is a unique (up to a
parametrisation) one-parameter semigroup of Möbius map
$\psi(t)\subset\mathrm{SL}_{2}(\mathbb{R}{})$, which maps $[x_{1},y_{1}]$ to
$[x_{2},y_{2}]$ and $[x_{3},y_{3}]$:

$\psi(t_{j})(x_{1})=x_{j},\quad\psi(t_{j})(y_{1})=y_{j},\qquad\text{ for some
}t_{j}\in\mathbb{R}{}\text{ and }j=2,3.$

  2. 2.

Let $\phi_{XY}$ have two fixed points $x<y$ and ${C_{xy}}$ be the orientation
inverting Möbius transformation with the matrix (8). For $j=1$, $2$, $3$, we
define:

$\displaystyle x^{\prime}_{j}$ $\displaystyle=x_{j},$ $\displaystyle
y^{\prime}_{j}$ $\displaystyle=y_{j},$ $\displaystyle x^{\prime\prime}_{j}$
$\displaystyle={C_{xy}}x_{j},$ $\displaystyle y^{\prime\prime}_{j}$
$\displaystyle={C_{xy}}y_{j}$ $\displaystyle\text{ if }x<x_{j}<y;$
$\displaystyle x^{\prime}_{j}$ $\displaystyle={C_{xy}}x_{j},$ $\displaystyle
y^{\prime}_{j}$ $\displaystyle={C_{xy}}y_{j},$ $\displaystyle
x^{\prime\prime}_{j}$ $\displaystyle=x_{j},$ $\displaystyle
y^{\prime\prime}_{j}$ $\displaystyle=y_{j},$ $\displaystyle\text{ otherwise}.$

Then, there is a one-parameter semigroup of Möbius map
$\psi(t)\subset\mathrm{SL}_{2}(\mathbb{R}{})$, and $t_{2}$,
$t_{3}\in\mathbb{R}{}$ such that:

$\psi(t_{j})(x^{\prime}_{1})=x^{\prime}_{j},\quad\psi(t_{j})(x^{\prime\prime}_{1})=x^{\prime\prime}_{j},\quad\psi(t_{j})(y^{\prime}_{1})=y^{\prime}_{j},\quad\psi(t_{j})(y^{\prime\prime}_{1})=y^{\prime\prime}_{j},$

where $j=2,3$.

###### Proof.

Consider the one-parameter subgroup of
$\psi(t)\subset\mathrm{SL}_{2}(\mathbb{R}{})$ such that $\phi_{XY}=\psi(1)$.
Note, that $\psi(t)$ and $\phi_{XY}$ have the same fixed points (if any) and
no point $x_{j}$ is fixed since $x_{j}\neq y_{j}$. If the number of fixed
points is less than $2$, then $\psi(t)x_{1}$, $t\in\mathbb{R}{}$ produces the
entire real line except a possible single fixed point. Therefore, there are
$t_{2}$ and $t_{3}$ such that $\psi(t_{2})x_{1}=x_{2}$ and
$\psi(t_{3})x_{1}=x_{3}$. Since $\psi(t)$ and $\phi_{XY}$ commute for all $t$
we also have:

$\psi(t_{j})y_{1}=\psi(t_{j})\phi_{XY}x_{1}=\phi_{XY}\psi(t_{j})x_{1}=\phi_{XY}x_{j}=y_{j},\quad\text{
for }j=2,3.$

If there are two fixed points $x<y$, then the open interval $(x,y)$ is an
orbit for the subgroup $\psi(t)$. Since all $x^{\prime}_{1}$, $x^{\prime}_{2}$
and $x^{\prime}_{3}$ belong to this orbit and ${C_{xy}}$ commutes with
$\phi_{XY}$ we may repeat the above reasoning for the dashed intervals
$[x^{\prime}_{j},y^{\prime}_{j}]$. Finally,
$x^{\prime\prime}_{j}={C_{xy}}x^{\prime}_{j}$ and
$y^{\prime\prime}_{j}={C_{xy}}y^{\prime}_{j}$, where ${C_{xy}}$ commutes with
$\phi$ and $\psi(t_{j})$, $j=2$, $3$. Uniqueness of the subgroup follows from
Lemma 13. ∎

The group $\mathrm{SL}_{2}(\mathbb{R}{})$ acts transitively on collection of
all cycles ${C_{xy}}$, thus this is a
$\mathrm{SL}_{2}(\mathbb{R}{})$-homogeneous space. It is easy to see that the
fix-group of the cycle ${C_{-1,1}}$ is $A\\!^{\prime}$ (21). Thus the
homogeneous space of cycles is isomorphic to
$\mathrm{SL}_{2}(\mathbb{R}{})/A\\!^{\prime}$.

