Manfred Evers]Bendenkamp 21, 40880 Ratingen, Germany

# Geometry on Real Projective Cayley-Klein Spaces

Manfred Evers [<EMAIL_ADDRESS>
###### Abstract.

We investigate several topics of the geometry on real Cayley-Klein spaces. An
important concern for us is to define a distance function on the projective
space in such a way that the distance between two anisotropic subspaces of the
same dimension can be easily calculated and case distinctions are avoided as
far as possible.

## Introduction

We introduce a real semi Cayley-Klein space as a pair
$(\textrm{P}\boldsymbol{V},\beta)$ consisting of a projective space
$\textrm{P}\boldsymbol{V}$ over a real vector space $\boldsymbol{V}$ and a
symmetric bilinear form
$\beta{:\,}\boldsymbol{V}\times\boldsymbol{V}\to\mathbb{R}$ different from the
zero function. A point $P{\,\in\,}\textrm{P}\boldsymbol{V}$ is an isotropic
point of $(\textrm{P}\boldsymbol{V},\beta)$ if for any nonzero vector
$\boldsymbol{p}\in\boldsymbol{V}$ representing $P$ ( we write
$P=\mathbb{R}\boldsymbol{p}\,$) the equation
$\beta(\boldsymbol{p},\boldsymbol{p})=0$ applies, otherwise $P$ is
anisotropic. Let $\mathcal{Q}$ be the set of isotropic points of
$(\textrm{P}\boldsymbol{V},\beta)$.

A reflection $\phi$ in an anisotropic point $P=\mathbb{R}\boldsymbol{p}$ of
$(\textrm{P}\boldsymbol{V},\beta)$ is an automorphism of
$\textrm{P}\boldsymbol{V}$ which is different from the identical mapping and
maps any point $Q=\mathbb{R}\boldsymbol{q}$ of $\textrm{P}\boldsymbol{V}$ to a
point
$Q^{\prime}=\mathbb{R}\big{(}\beta(\boldsymbol{p},\boldsymbol{p})\boldsymbol{q}-2\beta(\boldsymbol{p},\boldsymbol{q})\boldsymbol{p}\big{)}$.
These point reflections are idempotent mappings. Points that stay fixed under
all point reflections are called singular points in
$(\textrm{P}\boldsymbol{V},\beta)$. All singular points are isotropic, but not
all isotropic points have to be singular. The singular points form the radical
of $\beta$, denoted by rad$(\beta)$. This is a projective subspace of
$\textrm{P}\boldsymbol{V}$, which might be empty, but on the other hand it can
also be a hyperplane in $\textrm{P}\boldsymbol{V}$.

If $P=\mathbb{R}\boldsymbol{p}$ and $Q=\mathbb{R}\boldsymbol{q}$ are two
anisotropic points in $(\textrm{P}\boldsymbol{V},\beta)$, then $\xi(P,Q):=$
$1{-}\beta(p,q)/(\beta(p,p)\beta(q,q))$ is a real number in the interval
$[0,1]$, which can be interpreted as a distance between the points $P$ and $Q$
and is called quadrance of the points $P$ and $Q$.111) We adopt the definition
of the function and the name quadrance from N. Wildberger [30, 31]. ) If
$\phi$ is an arbitrary reflection in an anisotropic point, then
$\xi(P,Q)=\xi(\phi(P),\phi(Q))$ for all anisotropic points $P$ and $Q$.

The quadrance function
$\xi{:\,}(\textrm{P}\boldsymbol{V}{\smallsetminus}{\smallsetminus}\,\mathcal{Q})\times(\textrm{P}\boldsymbol{V}{\smallsetminus}{\smallsetminus}\,\mathcal{Q})\to\mathbb{R}$
can strongly degenerate. For example, if the set rad$(\beta)$ of singular
points is a hyperplane in $\textrm{P}\boldsymbol{V}$, then $\xi$ is the zero
function. Of course, in this case the distance function is too coarse to allow
meaningful distance measurement. But if we define on rad$(\beta)$ another
bilinear form $\beta^{\prime}$, we can significantly improve the situation: We
can now introduce reflections in points which are anisotropic in the semi
Cayley-Klein space $(\textrm{rad}(\beta),\beta^{\prime})$. But more
importantly, by the help of $\beta^{\prime}$ the function $\xi$ can be
modified to a function
$\xi^{\prime}{:\,}(\textrm{P}\boldsymbol{V}{\smallsetminus}{\smallsetminus}\,\mathcal{Q})\times(\textrm{P}\boldsymbol{V}{\smallsetminus}{\smallsetminus}\,\mathcal{Q})\to\mathbb{R}$
such that $\xi^{\prime}(P,Q)\neq 0$ for almost all distinct anisotropic points
$P,Q$ and $\xi^{\prime}(P,Q)=\xi(P,Q)$ for all anisotropic points
$P,Q\in\textrm{P}\boldsymbol{V}$ with $\xi(P,Q)\neq 0$. And if $\phi$ is a
reflection in an anisotropic point of $(\textrm{P}\boldsymbol{V},\beta)$, then
$\xi^{\prime}(P,Q)=\xi^{\prime}(\phi(P),\phi(Q))$ for all anisotropic points
$P$ and $Q$.

We now come to the notion of a real Cayley-Klein space. A Cayley-Klein space
with underlying projective space $\textrm{P}\boldsymbol{V}$ is a nested
sequence of finitely many, say $\rho+1$, semi Cayley-Klein spaces
$(A_{i},{{\beta_{i}}})$, such that $A_{0}=\textrm{P}\boldsymbol{V}$, $A_{i+1}$
is a nonempty radical rad(${\beta_{i}}$), $0\leq i\leq\rho-1$, and
$A_{\rho+1}{:=\textrm{rad}(\beta_{\rho})}=\emptyset.$

The first section gives a short introduction to the subject. We assume that
the reader is familiar with projective geometry, but in order to introduce the
terminology and fix notations, we give some basic definitions, rules and
theorems. As a main basis serves the book Vorlesungen über höhere Geometrie
[10] by O. Giering, which was published in 1982 and deals with the topic of
Cayley-Klein spaces in detail. There are a number of more recent works on this
subject; we cite [15, 17, 21, 22, 27, 28]. An outline of the historical
development is given in [1].

In the second section, we study the geometry on semi Cayley-Klein spaces. This
includes the determination of distances between projective subspaces. Another
focus is on the investigation of geometric figures such as circles and
simplices.

In the third section, the geometry on Cayley-Klein spaces is examined. In
contrast to the calculations in the second section, distance measurement
exhibits a fine structure.

In Section 4 it is shown how geometric algebra (GA) can be used to calculate
reflections and their compositions.

##  1\. Fundamentals / terminology and notation

### 1.1.

Real and complex projective spaces.  Let $\mathbb{F}$ be the field of real or
complex numbers and $\boldsymbol{V}$ be a vector space of finite dimension
$n{+}1$ over $\mathbb{F}$. We introduce the projective space of
$\boldsymbol{V}$ by
$\textrm{P}\boldsymbol{V}=\\{\,\mathbb{F}\boldsymbol{v}\;|$
$\;\boldsymbol{v}\in\boldsymbol{V}{\smallsetminus}{\smallsetminus}\,\\{\boldsymbol{0}\\}\\}$.
The dimension $n$ of this projective space is defined by
$n=\textrm{dim}(\boldsymbol{V})-1$. A subset ${U}$ of
$\textrm{P}\boldsymbol{V}$ is called a subspace if there exists a linear
subspace $\boldsymbol{U}$ of $\boldsymbol{V}$ with
${U}=\textrm{P}\boldsymbol{U}$; we write ${U}\leq\textrm{P}\boldsymbol{V}$. In
particular, the empty set is a projective subspace of
$\textrm{P}\boldsymbol{V}$. It is quite common to notate this subspace by $0$;
the dimension of $0$ is $\textrm{-}1$. If
${U}_{1}=\textrm{P}\boldsymbol{U}_{1}$ and
${U}_{2}=\textrm{P}\boldsymbol{U}_{2}$ are subspaces of
$\textrm{P}\boldsymbol{V}$, their join is the projective subspace
${\textrm{P}(\boldsymbol{U}_{1}+\boldsymbol{U}_{2})}$; we denote it by
${U}_{1}\sqcup{U}_{2}$. The intersection of two subspaces ${U}_{1},{U}_{2}$ of
$\textrm{P}\boldsymbol{V}$ is always a subspace, which is called the meet of
${U}_{1}$ and ${U}_{2}$ and which we denote by ${U}_{1}\sqcap{U}_{2}$222) The
symbols $\sqcap$ and $\sqcup$ are preferred to $\wedge$ and $\vee$,
since the latter are often used for exterior products (in Grassmann algebras,
see section 4). ). The elements of $\textrm{P}\boldsymbol{V}$ are called
points. Let $\boldsymbol{B}:=(\boldsymbol{b}_{1},\cdots,\boldsymbol{b}_{n+1})$
be an ordered basis of $\boldsymbol{V}$. Given a vector
$\boldsymbol{v}\in\boldsymbol{V}$ with coordinates
$({v}_{1}{,\dots\,}{v}_{n+1})$ with respect to this basis, we denote the
vector $\boldsymbol{v}$ by $[v_{1},\dots,v_{n+1}]_{\boldsymbol{B}}$ and the
point $\mathbb{F}\boldsymbol{v}$ by
${[v_{1}:\dots:v_{n+1}]_{\boldsymbol{B}}}$.
If a subspace contains just one point $P$, then we follow a convention and
denote this subspace by $P$ instead of $\\{P\\}$. A $k$-dimensional subspace
of $\textrm{P}\boldsymbol{V}$ is also called a $k$-plane (in $\textrm{P}V$).
There are separate names for two special cases: A hyperplane of
$\textrm{P}\boldsymbol{V}$ is a subspace (a plane) of dimension $n{\,-\,}1$,
and if $P$ and $Q$ are different points in $\textrm{P}\boldsymbol{V}$, the
projective one-dimensional subspace $P\sqcup Q$ is called a line passing
through $P$ and $Q$. If ${U}$ is a subspace of $\textrm{P}\boldsymbol{V}$,
then a subspace ${U^{\prime}}\leq\textrm{P}\boldsymbol{V}$ with
${U}\sqcup{U}^{\prime}=\textrm{P}\boldsymbol{V}$ and
${U}\sqcap{U}^{\prime}=\emptyset$ is called a projective complement of ${U}$.
Given a subset ${S}$ of $\textrm{P}\boldsymbol{V}\\!$, the set
$\textrm{span}({S}):=\bigcap\\{{U}\,|\,{U}\;\textrm{is a subspace
of}\;\textrm{P}\boldsymbol{V}\textrm{and}\;{{S}\subseteq{U}\,}\\}$ is a
projective subspace of $\textrm{P}\boldsymbol{V}$, called the span of ${S}$.
Put $\text{dim}(S):=\text{dim}(\text{span}(S))$. If
$U\leq\textrm{P}\boldsymbol{V}$ is a $k$-plane, a set of $k{+}1$ points
spanning $U$ is called a minimal generating set of $U$. We say that the points
of a set ${S}\subseteq\textrm{P}\boldsymbol{V}\\!$ are in a general position
if $\text{dim}(\tilde{\mathcal{S}})=k-1$ for each subset $\tilde{{S}}$ of
$\mathcal{S}$ with $\\#\tilde{{S}}=k\leq n{+}1$. An $(n{+}2)$-tuple
$(P_{0},\dots,P_{n+1})$ of points is called a projective frame of
$\textrm{P}\boldsymbol{V}$ if these points are in general position. We give an
example: Suppose $\boldsymbol{V}=\mathbb{R}^{n+1}$ with canonical basis
$\boldsymbol{B}=(\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{n+1})$. Put
$E_{i}:=\mathbb{R}\boldsymbol{e}_{i},1\leq i\leq n{+}1$, and
$E_{0}:=\mathbb{R}(\boldsymbol{e}_{1}+\cdots+\boldsymbol{e}_{n+1}).$ Then
$(E_{0},E_{1},\dots,E_{n+1})$ is a projective frame. It is called the
canonical frame of $\textrm{P}\mathbb{R}^{n+1}$. The point
$E_{0}=[1:1:\dots:1]_{\boldsymbol{B}}$ is called unit-point of the frame. Each
$(n{+}1)$-elementary subset of $\\{E_{0},E_{1},\dots,E_{n+1}\\}$ is a minimal
generating set of $\textrm{P}\boldsymbol{V}$.