######  Lemma 13.

Let $H$ be a one-parameter continuous subgroup of
$\mathrm{SL}_{2}(\mathbb{R}{})$ and $X=\mathrm{SL}_{2}(\mathbb{R}{})/H$ be the
corresponding homogeneous space. If two orbits of one-parameter continuous
subgroups on $X$ have at least three common points then these orbits coincide.

###### Proof.

Since $H$ is conjugated either to $A\\!^{\prime}$, $N^{\prime}$ or $K$, the
homogeneous space $X=\mathrm{SL}_{2}(\mathbb{R}{})/H$ is isomorphic to the
upper half-plane in double, dual or complex numbers [5, § 3.3.4]. Orbits of
one-parameter continuous subgroups in $X$ are conic sections, which are
circles, parabolas (with vertical axis) or equilateral hyperbolas (with
vertical axis) for the respective type of geometry. Any two different orbits
of the same type intersect at most at two points, since an analytic solution
reduces to a quadratic equation. ∎

Alternatively, we can reformulate Prop. 12 as follows: three different cycles
${C_{x_{1}y_{1}}}$, ${C_{x_{2}y_{2}}}$, ${C_{x_{3}y_{3}}}$ define a one-
parameter subgroup, which generate either one orbit or two related orbits
passing the three cycles.

We have seen that the number of fixed points is the key characteristics for
the map $\phi_{XY}$. The next result gives an explicit expression for it.

######  Proposition 14.

The map $\phi_{XY}$ has zero, one or two fixed points if the expression

$\det\begin{pmatrix}1&x_{1}y_{1}&y_{1}-x_{1}\\\ 1&x_{2}y_{2}&y_{2}-x_{2}\\\
1&x_{3}y_{3}&y_{3}-x_{3}\end{pmatrix}^{2}-4\det\begin{pmatrix}x_{1}&1&y_{1}\\\
x_{2}&1&y_{2}\\\
x_{3}&1&y_{3}\end{pmatrix}\cdot\det\begin{pmatrix}x_{1}&-x_{1}y_{1}&y_{1}\\\
x_{2}&-x_{2}y_{2}&y_{2}\\\ x_{3}&-x_{3}y_{3}&y_{3}\end{pmatrix}$ (23)

is negative, zero or positive respectively.

###### Proof.

If a Möbius transformation $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ maps
$x_{1}\mapsto y_{1}$, $x_{2}\mapsto y_{2}$, $x_{3}\mapsto y_{3}$ and $s\mapsto
s$, then we have a homogeneous linear system, cf. [7, Ex. 13.2.4]:

$\begin{pmatrix}x_{1}&1&-x_{1}y_{1}&-y_{1}\\\ x_{2}&1&-x_{2}y_{2}&-y_{2}\\\
x_{3}&1&-x_{3}y_{3}&-y_{3}\\\ s&1&-s^{2}&-s\end{pmatrix}\begin{pmatrix}a\\\
b\\\ c\\\ d\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\end{pmatrix}.$ (24)

A non-zero solution exists if the determinant of the $4\times 4$ matrix is
zero. Expanding it over the last row and rearranging terms we obtain the
quadratic equation for the fixed point $s$:

$s^{2}\det\begin{pmatrix}x_{1}&1&y_{1}\\\ x_{2}&1&y_{2}\\\
x_{3}&1&y_{3}\end{pmatrix}+s\det\begin{pmatrix}1&x_{1}y_{1}&y_{1}-x_{1}\\\
1&x_{2}y_{2}&y_{2}-x_{2}\\\
1&x_{3}y_{3}&y_{3}-x_{3}\end{pmatrix}+\det\begin{pmatrix}x_{1}&-x_{1}y_{1}&y_{1}\\\
x_{2}&-x_{2}y_{2}&y_{2}\\\ x_{3}&-x_{3}y_{3}&y_{3}\end{pmatrix}=0.$

The value (23) is the discriminant of this equation. ∎

######  Remark 15.

It is interesting to note, that the relation $ax+b-cxy-dy=0$ used in (24) can
be stated as e-orthogonality of the cycle $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$ and the isotropic bilinear form $\begin{pmatrix}x&-xy\\\
1&-y\end{pmatrix}$.