### 1.2.

The cross ratio.  Given four vectors
$\boldsymbol{a},\boldsymbol{b},\boldsymbol{c},\boldsymbol{d}\in\mathbb{F}^{2}\,{\smallsetminus}{\smallsetminus}\,\\{\boldsymbol{0}\\}$,
the cross ratio of the four points
$\mathbb{F}\boldsymbol{a},\mathbb{F}\boldsymbol{b},\mathbb{F}\boldsymbol{c},\mathbb{F}\boldsymbol{d}\in\textrm{P}\mathbb{F}^{2}$
is given by the point
$P=\hskip
2.84526pt(\mathbb{F}\boldsymbol{a},\mathbb{F}\boldsymbol{b};\mathbb{F}\boldsymbol{c},\mathbb{F}\boldsymbol{d})$
$\hskip 4.55244pt:=\Big{(}\det\\!\left(\begin{array}[]{rr}a_{1}&a_{2}\\\
d_{1}&d_{2}\end{array}\right)\det\left(\begin{array}[]{rr}c_{1}&c_{2}\\\
b_{1}&b_{2}\end{array}\right){\,:\,}\det\left(\begin{array}[]{rr}a_{1}&a_{2}\\\
c_{1}&c_{2}\end{array}\right)\det\left(\begin{array}[]{rr}d_{1}&d_{2}\\\
b_{1}&b_{2}\end{array}\right)\,\Big{)}{\;\in\;}\textrm{P}\mathbb{F}^{2}\,$,
cf. [25, Ch. 13] for the case $\mathbb{F}=\mathbb{R}$.
If $\varphi{:\,}\mathbb{F}^{2}\to\mathbb{F}^{2}$ is an automorphism, then

$(\mathbb{F}\boldsymbol{a},\mathbb{F}\boldsymbol{b};\mathbb{F}\boldsymbol{c},\mathbb{F}\boldsymbol{d})=(\mathbb{F}\varphi(\boldsymbol{a}),\mathbb{F}\varphi(\boldsymbol{b});\mathbb{F}\varphi(\boldsymbol{c}),\mathbb{F}\varphi(\boldsymbol{d}))$.

Remark: Usually the double ratio is represented as the ratio

$\big{(}\det\left(\begin{array}[]{rr}a_{1}&a_{2}\\\
c_{1}&c_{2}\end{array}\right)\det\left(\begin{array}[]{rr}d_{1}&d_{2}\\\
b_{1}&b_{2}\end{array}\right)\big{)}:\big{(}{\det\left(\begin{array}[]{rr}a_{1}&a_{2}\\\
d_{1}&d_{2}\end{array}\right)\det\left(\begin{array}[]{rr}c_{1}&c_{2}\\\
b_{1}&b_{2}\end{array}\right)\big{)}}$

of two numbers. It should be noted that the divisor can be zero.

### 1.3.

Projective collineations.  Let
$\varphi{:\,}\boldsymbol{V}\to\boldsymbol{V}^{\prime}$ be an injective linear
map between two $\mathbb{F}$-vector spaces $\boldsymbol{V}$ and
$\boldsymbol{V}^{\prime}$. Then $\varphi$ induces a map
${\phi}{:\,}\textrm{P}\boldsymbol{V}\to\textrm{P}\boldsymbol{V}^{\prime}$
between the projective spaces $\textrm{P}\boldsymbol{V}$ and
$\textrm{P}\boldsymbol{V}^{\prime}$ by
${\phi}(\mathbb{F}\boldsymbol{v})=\mathbb{F}{\varphi(\boldsymbol{v})}$. Since
${\phi}{:\,}\textrm{P}V\to\textrm{P}V^{\prime}$ maps a set of collinear points
of $\textrm{P}V$ to a set of collinear points of $\textrm{P}V^{\prime}$,
${\phi}$ is called a projective collineation.
If $\varphi{:\,}V\to V^{\prime}$ is bijective,
${\phi}{:\,}\textrm{P}V\to\textrm{P}V^{\prime}$ is an isomorphism.
If $A,B,C,D$ are four points in a $1$-dimensional projective space
$\textrm{P}\boldsymbol{V}$ and
$\phi{:\,}\textrm{P}\boldsymbol{V}\to\textrm{P}\mathbb{F}^{2}$ is an
isomorphism, then we put $(A,B;C,D):=(\phi(A),\phi(B);\phi(C),\phi(D))$. It
can be easily checked that the point $(A,B;C,D)$ (the cross ratio of the four
points) does not depend on the special choice of the isomorphism.
Four collinear points $A,B,C,D$ in a projective space form a harmonic range if
$(A,B;C,D)$ $=(1:-1).$

The automorphism group Aut$(\textrm{P}\boldsymbol{V})$ of $\textrm{P}V$ can be
identified with the projective linear group PGL$(\boldsymbol{V})$ =
GL$(\boldsymbol{V})/$Z, where the center Z of GL$(\boldsymbol{V})$ consists of
all nonzero multiples of the identity. Given a basis $\boldsymbol{B}$ of
$\textrm{P}\boldsymbol{V}$, an element $\phi$ of
$\textrm{Aut}(\textrm{P}\boldsymbol{V})$ can be represented by an invertible
$(n{+}1)\times(n{+}1)$-matrix
$\mathfrak{A}_{\phi}{\,=\,}(\mathfrak{a}_{ij})_{1\leq i,j\leq n{+}1}$ which
acts on a row $(x_{1},\dots,x_{n+1})$ of $n{+}1$ entries by matrix
multiplication from the right:
$\phi([x_{1}:\dots:x_{n+1}]_{\boldsymbol{B}})=[y_{1}:\dots:y_{n+1}]_{\boldsymbol{B}}\,,\,(x_{1},\dots,x_{n+1})\mathfrak{A}_{\phi}{\,=\,}(y_{1},\dots,y_{n+1})$.
All these automorphisms are collineations. Furthermore, these mappings
preserve the cross ratio of quadruples of collinear points. Two subsets of
$\textrm{P}\boldsymbol{V}$ are called projectively equivalent if one can be
mapped onto the other by an automorphism.
Given two projective frames $(P_{0},\dots,P_{n+1})$, $(Q_{0},\dots,Q_{n+1})$,
there exists precisely one automorphism
$\phi\in\text{Aut}(\textrm{P}\boldsymbol{V})$ with $\phi(P_{i})=Q_{i}$, $0\leq
i\leq n{+}1$.
The only automorphism that fixes a projective frame is the identity; any other
automorphism has at most $n{+}1$ independent fixed points. An automorphism
$\phi\neq\textrm{id}$ is called biaxial collineation with axes $A$ and $B$ if
$A$ and $B$ are nonempty complementary subspaces of $\textrm{P}\boldsymbol{V}$
and $\phi$ fixes all points of $A$ and of $B$. Such a collineation also fixes
all lines incident with $A$ and $B$. If $A$ is a hyperplane and $B$ consists
of one point, $\phi$ is called dilation with axis $A$ and center $B$. A
dilation $\phi$ is a special case of a central collineation (also called
perspectivity), all the hyperplanes through one point, its center, are
invariant under $\phi$. There is another kind of central collineation which is
called elation. As a dilation, an elation also leaves a hyperplane, its axis,
pointwise invariant and fixes all hyperplanes through its center; but the
center of an elation is a point on its axis. Given any central collineation
$\phi$ with center $Z=\mathbb{R}\boldsymbol{z}$ and axis
$U=\mathbb{R}\boldsymbol{U}$, we can find a linear form
$m\\!:\boldsymbol{V}\to\mathbb{R}$ with $\boldsymbol{U}=\textrm{ker}(m)$ and
$m(\boldsymbol{z})\neq-1$ such that

$\phi(\mathbb{R}\boldsymbol{p})=\mathbb{R}(\boldsymbol{p}+m(\boldsymbol{p}){{}_{\,}}\boldsymbol{z})\;$for
all $\boldsymbol{p}\in\boldsymbol{V}$.

The points $P{\,=\,}\mathbb{R}\boldsymbol{p},Z,\phi(P)$ are collinear. And
$\phi(P)=P$ precisely when $P=Z$ or $m(\boldsymbol{p})=0$.
Aut$(\textrm{P}\boldsymbol{V})$ is generated by central collineations; each
automorphism is the product of at most $n{+}1$ central collineations.

We denote the set of all projective subspaces of $\textrm{P}\boldsymbol{V}\\!$
by $\textrm{sub}(\textrm{P}\boldsymbol{V})$. An automorphism
$\phi\in\text{Aut}(\text{P}\boldsymbol{V})$ induces a mapping
$\text{sub}(\text{P}\boldsymbol{V})\to\text{sub}(\text{P}\boldsymbol{V})$,
which we also denote by $\phi$, with
$\phi({U}_{1}\sqcup{U}_{2})=\phi({U}_{1})\sqcup\phi({U}_{2})$ and
$\phi({U}_{1}\sqcap{U}_{2})=\phi({U}_{1})\sqcap\phi({U}_{2})$.
A mapping ${\kappa{:}\textrm{sub}(\textrm{P}\boldsymbol{V})\\!\to}$
$\textrm{sub}(\textrm{P}\boldsymbol{V})$ is called a correlation if it maps
subspaces of dimension $k$ to subspaces of dimension $n{-}k{-}1$ such that
$\kappa({U}_{1}\sqcup{U}_{2})=\kappa({U}_{1})\sqcap\kappa({U}_{2})$ and
$\kappa({U}_{1}\sqcap{U}_{2})=\kappa({U}_{1})\sqcup\kappa({U}_{2})$.

### 1.4.

Quadrics.  We already introduced $\boldsymbol{V}$ as an $\mathbb{F}$-vector
space of dimension $n{+}1$. Let
$\beta{:\,}\boldsymbol{V}\times\boldsymbol{V}\to\mathbb{F}$ be a symmetric
bilinear form. Given a basis
$\boldsymbol{B}=(\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{n+1})$, we define a
matrix
$\mathfrak{A}=(\mathfrak{a}_{ij})\in\mathbb{F}^{n+1}\times\mathbb{F}^{n+1}$ by
$\mathfrak{a}_{ij}{\,:=\,}\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{j})$. The
rank of this matrix does not depend on the special choice of the basis
$\boldsymbol{B}$, so we can assign this number to the bilinear form $\beta$.
It is always possible to find an orthogonal basis
$(\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{n+1})$ for $\beta$, i.e. a basis of
$\boldsymbol{V}$ with
$\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{j})=0\;\text{for}\,i\neq j.$ If
$\boldsymbol{B}$ is an orthogonal basis, the associated matrix is diagonal. In
the following we always assume that rank$(\beta)=\textrm{rank}(\mathfrak{A})$
is not zero.
The set
$Q_{\beta}:=\\{P{\,=\,\mathbb{F}\boldsymbol{b}}\in\text{P}\boldsymbol{V}\,|\,\beta(\boldsymbol{b},\boldsymbol{b})=0\\}$
is called the quadric associated with $\beta$. In case of
$\mathbb{F}=\mathbb{R}$ this quadric can be the empty set, but if
$\mathbb{F}=\mathbb{C}$ this quadric is always nonempty.
Two points
$P=\mathbb{F}\boldsymbol{p},Q=\mathbb{F}\boldsymbol{q}\in\textrm{P}\boldsymbol{V}$
are called conjugate with respect to $\beta$, if
$\beta(\boldsymbol{p},\boldsymbol{q})=0$. Two sets
$\mathcal{S}_{1},\mathcal{S}_{2}\subset\textrm{P}\boldsymbol{V}$ are conjugate
with respect to $\beta$, if $P_{1}\,\textrm{is conjugate to}\,P_{2}$ for all
points $P_{1}{\,\in\,}\mathcal{S}_{1}$ and $P_{2}{\,\in\,}\mathcal{S}_{2}$. If
$\mathcal{S}$ is a subset of $\textrm{P}\boldsymbol{V}$, the set
$\textrm{polar}_{\beta}(\mathcal{S}):=\\{P\in\textrm{P}\boldsymbol{V}\,$
$|\,P\,\textrm{is conjugate to}\,\mathcal{S}\\}$ is a projective subspace of
$\textrm{P}\boldsymbol{V}$ and is called the polar of $\mathcal{S}$ with
respect to $\beta$. Obviously,
$\mathcal{S}\subseteq\textrm{polar}_{\beta}(\textrm{polar}_{\beta}(\mathcal{S}))$.
The mapping
$\textrm{polar}_{\beta}{:\textrm{sub}(\textrm{P}\boldsymbol{V})\\!\to}$
$\textrm{sub}(\textrm{P}\boldsymbol{V})$ is a correlation iff
rank$(\beta)=n{+}1$.

### 1.5.

Real semi Cayley-Klein spaces. We are especially interested in the case
$\mathbb{F}=\mathbb{R}$. We choose a symmetric bilinear form
$\beta{:\,}\boldsymbol{V}\times\boldsymbol{V}\to\mathbb{R}$ with
rank$(\beta)>0$ and call it the absolute bilinear form of $\boldsymbol{V}$. If
$\boldsymbol{B}=(\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{n+1})$ is an
orthogonal basis for $\beta$, we put
$\beta_{+}{:=\,}\\#\\{\boldsymbol{b}_{i}|\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{i}){>}0\\}$,
$\beta_{-}{:=\,}\\#\\{\boldsymbol{b}_{i}|\,\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{i})<0\\}$,
$\beta_{0}{:=\,}\\#\\{\boldsymbol{b}_{i}|\,\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{i})=0\\}$.
These three numbers do not change if we change the basis for another
orthogonal basis (Sylvester’s law of inertia). The triple
$(\beta_{+},\beta_{-},\beta_{0})$ is called the signature of $\beta$. Now it
is obvious that we can even find an orthogonal basis
$\boldsymbol{B}=(\boldsymbol{b}_{1},\dots,\boldsymbol{b}_{n+1})$ such that
$\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{i})=1$ for all
$i{\,\in\,}\\{1,\dots,\beta_{+}\\}$,
$\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{i})=-1$ for all
$i{\,\in\,}\\{\beta_{+}{+\,}1,\dots,\beta_{+}{+\,}\beta_{-}\\}$ and
$\beta(\boldsymbol{b}_{i},\boldsymbol{b}_{i})=0$ for all
$i>\textrm{rank}(\beta)$.
$\mathrm{P}\boldsymbol{V}$ can now be decomposed into three disjoint subsets:
$\textrm{P}\boldsymbol{V}=Q_{\beta}{\,\dot{\cup}\,}Q_{\beta}^{\,+}{\,\dot{\cup}\,}Q_{\beta}^{\,-}$
where $Q_{\beta}$ is the quadric associated with $\beta$,
$Q_{\beta}^{\,+}{:=\,}\\{\mathbb{R}\boldsymbol{v}\,|\,\beta(v,v)>0\\}$ and
$Q_{\beta}^{\,-}:=\\{\mathbb{R}\boldsymbol{v}\,|\,\beta(v,v)<0\\}$. Since
$\beta{:\,}\boldsymbol{V}\times\boldsymbol{V}\to\mathbb{R}$ is a nonzero
function,
$\mathcal{Q}_{\beta}^{\;\times}:=\mathcal{Q}_{\beta}^{\;+}\cup\mathcal{Q}_{\beta}^{\;-}$
is a dense subset of $\textrm{P}\boldsymbol{V}$ (with respect to the topology
on $\textrm{P}\boldsymbol{V}$, which is the quotient topology of the canonical
topology on $\boldsymbol{V}$). We call $(\textrm{P}\boldsymbol{V},\beta)$ a
semi Cayley-Klein space.

Let $\boldsymbol{U}$ be a linear subspace of $\boldsymbol{V}$. We assume that
the restriction ${\beta^{\prime}}:=\beta|\boldsymbol{U}\times\boldsymbol{U}$
is a nonzero function. Then $(\textrm{P}{\boldsymbol{U}},{\beta}^{\prime})$ is
a semi CK subspace of $(\textrm{P}\boldsymbol{V}\\!,\beta)$.

Let $U=\textrm{P}\boldsymbol{U}\leq\textrm{P}\boldsymbol{V}$ be a plane of
dimension $s\leq n$ and let $\boldsymbol{u}_{1},\dots,\boldsymbol{u}_{s+1}$ be
a basis of $\boldsymbol{U}$. Define a matrix
$\mathfrak{N}=(\mathfrak{n}_{ij})$ by
$\mathfrak{n}_{ij}:=\beta(\boldsymbol{u}_{i},\boldsymbol{u}_{j})$. The rank of
this matrix $\mathfrak{N}$ does not dependent of the special choice of the
basis. If $\det(\mathfrak{N})=0$, we call $U$ isotropic, otherwise
anisotropic. As a special case, a point $P\in\mathbb{R}\boldsymbol{V}$ is
isotropic iff $P\in\mathcal{Q}_{\beta}$.
If $P$ is an anisotropic point, then
$P\cap\textrm{polar}_{\beta}(P)=\emptyset$.
$P$ is called double point of a plane $U$ if
$U\subseteq\textrm{polar}_{\beta}(P)$.
A plane $U$ is anisotropic iff it does not contain any double point.
Proof: We assign a matrix $\mathfrak{N}$ to $U$ as described above.
If there is a double point $P$ in $U$, we may assume that it is
$\mathbb{R}\boldsymbol{u}_{1}$. In this case, the first row of the matrix
$\mathfrak{N}$ is a zero-row and $\det(\mathfrak{N})=0$.
Now we assume that $U$ is isotropic and $\det(\mathfrak{N})=0$. We may also
assume that the matrix $\mathfrak{N}$ is a diagonal matrix. Then there exists
a vector $\boldsymbol{u}_{i}$ with
$\beta(\boldsymbol{u}_{i},\boldsymbol{u}_{j})=0$ for all $j$, and
$\mathbb{R}\boldsymbol{u}_{i}$ must be a double point of $U$. $\Box$
A point $P{\,=\,}\mathbb{R}(\boldsymbol{p})\in\textrm{P}\boldsymbol{V}$ is
called singular if it is a double point of $\textrm{P}\boldsymbol{V}$. The
singular points form a plane in $\textrm{P}\boldsymbol{V}$ which is called the
vertex of $\mathcal{Q}_{\beta}$ or the radical rad$(\beta)$ of $\beta$.
Remark: A point $P$ is a double point of the $0$-dimensional space $P$ iff
$P\in\mathcal{Q}_{\beta}$.
Properties of polars: (1) The polar of a singular point is
$\textrm{P}\boldsymbol{V}$, the polar of a nonsingular point is a hyperplane
in $\textrm{P}\boldsymbol{V}$. (2) $A\leq\textrm{P}\boldsymbol{V}$ is an
anisotropic plane iff $\textrm{polar}_{\beta}(A){:=\,}\bigcap_{P\in
A}\textrm{polar}_{\beta}(P)$ is a projective complement of $A$ in
$\textrm{P}\boldsymbol{V}$.
A subspace ${U}\leq\textrm{P}\boldsymbol{V}$ of positive dimension located
entirely in the quadric $\mathcal{Q}_{\beta}$ is called totally isotropic. It
is obvious, that all subspaces of $\textrm{rad}(\beta)$ are totally isotropic
if their dimension is positive. But even if rad$(\beta)$ is empty,
$(\textrm{P}\boldsymbol{V},\beta)$ can have totally isotropic subspaces. An
example is $(\,\textrm{P}\mathbb{R}^{4},\beta\,)$ with
$\beta\big{(}(v_{1},v_{2},v_{3},v_{4}),(w_{1},w_{2},w_{3},w_{4})\big{)}=v_{1}w_{1}+v_{2}w_{2}-v_{3}w_{3}-v_{4}w_{4}.$
The maximal dimension of a totally isotropic subspace is
$n-\text{max}(\beta_{+},\beta_{-})$.

### 1.6.

Automorphisms on a semi CK space $(\,\mathrm{P}\boldsymbol{V},\beta\,)$. Let
$(\textrm{P}\boldsymbol{V}\\!,\beta),(\textrm{P}\tilde{\boldsymbol{V}}\\!,\tilde{\beta})$
semi CK spaces. Let
$\phi{:\,}\textrm{P}\boldsymbol{V}\to\textrm{P}\tilde{\boldsymbol{V}}$ be a
projective collineation and
$\varphi{:\;}\boldsymbol{V}\to\tilde{\boldsymbol{V}}$ a representative of
$\phi$. Then $\phi$ is called a projective collineation from the semi CK space
$(\textrm{P}\boldsymbol{V}\\!,\beta)$ to the semi CK space
$(\textrm{P}\boldsymbol{\tilde{\boldsymbol{V}}}\\!,\tilde{\beta})$ if
$\tilde{\beta}({\varphi}(\boldsymbol{p}),{\varphi}(\boldsymbol{q})){\,=\,}0\Leftrightarrow\beta(\boldsymbol{p},\boldsymbol{q}){\,=\,}0$
for all points $P=\mathbb{R}\boldsymbol{p}$ and $Q=\mathbb{R}\boldsymbol{p}$.
If this mapping $\phi$ is bijective, then $\phi$ is an isomorphism between
semi CK spaces and, in case of $(\textrm{P}\boldsymbol{V}\\!,\beta)=$
$(\textrm{P}\boldsymbol{\tilde{\boldsymbol{V}}}\\!,\tilde{\beta})$, $\phi$ is
an automorphism on $(\textrm{P}\boldsymbol{V}\\!,\beta)$.

Special automorphisms on a semi CK space are reflections: Let
$Z=\mathbb{R}\boldsymbol{z}\in\textrm{P}\boldsymbol{V}$ be an anisotropic
point and let $\phi\in\textrm{Aut}(\textrm{P}\boldsymbol{V})$ be a central
collineation with center $Z$ and axis $\textrm{polar}_{\beta}(Z)$. Then
$\textrm{polar}_{\beta}(Z)$ is a hyperplane not incident with $Z$, and $\phi$
is a dilation. Because rad$(\beta)\leq\textrm{polar}_{\beta}(Z)$, all points
of rad$(\beta)$ are fixed points. There exists some number
$\lambda\in\mathbb{R}{\smallsetminus}{\smallsetminus}\\{0,1\\}$ such that
$\phi$ maps a point $P=\mathbb{R}\boldsymbol{p}\in\textrm{P}\boldsymbol{V}$ to
a point

$P^{\prime}=\mathbb{R}\big{(}\lambda\beta(\boldsymbol{z},\boldsymbol{z})\boldsymbol{p}+(1-\lambda)\beta(\boldsymbol{z},\boldsymbol{p})\boldsymbol{z}\big{)}$.

If $\lambda=-1$, $\phi$ is an involution and $\phi$ is called a reflection in
the point $Z$. It can be easily checked that a reflection maps an anisotropic
point to an anisotropic point lying in the same connected component of
$\mathcal{Q}_{\beta}^{\;\times}$. If $P$ is the only isotropic point on the
(isotropic) line $P\sqcup Z$, then $P$ is a fixed point of the reflection
$\phi$, and if the line $P\sqcup Z$ contains still another isotropic point
$Q$, then $P$ and $Q$ are interchanged by $\phi$.
Given an anisotropic plane $U$, then a biaxial collineation $\phi$ with axes
$U$ and $\textrm{polar}_{\beta}(U)$ is called a reflection in $U$. If $Z$ is
an anisotropic point of $U$ and $P$ is any point of
$\textrm{P}\boldsymbol{V}$, then $\phi(P)$ is the image of $P$ under a
reflection in $Z$.