If $y=g_{0}\cdot x$ for some $g_{0}\in H_{\tau}$, then for any $g\in H_{\tau}$
we also have $y_{g}=g_{0}\cdot x_{g}$, where $x_{g}=g\cdot x_{g}$ and
$y_{g}=g\cdot y_{g}$. Thus, we demonstrated the first part of the following
result.

######  Lemma 16.

Let $\tau=1$, $0$ or $-1$ and a real constant $t\neq 0$ be such that $1-\tau
t^{2}>0$.

  1. 1.

The collections of intervals:

$I_{\tau,t}=\left\\{[x,\textstyle\frac{x+\tau
t}{tx+1}]\,\mid\,x\in\mathbb{R}{}\right\\}$ (25)

is preserved by the actions of subgroup $H_{\tau}$. Any three different
intervals from $I_{\tau,t}$ define the subgroup $H_{\tau}$ in the sense of
Prop. 12.

  2. 2.

All $H_{\tau}$-invariant bilinear forms compose the family
$P_{\tau,t}=\left\\{\begin{pmatrix}a&\tau b\\\ b&a\end{pmatrix}\right\\}$.

The family $P_{\tau,t}$ consists of the eigenvectors of the $4\times 4$ matrix
from (15) with suitably substituted entries. There is (up to a factor) exactly
one $\tau$-isotropic form in $P_{\tau,t}$, namely $\begin{pmatrix}1&\tau\\\
1&1\end{pmatrix}$. We denote this form by $\iota$. It corresponds to the point
$(0,1)\in\mathbb{R}^{2}{}$ as discussed after Defn. 7. We may say that the
subgroup $H_{\tau}$ fixes the point $\iota$, this will play an important rôle
below.

##  7 Geometrisation of cycles

We return to the geometric version of the Poincaré extension considered in
Sec. 2 in terms of cycles. Cycles of the form $\begin{pmatrix}x&-x^{2}\\\
1&-x\end{pmatrix}$ are $\tau$-isotropic for any $\tau$ and are parametrised by
the point $x$ of the real line. For a fixed $\tau$, the collection of all
$\tau$-isotropic cycles is a larger set containing the image of the real line
from the previous sentence. Geometrisation of this embedding is described in
the following result.

######  Lemma 17.

  1. 1.

The transformation $x\mapsto\frac{x+\tau t}{tx+1}$ from the subgroup
$H_{\tau}$, which maps $x\mapsto y$, corresponds to the value
$t=\frac{x-y}{xy-\tau}$.

  2. 2.

The unique (up to a factor) bilinear form $Q$ orthogonal to ${C_{xx}}$,
${C_{yy}}$ and $\iota$ is

$Q=\begin{pmatrix}\frac{1}{2}(x+y+xy-\tau)&-xy\\\
1&\frac{1}{2}(-x-y+xy-\tau)\end{pmatrix}.$

  3. 3.

The defined above $t$ and $Q$ are connected by the identity:

$\frac{\left\langle
Q,\mathbb{R}{}\right\rangle_{\tau}}{\sqrt{\left|\left\langle
Q,Q\right\rangle_{\tau}\right|}}=\frac{\tau}{\sqrt{\left|t^{2}-\tau\right|}}.$
(26)

Here, the real line is represented by the bilinear form
$\mathbb{R}{}=\begin{pmatrix}2^{-1/2}&0\\\ 0&2^{-1/2}\end{pmatrix}$ normalised
such that $\left\langle\mathbb{R}{},\mathbb{R}{}\right\rangle_{\tau}=\pm 1$.

  4. 4.

For a cycle $Q=\begin{pmatrix}l+n&-m\\\ k&-l+n\end{pmatrix}$, the value
$\frac{\left\langle Q,\mathbb{R}{}\right\rangle_{e}}{\sqrt{\left|\left\langle
Q,Q\right\rangle_{e}\right|}}=-\frac{n}{\sqrt{\left|l^{2}+n^{2}-km\right|}}$
is equal to the cosine of the angle between the curve $k(u^{2}+\tau
v^{2})-2lu-2nv+m=0$ (18) and the real line, cf. [5, Ex. 5.23].