### 1.7.

Barycentric coordinates of points in a semi CK space. Let from now on
$\boldsymbol{V}$ be the real vector space $\mathbb{R}^{n+1}$ with canonical
basis $\boldsymbol{B}=(\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{n+1})$. Given
a vector
$\boldsymbol{v}=[v_{1},\dots,v_{n+1}]_{\boldsymbol{B}}\in\boldsymbol{V}$, we
write $\boldsymbol{v}=(v_{1},\dots,v_{n+1})$, and for the point
$\mathbb{R}\boldsymbol{v}\in\textrm{P}\boldsymbol{V}$ we use the notation
$(v_{1}{\,:\,}\dots\,:\,v_{n+1})$. Let $(\beta_{+},\beta_{-},\beta_{0})$ be a
triple of nonnegative integers with
$\beta_{+}{\,+\,}\beta_{-}{\,+\,}\beta_{0}=n{+}1$ and let
$\mathfrak{A}^{(\beta_{+},\beta_{-},\beta_{0})}=\mathfrak{A}=(\mathfrak{a}_{ij})$
denote the $(n{+}1)\times(n{+}1)$-diagonal matrix with
$\mathfrak{a}_{ii}{\,=\,}1$ for $1\leq i\leq\beta_{+}$,
$\mathfrak{a}_{ii}{\,=\,}{-}1$ for
$\beta_{+}{\,<\,}i\leq\beta_{+}{+\,}\beta_{-}$ and $\mathfrak{a}_{ii}{\,=\,}0$
for $i{\,>\,}\beta_{+}{+\,}\beta_{-}$. Associated with $\mathfrak{A}$ is the
bilinear form
$\beta{:\,}\mathbb{R}^{n+1}{\times\,}\mathbb{R}^{n+1}\to\mathbb{R}$,
$\beta(\boldsymbol{e}_{i},\boldsymbol{e}_{j})=\mathfrak{a}_{ij}$.
$(\textrm{P}\boldsymbol{V},\beta)$ is a semi CK space.
Remark: Every real semi CK space can be subsumed here (up to isomorphism).
Instead of $\beta(\boldsymbol{v},\boldsymbol{w})$ we will usually write
$\boldsymbol{v}{\scriptstyle{[\mathfrak{A}]}}\boldsymbol{w}$. (We regard
${\scriptstyle{[\mathfrak{A}]}}$ as a generalized inner product.) For
$\mathcal{Q}_{\beta}$ we write $\mathcal{Q}_{\mathfrak{A}}$, the polar of a
set $\mathcal{S}\subset\mathrm{P}\boldsymbol{V}$ with respect $\beta$ to will
be denoted by $\mathcal{S}^{\mathfrak{A}}$ instead of
polar${}_{\beta}(\mathcal{S})$, and the semi CK space
$(\textrm{P}\boldsymbol{V},\beta)$ we denote by
$(\textrm{P}\boldsymbol{V},\mathfrak{A})$.

We assign a vector $P^{\circ}\in\boldsymbol{V}$ to each anisotropic point $P$:
First, we define a function $\chi{:}\;\boldsymbol{V}\rightarrow\\{-1,0,1\\}$
by
$\chi(p_{0}{,\dots,}p_{n+1})=\begin{cases}\;\,0,\;\text{if
}(p_{1},\dots,p_{n+1})=(0,\dots,0)\;,\\\ \;\,1,\;\text{if
}(p_{1},\dots,p_{n+1})>(0,\dots,0)\;\textrm{with respect to the lexicographic
order,}\\\ -1,\text{if }(p_{1},\dots,p_{n+1})<(0,\dots,0)\;\textrm{with
respect to the lexicographic order},\\\ \end{cases}$
then we put $\displaystyle
P^{\circ}:=\frac{\chi(\boldsymbol{p})}{\sqrt{|{\boldsymbol{p}{\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}|}}}$.
We introduce a function sgn${:\,}\textrm{P}\boldsymbol{V}\to\\{-1,0,1\\}$ by
sgn$(P)=0$ if $P$ is isotropic, and
sgn$(P)=P^{\circ}{\scriptstyle{[\mathfrak{A}]}\,}P^{\circ}$ if $P$ is
anisotropic.
Since the matrix $\mathfrak{A}$ is nonzero, there exist $\,n{+}1$ independent
anisotropic points $P_{1},\dots,P_{n+1}$ generating
$\textrm{P}\boldsymbol{V}$. Given a point $Q=\mathbb{R}\boldsymbol{q}$, there
exists an $(n{+}1)$-tuple $(\tilde{q}_{1},\dots,\tilde{q}_{n+1})$ of real
numbers such that
$\boldsymbol{q}=\tilde{q}_{1}P_{1}^{\circ}+\cdots+\tilde{q}_{n+1}P_{n{+}1}^{\circ}$.
The tuple $(\tilde{q}_{1},\dots,\tilde{q}_{n+1})$ is uniquely determined by
the point $Q$ and the tuple $(P_{1},\dots,P_{n+1})$ except for multiplication
by a nonzero real number, and is called tuple of homogeneous coordinates of
$Q$ with respect to $(P_{1},\dots,P_{n+1})$. After having fixed the tuple
$(P_{1},\dots,P_{n+1})$, we write $Q=[\tilde{q}_{1}:\dots:\tilde{q}_{n+1}]$.
If $Q$ is anisotropic, there exists a uniquely determined $(n{+}1)$-tuple
$(\tilde{q}_{1},\dots,\tilde{q}_{n+1})$ such that
$Q^{\circ}=\tilde{q}_{1}P_{1}^{\circ}+\cdots+\tilde{q}_{n+1}P_{n{+}1}^{\circ}$.
This tuple is the tuple of absolute coordinates of $Q$ with respect to
$(P_{1},\dots,P_{n+1})$.

### 1.8.

The dual of a semi CK space. Let
$\boldsymbol{V}^{\ast}=\textrm{Hom}(\boldsymbol{V},\mathbb{R})$ be the vector
space dual to $\boldsymbol{V}$ and let
$\boldsymbol{e}^{1},\dots,\boldsymbol{e}^{n+1}$ be the dual basis of
$\boldsymbol{e}_{1}{,\dots,}\boldsymbol{e}_{n+1}$. Put
$E^{i}{:=\,}\mathbb{R}\boldsymbol{e}^{i},i{\,=\,}1,\dots,n{+}1$. The
projective space $\textrm{P}\boldsymbol{V}^{\ast}$ is called the dual of
$\textrm{P}\boldsymbol{V}$.

Let $(\textrm{P}\boldsymbol{V},\mathfrak{A})$ be a semi CK space, and let
$\mathfrak{A}^{\sharp}=\textrm{adj}(\mathfrak{A})$ denote the adjugate of
$\mathfrak{A}$. We call the semi CK space
$(\textrm{P}\boldsymbol{V}^{\ast},{{\mathfrak{A^{\sharp}}}})$ the dual of
$(\textrm{P}\boldsymbol{V},{{\mathfrak{A}}})$.

We now assume that rank$(\mathfrak{A})=n{+}1.$ If $P_{1},P_{2}$ are
anisotropic points in $\textrm{P}\boldsymbol{V}$, then
$H_{1}{\,:=\,}P_{1}^{\mathfrak{A}},H_{2}{\,:=\,}P_{2}^{\mathfrak{A}}$ are
anisotropic hyperplanes in $\textrm{P}\boldsymbol{V}$ and there are uniquely
determined anisotropic elements
$x_{1},x_{2}\in\textrm{P}\boldsymbol{V}^{\ast}$ with
$x_{1}(Q_{1})=x_{2}(Q_{2})=0$ for all $Q_{1}\in H_{1}$ and all $Q_{2}\in
H_{2}$. For $x_{1},x_{2}$ and $P_{1},P_{2}$ the equation
$\displaystyle\frac{(x_{1}{\scriptstyle{[\mathfrak{A}^{-1}]}}x_{2})^{2}}{(x_{1}{\scriptstyle{[\mathfrak{A}^{-1}]}}x_{1})(x_{2}{\scriptstyle{[\mathfrak{A}^{-1}]}}x_{2})}{\,=\,}\frac{(P_{1}{\scriptstyle{[\mathfrak{A}]}}P_{2})^{2}}{(P_{1}{\scriptstyle{[\mathfrak{A}]}}P_{1})(P_{2}{\scriptstyle{[\mathfrak{A}]}}P_{2})}$
applies.

### 1.9.

Perspectivity and orthology. Let $U\leq\textrm{P}\boldsymbol{V}$ be a plane of
dimension $s>1$. $U$ is arguesian: Let $(P_{1},\dots,P_{s{+}1})$,
$(Q_{1},\dots,Q_{s{+}1})$ be two systems of points, each generating $U$.
$(P_{1},\dots,P_{s{+}1})$ and $(Q_{1},\dots,Q_{s{+}1})$ are perspective if
there exists exactly one point $Z$ such that all the sets $P_{i}\sqcup
Q_{i}\sqcup Z\,$, $1\leq{i\leq{s{+}1},}$ are lines. If such a point $Z$
exists, it is called the perspector of $(P_{1},\dots,P_{s{+}1})$ and
$(Q_{1},\dots,Q_{s{+}1})$.
Suppose this perspector $Z$ is different from all the points
$P_{i},Q_{i},\;i=1,\dots,s{+}1$, then there exists a uniquely determined
central collineation $\phi{:\,}U{\,\to\,}U$ mapping
$(P_{1},\dots,P_{s{+}1},Z)$ onto $(Q_{1},\dots,Q_{s{+}1},Z)$. The axis of
$\phi$, a hyperplane of $U$, is called perspectrix of
$(P_{1},\dots,P_{s{+}1})$ and $(Q_{1},\dots,Q_{s{+}1})$.
A perspectrix can be assigned to these two tuples even if the perspector is
one of the points $P_{i},Q_{i},1\leq i\leq s{+}1.$ For example, if
$Z=P_{s+1}\neq Q_{s+1}$, then $Q_{1}\sqcup\dots\sqcup Q_{s}$ is the
perspectrix.

We now assume that $U$ is anisotropic. Given an $(s{+}1)$-tuple
$R_{1},\dots,R_{s+1}$ of points spanning $U$, put
$R_{i}^{\;\delta}{:=\,}(\bigsqcup_{j\neq i}R_{j})^{\mathfrak{A}}\,\sqcap\,U$,
$1\leq i\leq s{+}1$. If $(P_{1},\dots,P_{s{+}1})$ and
$(Q_{1},\dots,Q_{s{+}1})$ are perspective with center $Z$ and perspectrix
$S\leq U$, then
$(P_{1}^{\;\delta},\dots,P_{s+1}^{\;\,\delta}),(Q_{1}^{\;\delta},\dots,Q_{s+1}^{\;\,\delta})$
are also perspective; their perspector is $S^{\mathfrak{A}}\sqcap U$ and their
perspectrix is $Z^{\mathfrak{A}}\sqcap A$.
Two tuples $(P_{1},\dots,P_{s+1})$ and $(Q_{1},\dots,Q_{s+1})$ are called
orthologic with orthology center $Z$ if $(P_{1},\dots,P_{k+1})$ and
$(Q_{1}^{\;\delta},\dots,Q_{k+1}^{\;\,\delta})$ are perspective at $Z$.
If $(P_{1}{,\dots,}P_{s+1})$ and $(Q_{1}{,\dots,}Q_{s+1})$ are orthologic,
then $(Q_{1}{,\dots,}Q_{s+1})$ and $(\\!P_{1}{,\dots,}P_{s+1})$ are also
orthologic. If $S$ is the perspectrix of $(\\!P_{1}{,\dots,}P_{s+1})$ and
$(Q_{1}^{\;\delta}{,\dots,}Q_{s+1}^{\;\,\delta})$, then
$S^{\mathfrak{A}}\sqcap A$ is the associated orthology center.

##  2\. Metric structures on $\mathrm{P}V$

### 2.1.

Metric structures on the projective line. K. von Staudt [24], E. Laguerre [18]
and A. Cayley [2] showed us how quadrics in $\textrm{P}\boldsymbol{V}$ can be
used to introduce metric structures on $\textrm{P}\boldsymbol{V}$. An
important role within the connection between quadrics and distances plays the
cross ratio of points and of lines. Later, systematic studies by Felix Klein
[16] led to a classification of metric geometries. We give a rough
illustration of the ideas of these great mathematicians for the simple case
$n=1$. We start with the inner product ${\scriptstyle{[\mathfrak{A}]}}\,$ on
the projective line $\textrm{P}\mathbb{R}^{2}$, which is given by the matrix
$\mathfrak{A}=\left(\begin{array}[]{rr}1&0\\\ 0&r\\\ \end{array}\right)$.
(Every quadric on the projective line is projectively equivalent to a quadric
$\mathcal{Q}_{\mathfrak{A}}$ with a suitable number $r\in\\{-1,0,1\\}$.) The
set of isotropic points is empty if $r>0$, and consists of the two points
$J_{-}=(-\sqrt{-r}:1)$ and $J_{+}=(\sqrt{-r}:1)$ if $r<0$. If $r=0$, there is
exactly one isotropic point. This point is a double point of
$\textrm{P}\mathbb{R}^{2}$; the points $J_{-}$ and $J_{+}$ merge, so to speak,
for $r=0$ to one point $J=(0:1)$. In the case of $r>0$, Laguerre [18]
encouraged us to think of $\textrm{P}\mathbb{R}^{2}$ embedded in the complex
projective space $\textrm{P}\mathbb{C}^{2}$ and to think of the points $J_{-}$
and $J_{+}$ as complex valued because these two complex valued points (in
fact, his focus was on the angle between lines rather than on the distance
between points) are helpful for introducing a metric. The following
classification comes from F. Klein: depending on whether $r>0$ or $r=0$ or
$r<0$, we speak of the elliptic or parabolic or hyperbolic case.

Let us first assume $r\neq 0$. In this case
$(\textrm{P}\mathbb{R}^{2},\mathfrak{A})$ is an anisotropic CK space. We take
two independent anisotropic points
$P_{1}=\mathbb{R}\boldsymbol{p}_{1}=\mathbb{R}(p_{11}{\,,\,}p_{12})$ and
$P_{2}=\mathbb{R}\boldsymbol{p}_{2}=\mathbb{R}(p_{21},p_{22})$,
$\chi(p_{11},p_{12})=\chi(p_{21},p_{22})$, and determine homogeneous
barycentric coordinates for $J_{-}$ and $J_{+}$ with respect to
$(P_{1},P_{2})$:

$\displaystyle
J_{\mp}=\mathbb{R}\big{(}{(p_{21}\pm\sqrt{-r}\,p_{22})}(p_{11},p_{12})-{(p_{11}\pm\sqrt{-r}\,p_{12})}(p_{21},p_{22})\big{)}$

$\hskip
66.86397pt\displaystyle=\frac{p_{21}\pm\sqrt{-r}\,p_{22}}{\sqrt{|p_{21}^{\;2}-rp_{22}^{\;2}}|}P_{1}-\frac{p_{11}\pm\sqrt{-r}\,p_{12}}{\sqrt{|p_{11}^{\;2}-rp_{12}^{\;2}}|}P_{2}.$
The cross ratio of the four points $P_{1},P_{2},J_{-},J_{+}$ is

$\displaystyle(P_{1},P_{2};J_{-},J_{+})={((p_{11}+\sqrt{-r}\,p_{12})(p_{21}-\sqrt{-r}\,p_{22}))}{\,:\,}{((p_{11}-\sqrt{-r}\,p_{12})(p_{21}+\sqrt{-r}\,p_{22}))}\,.$

In the elliptic case, this cross ratio is a complex number of modulus $1$ and,
following Laguerre, Cayley and Klein, we can define a distance
$d_{12}\in\mathbb{R}$ between the points $P_{1}$ and $P_{2}$ by
$d_{12}=c\,\ln(\sqrt{(P_{1},P_{2};J_{-},J_{+})}$ with a constant $c$ that can
be fixed to $c=\frac{1}{2\mathbf{i}}$ by requiring that the distance $d$ of
two points on the elliptic line is within the range $0\leq
d\leq\frac{1}{2}\pi$ and that the distance of these points is $\frac{1}{2}\pi$
precisely when one point is conjugate to the other. If we take this distance
function, we get the equation (see [17, ch. 5]

$\cos(d_{12})=P_{1}^{\,\circ}{\scriptstyle{[\mathfrak{A}]}\,}P_{2}^{\,\circ}\;\;\;\;\;\;(*).$

At the beginning of the $20^{\,th}$ century, the Hungarian mathematician C.
Vörös 333) I take this information from [13].) proposed to proceed in the
hyperbolic case quite analogously to the elliptic case: All hyperbolic lines
have the same length $\pi\mathbf{i}$, and the distance function satisfies the
equation

$\displaystyle\cosh^{2}(d_{12})=\,\frac{(\boldsymbol{p}_{1}{\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{2})^{2}}{({\boldsymbol{p}_{1}{\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{1})}({\boldsymbol{p}_{2}{\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{2}})}\,=\textrm{sgn}({P_{1}})\,\textrm{sgn}({P_{2}})\,(P_{1}^{\,\circ}{\scriptstyle{[\mathfrak{A}]}\,}P_{2}^{\,\circ})^{2}\,\;\;\;(**).$

If $\textrm{sgn}({P_{1}})\textrm{sgn}({P_{2}})=1$, then $\cosh(d_{12})>1$ and
$d_{12}$ is a real number. If $\textrm{sgn}({P_{1}})\textrm{sgn}({P_{2}})=-1$,
then $\cosh(d_{12})$ is purely imaginary. In this case, equation $(**)$ can be
satisfied by allowing $d_{12}$ to be a complex number of the form
$t{\,+\,}\frac{1}{2}{\pi}\mathbf{i}\,,\,t{\,\in\,}\mathbb{R}$.
We adopt Vörös’ proposal and, in order to achieve as much harmony as possible
in formulae that are valid in the elliptic plane and corresponding ones that
are valid in the hyperbolic plane, we replace the elliptic distance $d_{12}$
by $d_{12}\mathbf{i}\,.$ Now formula $(**)$ is valid also on the elliptic
line.
The points conjugate to $P_{1}^{\,}$ and $P_{2}^{\,}$ are
$P_{1}^{\,\mathfrak{A}}=(p_{12}{:}r\,p_{11})$ and
$P_{2}^{\,\mathfrak{A}}=(p_{22}{:}r\,p_{21})$, respectively. The cross ratio
$(P_{1}^{\,\mathfrak{A}},P_{2}^{\,\mathfrak{A}};J_{-},J_{+})$ is the same as
the cross ratio $(P_{1}^{\,},P_{2}^{\,};J_{-},J_{+})$.
We calculate the cross ratio of the points
$P_{1}^{\,},P_{2}^{\,\mathfrak{A}},P_{2}^{\,},P_{1}^{\,\mathfrak{A}}$ and get

$\displaystyle(P_{1}^{\,},P_{2}^{\,\mathfrak{A}};P_{2}^{\,},P_{1}^{\,\mathfrak{A}})=\,\frac{r\,(p_{11}p_{22}-p_{12}p_{21})^{2}}{(p_{11}^{\;2}+r\,p_{12}^{\;2})(p_{21}^{\;2}+r\,p_{22}^{\;2})}=-\sinh^{2}(d_{12})\;\;\;\;(***)\,.$

We adopt from N.J. Wildberger [31] the name quadrance of $P_{1}$ and $P_{2}$
for the real number

$\displaystyle\xi_{0}(P_{1},P_{2}):=-\sinh^{2}(d_{12})={1-\displaystyle\textrm{sgn}({P_{1}})\,\textrm{sgn}({P_{2}})\,{(P_{1}^{\,\circ}{\scriptstyle{[\mathfrak{A}]}\,}P_{2}^{\,\circ})^{2}}}$

$\hskip
163.88788pt\displaystyle=\,\frac{\det\\!\left(\begin{array}[]{rr}\boldsymbol{p}_{1}{\,\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{1}&\boldsymbol{p}_{1}{\,\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{2}\\\
\boldsymbol{p}_{2}{\,\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{1}&\boldsymbol{p}_{2}{\,\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{2}\\\
\end{array}\right)}{(\boldsymbol{p}_{1}{\,\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{1})(\boldsymbol{p}_{2}{\,\scriptstyle{[\mathfrak{A}]}\,}\boldsymbol{p}_{2})}$
.
If $Q=xP_{1}+y{\,}P_{2},x,y\in\mathbb{R}$, is anisotropic, then (see [8] for a
proof)

$x^{2}\,:\,y^{2}=({\textrm{sgn}({P_{2}})}\sinh^{2}(d(Q,P_{1}))){\;:\,}({\textrm{sgn}({P_{1}}})\sinh^{2}(d(Q,P_{2})))\;$.

Let us consider now the parabolic case. If $r=0$, there is precisely one
isotropic point, the double point $J=(0{\,:\,}1)$. Given two distinct
anisotropic points $P_{1},P_{2}\in\textrm{P}\mathbb{R}^{2}$, we can find real
numbers $p_{12},p_{22}$ such that $P_{1}=(1{\,:\,}p_{12})$ and
$P_{2}=(1{\,:\,}p_{22})$. If we now apply the formula
$\cosh(d(P_{1},P_{2}))=P_{1}^{\,\circ}{\scriptstyle{[\mathfrak{A}]}\,}P_{2}^{\,\circ}$,
we find out that in the parabolic case the projective line is a null line: The
distance $d(P_{1},P_{2})$ and the quadrance $\xi_{0}(P_{1},P_{2})$ of any two
anisotropic points $P_{1},P_{2}$ is 0. We can still postulate that the total
length of a projective line is $\pi\mathbf{i}$, we just have to demand that
the length of a line segment is $0$ iff the double point $J$ lies outside this
segment. We can accept this ”gross structure”, the distance of two anisotropic
points on a parabolic line being always 0, but it is possible to find a ”fine
structure” by using possibilities that Nonstandard Analysis offers us.
Excursus: Let $\mathbb{R}^{\ast}$ and $\mathbb{C}^{\ast}$ denote the hyperreal
and hypercomplex numbers. Any function $f{:\,}\mathbb{C}\to\mathbb{C}$ can be
extended to a function $f^{\ast}{:\,}\mathbb{C}^{\ast}\to\mathbb{C}^{\ast}$ in
a canonical way (s. [26]). In the following write $f$ instead of $f^{\ast}$;
we skip $\ast$ .
If $f$ is meromorphic in a neighborhood of $0$ in $\mathbb{C}$, we can find a
series $(a_{n})_{n{\,=\,}0,\dots,\infty}\in\mathbb{C}^{\mathbb{N}}$ and an
integer $n$, both uniquely determined, such that $a_{0}\neq 0$ and
$f(\varepsilon)=\varepsilon^{-n}(a_{0}+a_{1}\varepsilon+a_{2}\varepsilon^{2}+\cdots)$
for all infinitesimals $\varepsilon\neq 0$. Let
$\star_{\varepsilon}(f(\varepsilon)):=\varepsilon^{-n}a_{0}$ be the leading
term of $f(\varepsilon)$.

Now we define an $\mathbf{\varepsilon}$-quadrance of anisotropic points
$P_{1}=(p_{11}{:}p_{12})$ and $P_{2}=(p_{21}{:}p_{22})$
by

$\displaystyle\xi_{\mathbf{\varepsilon}}(P_{1},P_{2}):=\star_{\mathbf{\varepsilon}}\big{(}1-\frac{(P_{1}^{\,\circ}{\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]}\,}P_{2}^{\,\circ})^{2}}{({P_{1}^{\,\circ}{\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]}\,}P_{1}^{\,\circ})}{(P_{2}^{\,\circ}{\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}}]}\,P_{2}^{\,\circ})}}\big{)}$,

where $\mathbf{\varepsilon}$ is a nonzero infinitesimal real number and
$\displaystyle\mathfrak{A}_{\mathbf{\varepsilon}}:=\textrm{diag}(1,\mathbf{\varepsilon})$.
While the two functions $\xi_{\mathbf{\varepsilon}}$ and $\xi_{0}$ agree on
the elliptic and on the hyperbolic line, they differ on the parabolic line.
On the parabolic line the following applies:
Let $P_{1}$ and $P_{2}$ be two distinct anisotropic points, then

$\displaystyle\xi_{\mathbf{\varepsilon}}(P_{1},P_{2})=\mathbf{\varepsilon}\,(\frac{p_{21}}{p_{11}}-\frac{p_{22}}{p_{21}})^{2}\neq
0=\xi_{0}(P_{1},P_{2})\,.$

And if $Q=(q_{1}{\,:\,}q_{2})\in\textrm{P}\mathbb{R}^{2}$ is anisotropic and
$Q=xP_{1}{\,+\,}y{\,}P_{2},{x,y\in\mathbb{R}}$, then

${|x|\,:\,|y|}=|q_{1}p_{22}-q_{2}p_{21}|:|q_{1}p_{12}-q_{2}p_{11}|\,$.