###### Proof.

The first statement is verified by a short calculation. A form
$Q=\begin{pmatrix}l+n&-m\\\ k&-l+n\end{pmatrix}$ in the second statement may
be calculated from the homogeneous system:

$\begin{pmatrix}0&-2x&x^{2}&1\\\ 0&-2y&y^{2}&1\\\
-2&0&-\tau&1\end{pmatrix}\begin{pmatrix}n\\\ l\\\ k\\\
m\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\ 0\end{pmatrix},$

which has the rank $3$ if $x\neq y$. The third statement can be checked by a
calculation as well. Finally, the last item is a particular case of the more
general statement as indicated. Yet, we can derive it here from the implicit
derivative $\frac{dv}{du}=\frac{ku-l}{n}$ of the function $k(u^{2}+\tau
v^{2})-2lu-2nv+m=0$ (18) at the point $(u,0)$. Note that this value is
independent from $\tau$. Since this is the tangent of the intersection angle
with the real line, the square of the cosine of this angle is:

$\frac{1}{1+(\frac{dv}{du})^{2}}=\frac{n^{2}}{l^{2}+k^{2}u^{2}+n^{2}-2kul}=\frac{n^{2}}{l^{2}+n^{2}-km}=\frac{\left\langle
Q,\mathbb{R}{}\right\rangle^{2}}{\left\langle Q,Q\right\rangle_{e}},$

if $ku^{2}-2ul+m=0$. ∎

Also, we note that, the independence of the left-hand side of (26) from $x$
can be shown from basic principles. Indeed, for a fixed $t$ the subgroup
$H_{\tau}$ acts transitively on the family of triples $\\{x,\frac{x+\tau
t}{tx+1},\iota\\}$, thus $H_{\tau}$ acts transitively on all bilinear forms
orthogonal to such triples. However, the left-hand side of (26) is
$\mathrm{SL}_{2}(\mathbb{R}{})$-invariant, thus may not depend on $x$. This
simple reasoning cannot provide the exact expression in the right-hand side of
(26), which is essential for the geometric interpretation of the Poincaré
extension.

To restore a cycle from its intersection points with the real line we need
also to know its cycle product with the real line. If this product is non-zero
then the sign of the parameter $n$ is additionally required. At the cycles’
language, a common point of cycles ${C}$ and ${\tilde{C}}$ is encoded by a
cycle $\hat{C}$ such that:

$\left\langle{\hat{C}},{C}\right\rangle_{e}=\left\langle{\hat{C}},{\tilde{C}}\right\rangle_{e}=\left\langle{\hat{C}},{\hat{C}}\right\rangle_{\tau}=0.$
(27)

For a given value of $\tau$, this produces two linear and one quadratic
equation for parameters of $\hat{C}$. Thus, a pair of cycles may not have a
common point or have up to two such points. Furthermore, Möbius-invariance of
the above conditions (26) and (27) supports the geometrical construction of
Poincaré extension, cf. Lem. 1:

######  Lemma 18.

Let a family consist of cycles, which are $e$-orthogonal to a given
$\tau$-isotropic cycle ${\hat{C}}$ and have the fixed value of the fraction in
the left-hand side of (26). Then, for a given Möbius transformation $g$ and
any cycle ${C}$ from the family, $g{C}$ is $e$-orthogonal to the
$\tau$-isotropic cycle $g{\hat{C}}$ and has the same fixed value of the
fraction in the left-hand side of (26) as ${C}$.

Summarising the geometrical construction, the Poincaré extension based on two
intervals and the additional data produces two situations:

  1. 1.

If the cycles $C$ and $\tilde{C}$ are orthogonal to the real line, then a pair
of overlapping cycles produces a point of the elliptic upper half-plane, a
pair of disjoint cycles defines a point of the hyperbolic. However, there is
no orthogonal cycles uniquely defining a parabolic extension.

  2. 2.

If we admit cycles, which are not orthogonal to the real line, then the same
pair of cycles may define any of the three different types (EPH) of extension.

These peculiarities make the extension based on three intervals, described
above, a bit more preferable.

##  8 Concluding remarks

Based on the consideration in Sections 3– 7 we describe the following steps to
carry out the generalised extension procedure:

  1. 1.