Let $Q_{1},Q_{2}$ be two anisotropic points. We can find real numbers
$s_{11},s_{12},s_{21},s_{22}$ such that
$Q_{1}^{\;\circ}=s_{11}P_{1}^{\;\circ}{\,+\,}s_{12}P_{2}^{\;\circ}$ and
$Q_{2}^{\;\circ}=s_{21}P_{1}^{\;\circ}{\,+\,}s_{22}P_{2}^{\;\circ}$. Then,

$\xi_{\mathbf{\varepsilon}}(Q_{1},Q_{2})=\xi_{\mathbf{\varepsilon}}(P_{1},P_{2})(s_{11}-s_{21})(s_{12}-s_{22})\,$.

Remarks: (1) In the following we often use the name squared distance for
$\xi_{\mathbf{\varepsilon}}(P_{1},P_{2})$ if $P_{1},P_{2}$ are anisotropic
points in a parabolic plane. (2) Usually, the (squared) distance of
anisotropic points in a parabolic space (in a euclidean space or a Lorentz-
Minkowski space, for example) is given by a real number $d$ instead of an
infinitesimal number $d{\,\mathbf{\varepsilon}}$. (3) E. Study [29, §23] was
presumably the first to use an infinitesimal number (an infinitesimal dual
number ${\,\mathbf{\varepsilon}}$) for calculating distances between points in
a euclidean space. Considerations of this kind are already presented in
Klein’s paper [16, p. 612].

### 2.2.

Point reflections on a one-dimensional semi CK space
$(\mathrm{P}\mathbb{R}^{2},\mathfrak{A}\,).$ The reflection $\sigma_{Z}$ in an
anisotropic point $Z\in\textrm{P}\mathbb{R}^{2}$ fixes the points $Z$ and
$Z^{\mathfrak{A}}$. Every other point $P$ is mapped to a point $P^{\prime}$
such that the points $Z,P,Z^{\mathfrak{A}},P^{\prime}$ form a harmonic range.
It can be easily checked that
$P^{\prime}\,{\scriptstyle{[\mathfrak{A}]}}\,P^{\prime}=P\,{\scriptstyle{[\mathfrak{A}]}}\,P$
and that
$P^{\prime}\,{\scriptstyle{[\mathfrak{A}]}}\,Z=P\,{\scriptstyle{[\mathfrak{A}]}}\,Z$.
From this follows that $P$ and $P^{\prime}$ are points in the same connected
component of $\mathcal{Q}^{\times}$ and that $d(P^{\prime},Z)=d(P,Z)$. The
last equation applies for the elliptic and hyperbolic lines, but also for the
parabolic line if we regard it as a null line. The double point on the
parabolic line is a fixed point of $\sigma_{Z}$; the two isotropic points on
the hyperbolic line are interchanged by $\sigma_{Z}$.
If we now use a parabolic quadrance on the parabolic line given by
$\xi_{\mathbf{\varepsilon}}(P,Q)=\mathbf{\varepsilon}\,(p-q)^{2}$ for
anisotropic points $P=(1:p)$ and $Q=(1:q)$, we can write $Z=(1:z)$,
$P^{\prime}=(1:p^{\prime})$ and get
$(1:p^{\prime})=P^{\prime}=2Z-P=(1:2z{-}p)$. It follows
$\xi_{\mathbf{\varepsilon}}(P^{\prime},Z)=\xi_{\mathbf{\varepsilon}}(P,Z)$;
the parabolic ${\mathbf{\varepsilon}}$-quadrance of two points is preserved by
reflections.

In the following we will first use a gross distance measure. In Section 3 we
turn to geometries where parabolic lines represent a generic and not just a
singular case; this is when fine distance measurement comes into play.

### 2.3.

Gross measurement in a semi CK space. Let
$(\textrm{P}\boldsymbol{V},\mathfrak{A}\,)$ be a semi CK space. A line without
any isotropic point is an elliptic line. A line is hyperbolic iff it passes
through anisotropic points of different sign; in this case it contains
precisely two nonsingular isotropic points. A line that is neither hyperbolic
nor elliptic, must either be totally isotropic or it must pass through exactly
one isotropic point, which in this case must be a double point. A line with
just one isotropic point is a parabolic line.
We introduce line segments with anisotropic endpoints and the length of these
segments. Lines are compact sets, such as all subspaces of $\textrm{P}V$. We
give all lines, even those entirely located in $\mathcal{Q}_{\mathfrak{A}}$,
the same measure $\pi\mathbf{i}$, with imaginary unit $\mathbf{i}$. Given a
line ${L}\leq\textrm{P}\mathbf{V}$ and an anisotropic point $P$ on ${L}$, then
there is exactly one point $Q$, denoted by $\textrm{conj}(P,{L})$, which is
conjugate to $P$; it is the meet of ${L}$ and $P^{\mathfrak{A}}$. If
$Q=\textrm{conj}(P,{L})$ is anisotropic, then $P=\textrm{conj}(Q,{L})$. Given
two distinct anisotropic points $P$ and $Q$, we introduce two line segments
$[P,Q]_{+}:=\\{sP{\,+\,}t\,Q|\,s,t{\,\in\,}\mathbb{R},st\geq 0\\}$ and
$[P,Q]_{-}:=\\{sP{\,+\,}t\,Q|\,s,t{\,\in\,}\mathbb{R},st\leq 0\\}$. Thus,
$[P,Q]_{+}$ and $[P,Q]_{-}$ are the closures of the two connected components
of the set $P\sqcup Q-\\{P,Q\\}$.
We define lengths $\mu_{0}([P,Q]_{\pm})$ of these two segments as complex
numbers with imaginary parts in the interval $[\,0,\pi\,]$, determined by the
following conditions:
$\bullet\;\displaystyle\cosh^{2}(\mu_{0}([P,Q]_{+})={\textrm{sign}({P})\,\textrm{sign}({Q})\,(P^{\circ}{{\scriptstyle{[\mathfrak{A}]}\,}}Q^{\circ})^{2}}$
and $\mu_{0}(\,[P,Q]_{-})=\pi\mathbf{i}-\mu(\,[P,Q]_{+})$.
$\bullet$ If $R$ is an anisotropic inner point of $[P,Q]_{+}$, then
$\mu_{0}([P,R]_{+})+\mu_{0}([R,Q]_{+})=\mu_{0}([P,Q]_{+}).$
$\bullet$ $\mu_{0}(\,[P,Q]_{+})=\mu_{0}(\,[P,Q]_{-})=\frac{1}{2}\pi\mathbf{i}$
precisely when $P$ and $Q$ are mutually conjugate.
$\bullet$ If the line $P\sqcup Q$ is elliptic, then
$(P^{\circ}{\scriptstyle{[\mathfrak{A}]}}\,Q^{\circ})^{2}<1$ and
$\mu_{0}(\,[P,Q]_{+})$, $\mu_{0}(\,[P,Q]_{-})$ are purely imaginary.
$\bullet$ If $\xi_{0}(P,Q)=0$, then there is exactly one isotropic point $R$
on $P{\,\sqcup\,}Q\,$.
If $R^{\,}\in[P,Q]_{+}$, then $\mu_{0}([P,Q]_{+})$ $=\pi\mathbf{i}$;
otherwise, $\mu([P,Q]_{+})=0$.
$\bullet$ If the line $P\sqcup Q$ is hyperbolic, then it intersects
$\mathcal{Q}_{\mathfrak{A}}$ transversally in two isotropic points. Here, we
consider two cases:
Case 1: The points $P$ and $Q$ are in the same connected component of
$\mathcal{Q}_{\mathfrak{A}}^{\;\times}$. Both isotropic points on $P\sqcup Q$
are either in $[P,Q]_{+}$ or in $[P,Q]_{-}$. Let $I$ be that of these two
intervals without any isotropic point. Then $\mu_{0}(I)$ is one of two real
numbers with the same absolute value which satisfy the equation
$\cosh^{2}(\mu_{0}(I))=(P^{\circ}{\scriptstyle{[\mathfrak{A}]}}\,Q^{\circ})^{2}$.
Having the choice between a positive or a negative number for $\mu_{0}(I)$, we
decide for the negative number if $P$ and $Q$ are points in
$\mathcal{Q}_{\mathfrak{A}}^{\;-}$, and for the positive number if $P$ and $Q$
are in $\mathcal{Q}_{\mathfrak{A}}^{\;+}$. The length of the other interval is
$\pi\,\mathbf{i}-\mu_{0}(I)$.
Case 2: The points $P$ and $Q$ are in different connected components of
$\mathcal{Q}_{\mathfrak{A}}^{\;\times}$. Let us assume that
${P}\in\mathcal{Q}_{\mathfrak{A}}^{\;+}$ and
$Q\in\mathcal{Q}_{\mathfrak{A}}^{\;-}$. Then
$R\\!:=\text{conj}(P,P{\,\sqcup\,}Q)$ $\in\mathcal{Q}_{\mathfrak{A}}^{\;-}$ ,
and $S\\!:=\text{conj}(Q,P{\,\sqcup\,}Q)\in\mathcal{Q}_{\mathfrak{A}}^{\;+}.$
In this case,
$\mu_{0}(\,[P,Q]_{+})=\,\mu_{0}(\,[P,S]_{+})+\frac{1}{2}\pi\mathbf{i},\;\mu_{0}(\,[P,Q]_{-})=\,-\mu_{0}(\,[P,S]_{+})+\frac{1}{2}\pi\mathbf{i}$
if $S\in[P,Q]_{+}$, and
$\mu_{0}(\,[P,Q]_{+})=-\mu_{0}(\,[P,S]_{+})+\frac{1}{2}\pi\mathbf{i},\;\mu_{0}(\,[P,Q]_{-})=\mu_{0}(\,[R,Q]_{+})+\frac{1}{2}\pi\mathbf{i}$
if $S\in[P,Q]_{-}$.
An analysis of the different cases shows that by knowing the number
$P^{\circ}{{\,\scriptscriptstyle{[\mathfrak{A}]}\,}}\,Q^{\circ}$ and one of
the two numbers ${\textrm{sgn}({P})}$, ${\textrm{sgn}({Q})}$ we can determine
$\mu_{0}([P,Q]_{\pm})$.

We now define the lengths of the segments with one or two isotropic boundary
points, guided by the principle: Isotropic boundary points are distributed
equally to adjacent segments.
Given an anisotropic point and a double point on a parabolic line, then the
length of each of the two segments with these points as boundary points is
$\frac{1}{2}\pi\mathbf{i}$.
If $P$ is an anisotropic and $Q$ an isotropic point on a hyperbolic line, then
the line splits into two segments with these points as boundary points. One
contains still another isotropic point while the other does not. The length of
the first segment is $-\textrm{sgn}({P})\infty+\frac{3}{4}\pi\mathbf{i}$, the
length of the second $\textrm{sgn}({P})\,\infty+\frac{1}{4}\pi\mathbf{i}$.
A hyperbolic line ${L}$ is a disjoint union of the three sets
${L}^{0}:={L}\cap\mathcal{Q}_{\mathfrak{A}},{L}^{\pm}:={L}\cap\mathcal{Q}_{\mathfrak{A}}^{\;\pm}$.
The length of the open segment ${L}^{\pm}$ is $\pm\infty$, the length of the
closed segment ${L}^{\pm}\cup{L}_{0}$ is
$\;\pm\infty+\frac{1}{2}\pi\mathbf{i}$.

In order to define a distance of anisotropic points, we introduce an order
$\prec$ on the complex numbers by
$a_{1}+b_{1}\mathbf{i}\prec
a_{2}+b_{2}\mathbf{i}\;\;\;\text{iff}\;\;\;\begin{cases}&b_{1}<b_{2}\vspace*{-2
mm}\\\ \text{or}\vspace*{-2 mm}&\\\ &b_{1}=b_{2}\;\;\text{and
}\,a_{1}<a_{2}\;.\end{cases}$
Put
$\mathbb{D}:=\\{a{\,+\,}b{\,}\mathbf{i}\,|\,a\in{\mathbb{R}},b\in[0,\frac{1}{2}\pi]\\}$
and define the function
$d_{0}{:\,}{Q}_{\mathfrak{A}}^{\times}\times\mathcal{Q}_{\mathfrak{A}}^{\times}\rightarrow\mathbb{D}$
by

$d_{0}(P,Q)=\begin{cases}0,\;\;\text{if }P=Q\;,\\\
\mu_{0}([P,Q]_{+}),\;\text{if }\,P\neq
Q\;\,\textrm{and}\;\,\mu_{0}([P,Q]_{+})\prec\mu_{0}([P,Q]_{-})\;,\\\
\mu_{0}([P,Q]_{-}),\;\text{otherwise}\,.\end{cases}$

This function $d_{0}$ is continuous with respect to the canonical topology on
$\text{P}V$, and we call $d_{0}(P,Q)$ the gross distance between the
(anisotropic) points $P$ and $Q$.
Remarks: (1) On an elliptic line, the distance of two points is uniquely
determined by its value under $\textrm{cosh}.$ The situation is different for
a hyperbolic line $P{\,\sqcup\,}Q$; in order to determine the distance
$d_{0}(P,Q)$, we also need to know one of the numbers
$\textrm{sgn}({P}),\textrm{sgn}({Q})$.
(2) This distance function can be extended to isotropic points, as long as
both points are not points on a totally isotropic line. But this extension is
not continuous.
(3) So far, no distance has been defined yet between points on the total
vertex ${U}$ of $\mathcal{Q}$. This can be achieved by declaring a quadric in
${U}$ as the absolute quadric, such that $U$ becomes a CK space, see Section
3.

### 2.4.

The quadrance of two equidimensional anisotropic planes. Let $A_{1}$ and
$A_{2}$ be two anisotropic planes of the same dimension $s{\,\leq\,}n$
generated by points
$P_{1}=\mathbb{R}\boldsymbol{p}_{1}{,\dots,}P_{s+1}=\mathbb{R}\boldsymbol{p}_{s+1}$
and
$Q_{1}=\mathbb{R}\boldsymbol{q}_{1}{,\dots,}Q_{s+1}=\mathbb{R}\boldsymbol{q}_{s+1}$,
respectively. We define two real numbers

$\displaystyle\zeta_{0}(A_{1},A_{2}){\;:=\;}\frac{(\det(\boldsymbol{p}_{i}{\scriptstyle{[\mathfrak{A}]}}\boldsymbol{q}_{i})_{1\leq
i,j\leq
s+1})^{2}}{(\det(\boldsymbol{p}_{i}{\scriptstyle{[\mathfrak{A}]}}\boldsymbol{p}_{j})_{1\leq
i,j\leq
s+1})(\det(\boldsymbol{q}_{i}{\scriptstyle{[\mathfrak{A}]}}\boldsymbol{q}_{j})_{1\leq
i,j\leq s+1})}\;\;\;\;(\ddagger)$

and ${\hskip 30.72897pt}\xi_{0}(A_{1},A_{2}){\;:=\;}1-\zeta_{0}(A_{1},A_{2})$.
If $A_{1},A_{2}$ are points, $\xi_{0}(A_{1},A_{2})$ is the quadrance of the
two points. We adopt this name for higher dimensional planes of the same
dimension.

### 2.5.

Reflections on a semi CK space are isometries.
Let $\phi\in\textrm{Aut}(\textrm{P}\boldsymbol{V})$ be a reflection in an
anisotropic plane $U\leq\textrm{P}\boldsymbol{V}$ and let $A_{1}$,
$A_{2}\leq\textrm{P}\boldsymbol{V}$ be two anisotropic planes of the same
dimension, then $\xi_{0}(\phi(A_{1}),\phi(A_{2}))=\xi_{0}(A_{1},A_{2})$.
Proof. First, we give a proof for the special case
$\textrm{dim}(A_{1})=\textrm{dim}(A_{2})=0$. In this case, $\phi(A_{1})$ and
$\phi(A_{2})$ are anisotropic points. Put
$M{:=\,}U\sqcap(A_{1}\sqcup\phi(A_{1}))$,
$\tilde{M}{:=\,}U^{\mathfrak{A}}\sqcap(A_{1}\sqcup\phi(A_{1}))$,
$N{:=\,}U\sqcap(A_{2}\sqcup\phi(A_{2}))$,
$\tilde{N}{:=\,}U^{\mathfrak{A}}\sqcap(A_{2}\sqcup\phi(A_{2}))$. Only one of
the following four cases can occur:
(1)
$M=A_{1}+\phi(A_{1}),\tilde{M}=A_{1}-\phi(A_{1}),N=A_{2}+\phi(A_{2}),\tilde{N}=A_{2}-\phi(A_{2})$,
(2)
$M=A_{1}-\phi(A_{1}),\tilde{M}=A_{1}+\phi(A_{1}),N=A_{2}+\phi(A_{2}),\tilde{N}=A_{2}-\phi(A_{2})$,
(3)
$M=A_{1}+\phi(A_{1}),\tilde{M}=A_{1}-\phi(A_{1}),N=A_{2}-\phi(A_{2}),\tilde{N}=A_{2}+\phi(A_{2})$,
(4)
$M=A_{1}-\phi(A_{1}),\tilde{M}=A_{1}+\phi(A_{1}),N=A_{2}-\phi(A_{2}),\tilde{N}=A_{2}+\phi(A_{2})$.
These cases can be treated essentially the same, therefore we only look at the
first case: We have
$0=(A_{1}^{\,\circ}-\phi(A_{1})^{\circ}){\scriptstyle{[\mathfrak{A}]}}(A_{2}^{\,\circ}+\phi(A_{2})^{\circ})$
and
$0=(A_{1}^{\,\circ}+\phi(A_{1})^{\circ}){\scriptstyle{[\mathfrak{A}]}}(A_{2}^{\circ}-\phi(A_{2})^{\circ})$
and therefore
$\phi(A_{1})^{\circ}{\scriptstyle{[\mathfrak{A}]}}A_{2}^{\,\circ}=\phi(A_{2})^{\circ}{\scriptstyle{[\mathfrak{A}]}}A_{1}^{\,\circ}$.
From this follows that $\xi(A_{1},\phi(A_{2}))=\xi(A_{2},\phi(A_{1}))$. By
interchanging the roles of $A_{2}$ and $\phi(A_{2})$ we get
$\xi(\phi(A_{1}),\phi(A_{2}))=\xi(A_{1},A_{2}).$
We now assume that $A_{1}$ is generated by independent anisotropic points
$P_{1},\dots,P_{s}$ and $A_{2}$ is generated by independent anisotropic points
$Q_{1},\dots,Q_{s}$. Then $\xi_{0}(X,Y)=\xi_{0}(\phi(X),\phi(Y))$ for all
$X,Y{\in\,}\\{P_{1},\dots P_{s},Q_{1},\dots,Q_{s}\\}$ and
$\xi_{0}(A_{1},A_{2})=\xi_{0}(\phi(A_{1}),\phi(A_{2})).$$\,\Box$

### 2.6.

Angles between planes. We define: Two planes
$A_{1},A_{2}\leq\textrm{P}\boldsymbol{V}$ form dihedral angles if
$\textrm{dim}(A_{1}\sqcap A_{2})=\textrm{dim}(A_{1})-1=\textrm{dim}(A_{2})-1$.
Let $A_{1},A_{2}$ be two planes which form dihedral angles. Put
$H:=A_{1}\sqcup A_{2}$ and $U:=A_{1}\sqcap A_{2}$.
$H{\smallsetminus}{\smallsetminus}\;(A_{1}\cup A_{2})$ consists of two
connected components. We call the closure of these components (dihedral)
angles. $A_{1}$ and $A_{2}$ are the sides of these angles, and $U$ is their
vertex.
Example: Two distinct points form dihedral angles with vertex $0$.
If the vertex $U$ is nonempty and anisotropic, $U^{\mathfrak{A}}\sqcap H$ is a
line which meets $A_{1}{\smallsetminus}{\smallsetminus}\,U$ and
$A_{2}{\smallsetminus}{\smallsetminus}\,U$ in points $P_{1}$ and $P_{2}$,
respectively. The segment $[P_{1};P_{2}]_{+}$ belongs to one angle, the
segment $[P_{1},P_{2}]_{-}$ to the other. We notate the first angle by
$[A_{1},A_{2}]_{+}$, the second by $[A_{1},A_{2}]_{-}$. If $P_{1}$ and $P_{2}$
are anisotropic, then $\xi_{0}(A_{1},A_{2})=\xi_{0}(P_{1},P_{2})$. Thus, we
put $d_{0}(A_{1},A_{2}):=d_{0}(P_{1},P_{2})$.
If the vertex is isotropic while the two sides $A_{1},A_{2}$ are anisotropic,
then $\xi_{0}(A_{1},A_{2})=0$ and $d_{0}(A_{1},A_{2})=0$.
Two examples:
(1) If
$\boldsymbol{V}=\mathbb{R}^{4},\mathfrak{A}=\textrm{diag}(1,1,-1,-1),P_{1}=(1{:}1{:}0{:}0),P_{2}=(1{:}{-}1{:}1{:}0),$
$P_{3}=(0{:}0{:}1{:}0)$, then
$P_{1}{\scriptstyle{[\mathfrak{A}]}}P_{2}=0=P_{1}{\scriptstyle{[\mathfrak{A}]}}P_{3}$
and $\zeta_{0}(P_{1}\sqcup P_{2},P_{1}\sqcup P_{3})=-1$, $\mu([P_{1}\sqcup
P_{2},P_{1}\sqcup
P_{3}]_{+})=\mu([P_{2},P_{3}]_{+})=\ln(\sqrt{2}+1)+\frac{1}{2}\pi\mathbf{i},$
$d_{0}(P_{1}\sqcup P_{2},P_{1}\sqcup
P_{3})=\mu([P_{2},P_{3}]_{-})=\ln(\sqrt{2}-1)+\frac{1}{2}\pi\mathbf{i}$.
(2) If
$\boldsymbol{V}=\mathbb{R}^{3},\mathfrak{A}=\textrm{diag}(1,1,0),P_{1}=(1{:}1{:}1),P_{2}=(-1{:}1{:}1),P_{3}=(1{:}-1{:}1)$,
then $\mu([P_{1}\sqcup P_{2},P_{1}\sqcup
P_{3}]_{+})=\mu([P_{2},P_{3}]_{+})=\pi\mathbf{i}$ and $\xi_{0}(P_{1}\sqcup
P_{2},P_{1}\sqcup P_{3})=\xi_{0}(P_{2},P_{3})=0$.
The distance $d_{0}(A_{1},A_{2})$ between almost all planes $A_{1},A_{2}$
forming dihedral angles is nonzero as long as their dimension is less than
$\textrm{rank}{(\mathfrak{A})}-1$. If their dimension exceeds
$\textrm{rank}{(\mathfrak{A})}-2$, their distance is always 0.