Points of the extended space are equivalence classes of aligned triples of
cycles in $P\mathbb{R}^{1}{}$, see Defn. 11. The equivalence relation between
triples will emerge at step 3.

  2. 2.

A triple $T$ of different cycles defines the unique one-parameter continues
subgroup $S(t)$ of Möbius transformations as defined in Prop. 12.

  3. 3.

Two triples of cycles are equivalent if and only if the subgroups defined in
step 2 coincide (up to a parametrisation).

  4. 4.

The geometry of the extended space, defined by the equivalence class of a
triple $T$, is elliptic, parabolic or hyperbolic depending on the subgroup
$S(t)$ being similar $S(t)=gH_{\tau}(t)g^{-1}$,
$g\in\mathrm{SL}_{2}(\mathbb{R}{})$ (up to parametrisation) to $H_{\tau}$ (22)
with $\tau=-1$, $0$ or $1$ respectively. The value of $\tau$ may be identified
from the triple using Prop. 14.

  5. 5.

For the above $\tau$ and $g\in\mathrm{SL}_{2}(\mathbb{R}{})$, the point of the
extended space, defined by the the equivalence class of a triple $T$, is
represented by $\tau$-isotropic (see Defn. 7(2)) bilinear form
$g^{-1}\begin{pmatrix}1&\tau\\\ 1&1\end{pmatrix}g$, which is $S$-invariant,
see the end of Section 6.

Obviously, the above procedure is more complicated that the geometric
construction from Section 2. There are reasons for this, as discussed in
Section 7: our procedure is uniform and we are avoiding consideration of
numerous subcases created by an incompatible selection of parameters.
Furthermore, our presentation is aimed for generalisations to Möbius
transformations of moduli over other rings. This can be considered as an
analog of Cayley–Klein geometries [9, Apps. A–B] [10].

It shall be rather straightforward to adopt the extension for
$\mathbb{R}^{n}{}$. Möbius transformations in $\mathbb{R}^{n}{}$ are naturally
expressed as linear-fractional transformations in Clifford algebras [11] with
a similar classification of subgroups based on fixed points [12, 13]. The
Möbius invariant matrix presentation of cycles $\mathbb{R}^{n}{}$ is already
known [11, (4.12)] [14] [15, § 5]. Of course, it is necessary to enlarge the
number of defining cycles from $3$ to, say, $n+2$. This shall have a
connection with Cauchy–Kovalevskaya extension considered in Clifford analysis
[16, 17]. Naturally, a consideration of other moduli and rings may require
some more serious adjustments in our scheme.

Our construction is based on the matrix presentations of cycles. This
techniques is effective in many different cases [5, 15]. Thus, it is not
surprising that such ideas (with some technical variation) appeared
independently on many occasions [11, (4.12)] [14] [18, § 1.1] [19, § 4.2]. The
interesting feature of the present work is the complete absence of any
(hyper)complex numbers. It deemed to be unavoidable [5, § 3.3.4] to employ
complex, dual and double numbers to represent three different types of Möbius
transformations extended from the real line to a plane. Also (hyper)complex
numbers were essential in [5, 6] to define three possible types of cycle
product (17), and now we managed without them.

Apart from having real entries, our matrices for cycles share the structure of
matrices from [11, (4.12)] [14] [5] [6]. To obtain another variant, one
replaces the map $\mathbold{i}$ (7) by

$\mathbold{t}:\ \begin{pmatrix}x_{1}&y_{1}\\\
x_{2}&y_{2}\end{pmatrix}\mapsto\begin{pmatrix}y_{1}&y_{2}\\\
x_{1}&x_{2}\end{pmatrix}\,.$

Then, we may define symmetric matrices in a manner similar to (8):

${C^{t}_{xy}}=\frac{1}{2}M_{xy}\cdot\mathbold{t}(M_{xy})=\begin{pmatrix}xy&\frac{x+y}{2}\\\
\frac{x+y}{2}&1\end{pmatrix}\,.$

This is the form of matrices for cycles similar to [18, § 1.1] [19, § 4.2].
The property (10) with matrix similarity shall be replaced by the respective
one with matrix congruence: $g\cdot{C^{t}_{xy}}\cdot
g^{t}={C^{t}_{x^{\prime}y^{\prime}}}$. The rest of our construction may be
adjusted for these matrices accordingly.

## Acknowledgments

I am grateful to anonymous references for many useful suggestions.

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