### 2.7.

Projections and parallels. Consider some point $P=\mathbb{R}\boldsymbol{p}$
and some anisotropic plane ${U}=\mathbb{R}\boldsymbol{U}$ of dimension $k<n$
with $P\notin{U}^{\mathfrak{A}}$. The perpendicular from $P$ to ${U}$ is the
$(n{-}k)$-plane $\,\text{perp}(P,{U}):=P\sqcup{U}^{\mathfrak{A}}$. This plane
meets ${U}$ at the point $\text{ped}(P,{U}):=\text{perp}(P,{U})\sqcap{U}$, the
projection or pedal of $P$ in ${U}$. If $P\in{U}$, then $\text{ped}(P,{U})=P.$
Regardless of the choice of point $P$, at least one of the points
$\textrm{ped}(P,{U}),\textrm{ped}(P,{U}^{\mathfrak{A}})$ is well-defined. If
$P$ is a point neither on ${U}$ nor on ${U}^{\mathfrak{A}}$, then both are
well-defined and together with $P$ they form a collinear triad.
If $P\in{U}^{\mathfrak{A}}$, then $d_{0}(P,{U})=\frac{1}{2}\pi\,\mathbf{i}$,
otherwise $d_{0}(P,{U})=d_{0}(P;\textrm{ped}(P,U))$.
If ${U}$ is a hyperplane with ${U}^{\mathfrak{A}}=Q$ and $P\neq Q$, then
$\text{ped}(P,{U})=({Q}{\,\scriptstyle{[\mathfrak{A}]}\,}{P}){Q}-(Q{\,\scriptstyle{[\mathfrak{A}]}\,}Q){P}$.
Suppose ${A}$ is a second plane with $\textrm{dim}\,{A}\leq\textrm{dim}\,{U}$
and ${A}\sqcap{U}^{\mathfrak{A}}=\emptyset$. Then
ped$({A},{U}):=\bigcup_{P\in{A}}\textrm{ped}(P,{U})\leq{U}$ is the pedal of
${A}$ in ${U}$, this is a plane of the same dimension as ${A}$. As distance
$d_{0}({A},{U})$ we define the distance between ${A}$ and
$\textrm{ped}({A},{U})$.
Example: $n=4$, $\mathfrak{A}=\textrm{diag}(1,1,1,1,0)$.
Put $U=E_{1}\sqcup E_{2}\sqcup
E_{3},P=(1{:}0{:}0{:}1{:}0),Q=(0{:}1{:}0{:}1{:}0),A=P\sqcup Q$. Then:
ped$(A,U)=E_{1}\sqcup E_{2}$ and
$\displaystyle\xi_{0}(A,U)=\xi_{0}(A,E_{1}\sqcup E_{2})={2}/{3}$.

We call the $k$-plane $\text{par}(P,{U}):=({U}\sqcap P^{\mathfrak{A}})\sqcup
P$ the plane through $P$ parallel to ${U}$.
Caution: In hyperbolic geometry two planes are often defined to be parallel to
each other if their intersection is an isotropic set. Both concepts of
parallelism have to be kept entirely apart.

### 2.8.

The angle between a line and a plane. Let $H$ be an anisotropic plane and let
$L$ be an anisotropic line. We assume that $L$ meets $H$ in one point $P$. The
lines $L$ and $P\sqcup\textrm{ped}(L;H)$ form two angles with vertex $P$. If
$Q$ is a point on $L$ and $Q\neq P$, then

$\displaystyle\sinh(\,\mu(\angle_{1}))=\sinh(\,\mu(\angle_{2}))=\frac{\sinh(d_{0}(Q;\textrm{ped}(Q;H))}{\sinh(d_{0}(Q,P))}=\frac{\cosh(d_{0}(Q,H^{\mathfrak{A}})}{\sinh(d_{0}(Q,P))}\,\mathbf{i}$
.

If $L$ is a line totally inside the hyperplane $H$, then the angle distance
between $L$ and $H$ is put to $0$.

### 2.9.

Symmetry points, midpoints, segment bisectors and angle bisectors. Let ${U}$
be an anisotropic plane and ${S}$ be a nonempty subset of
$\textrm{P}\boldsymbol{V}$. If ${S}$ is invariant under the reflection in $U$,
then ${U}$ and $U^{\mathfrak{A}}$ are called symmetry axes of ${S}$. If a
symmetry axis consists of only one point, then this point is called symmetry
point of ${S}$.
Let $P$ and $Q$ be two distinct anisotropic points in
$\textrm{P}\boldsymbol{V}$. If $\textrm{sgn}(P)\neq\textrm{sgn}(Q)$, then
$\\{P,Q\\}$ has no symmetry points. If both points have the same sign, then
the points $P{\,+\,}Q$ and $P{\,-\,}Q$ are called the midpoints of
$\\{P,Q\\}$. $P{\,+\,}Q$ is called the midpoint of $[P,Q]_{+}$ and $P{\,-\,}Q$
the midpoint of $[P,Q]_{-}$. At least one of the two midpoints is anisotropic
and a reflection in this point leaves both midpoints fixed. Thus, both
midpoints are symmetry points of $\\{P,Q\\}$.
If $P\sqcup Q$ is an anisotropic line and $P{\,\pm\,}Q$ an anisotropic point,
$(P{\,\pm\,}Q)^{\mathfrak{A}}$ is a perpendicular bisector of the segment
$[P,Q]_{\mp}$.

Let $A_{1},A_{2}$ be two distinct anisotropic planes of the same dimension,
and let $\angle(A_{1},A_{2})$ be an angle with sides $A_{1}$, $A_{2}$ and
vertex $U{:=}A_{1}\sqcap A_{2}$ situated in a anisotropic plane $A_{1}\sqcup
A_{2}$. If $U$ is anisotropic, the line $L:=U^{\mathfrak{A}}\sqcap A$ meets
$A_{1}\cup A_{2}$ in two points. If the segment $\angle(A_{1},A_{2})\sqcap L$
has a midpoint $M$, $M\sqcup\,U$ is called angle bisector of
$\angle(H_{1},H_{2})$.

### 2.10.

Distance of two equidimensional anisotropic planes.
Let $A_{1},A_{2}$ $\leq\mathrm{P}\boldsymbol{V}$ be two distinct anisotropic
planes of the same dimension $m>0$. We put $E{\,:=\,}A_{1}\sqcup A_{2}$,
$A_{1}^{\prime}:=E\sqcap A_{1}^{\mathfrak{A}}$, $A_{2}^{\prime}:=E\sqcap
A_{2}^{\mathfrak{A}}$ and assume that $E$ is anisotropic and that $A_{1}\sqcap
A_{2}^{\prime}=A_{2}\sqcap A_{1}^{\prime}=0$.
Then there are points $Q_{1},\dots,Q_{m+1}\in A_{1}$ and
$R_{1},\dots,R_{m+1}\in A_{2}$ such that
(1)
${\textrm{cosh}(d_{0}(Q_{i},Q_{j}))\;=\textrm{cosh}(d_{0}(R_{i},R_{j}))=0,\,1\leq
i<j\leq m+1}$.
(2) ${\textrm{cosh}(d_{0}(Q_{i},R_{j}))=0\,,\,i\neq j}$.
(3) If $Q_{i}\neq R_{i}$, then $Q_{i}\sqcup R_{i}$ is a line that meets both
planes, $A_{1}$ and $A_{2}$, perpendicularly.
(4)
$\zeta_{0}(A_{1},A_{2})=\prod_{i=1}^{m+1}(\textrm{cosh}(d_{0}(Q_{i},R_{i})))^{2}$.
Proof. The proof will be given in several steps.
We first assume that $A_{1}$ and $A_{2}$ are disjoint planes.
The theorem is true if $A_{1}$ and $A_{2}$ are lines that do not intersect.
The proof of this statement is rather technical and we use a CAS-system for
calculations. Let $P_{1},P_{2},P_{3},P_{4}$ be four anisotropic points and
$A_{1}=P_{1}\sqcup P_{2},A_{2}=P_{3}\sqcup P_{4}$. Put
$c_{ij}:=P_{i}^{\circ}{\scriptstyle{[\mathfrak{A}]}}P_{j}^{\circ},1\leq
i<j\leq 4$. For our purposes we may assume that $c_{12}=c_{34}=0$. We
introduce the number
$w:=\sum\limits_{i=1}^{2}\sum\limits_{k=3}^{4}\big{(}c_{{ii}}^{\;2}c_{kk}^{\;2}c_{jl}^{4}+2c_{ii}^{\;2}c_{kk}c_{ll}c_{jk}^{\;2}c_{jl}^{\;2}+2c_{kk}^{\;2}c_{ii}c_{jj}c_{il}^{\;2}c_{jl}^{\;2}\big{)}\\\
\hskip
156.49014pt+\;2c_{11}c_{22}c_{33}c_{44}\big{(}c_{13}^{\;2}c_{24}^{\;2}-4c_{13}c_{14}c_{23}c_{24}+c_{14}^{\;2}c_{23}^{\;2}\big{)}$,
where the indices are chosen such that $\\{i,j\\}=\\{1,2\\}$ and
$\\{k,l\\}=\\{3,4\\}$.
We show that this number $w$ is a non-negative real number by examining four
different cases.
(1) Both lines $A_{1}=P_{1}\sqcup P_{2}$ and $A_{2}=P_{3}\sqcup P_{4}$ are
elliptic lines, and both lie in $\mathcal{Q}^{+}$ or both in
$\mathcal{Q}^{-}$. Thus, $c_{11}=c_{22}=c_{33}=c_{44}\in\\{-1,1\\}$ and
$c_{13},c_{14},c_{23},c_{24}\in\mathbb{R}$. In this case,

$w=\big{(}(c_{13}+c_{23})^{2}+(c_{14}+c_{24})^{2}\big{)}\big{(}(c_{13}-c_{23})^{2}+(c_{14}-c_{24})^{2}\big{)}\in\mathbb{R}^{\geq
0}$.

(2) Both lines $A_{1}$ and $A_{2}$ are elliptic lines, $P_{1}\sqcup
P_{2}\subseteq\mathcal{Q}^{+}$, $P_{3}\sqcup P_{4}\subseteq\mathcal{Q}^{-}$.
Here, $c_{11}=c_{22}=1,c_{33}=c_{44}=-1$ and
$c_{13},c_{14},c_{23},c_{24}\in\mathbb{R}\mathbf{i}$. With
$c_{13}=c_{13}^{*}\mathbf{i},c_{14}=c_{14}^{*}\mathbf{i},c_{23}=c_{23}^{*}\mathbf{i},c_{24}=c_{24}^{*}\mathbf{i}$,
we get

$w=\big{(}(c_{13}^{*}+c_{23}^{*})^{2}+(c_{14}^{*}+c_{24}^{*})^{2}\big{)}\big{(}(c_{13}^{*}-c_{23}^{*})^{2}+(c_{14}^{*}-c_{24}^{*})^{2}\big{)}\in\mathbb{R}^{\geq
0}$.

(3) $A_{1}$ is elliptic, $A_{2}$ hyperbolic;
$c_{11}{\,=\,}c_{22}{\,=\,}c_{33}{\,=\,}1,c_{44}{\,=\,}-1;c_{13},c_{23}\in\mathbb{R}$,
$c_{14},c_{24}\in\mathbb{R}\mathbf{i}$. Then,

$w=\big{(}(c_{13}+c_{24}\mathbf{i})^{2}+(c_{23}-c_{14}\mathbf{i})^{2}\big{)}\big{(}(c_{13}-c_{24}\mathbf{i})^{2}+(c_{23}+c_{14}\mathbf{i})^{2}\big{)}\in\mathbb{R}^{\geq
0}$.

(4) Both lines $A_{1}$ and $A_{2}$ are hyperbolic lines;
$c_{11}{\,=\,}c_{33}{\,=\,}1,c_{22}{\,=\,}c_{44}{\,=\,}-1$;
$c_{13},c_{24}\in\mathbb{R},$ $c_{14},c_{23}\in\mathbb{R}\mathbf{i}$. With
$c_{14}=c_{14}^{*}\mathbf{i},c_{23}=c_{24}^{*}\mathbf{i}$, we get

$w=\big{(}(c_{13}+c_{24})^{2}+(c_{14}^{*}+c_{23}^{*})^{2}\big{)}\big{(}(c_{13}-c_{24})^{2}+(c_{14}^{*}-c_{23}^{*})^{2}\big{)}\in\mathbb{R}^{\geq
0}$.

Define points $Q_{1},Q_{2}$ on $P_{1}\sqcup P_{2}$ and $R_{1},R_{2}$ on
$P_{3}\sqcup P_{4}$ by
$Q_{1}:=(-\sqrt{w}+q+2(c_{13}c_{23}c_{22}c_{44}+c_{14}c_{24}c_{22}c_{33})P_{1}$
$+\;(+\sqrt{w}+q-2(c_{13}c_{23}c_{11}c_{44}+c_{14}c_{24}c_{11}c_{33})P_{2}$,
$Q_{2}:=(+\sqrt{w}+q+2(c_{13}c_{23}c_{22}c_{44}+c_{14}c_{24}c_{22}c_{33})P_{1}$
$+\;(-\sqrt{w}+q-2(c_{13}c_{23}c_{11}c_{44}+c_{14}c_{24}c_{11}c_{33})P_{2}$,

with
$q:=c_{11}(c_{33}c_{24}^{\;2}+c_{44}c_{23}^{\;2})-c_{22}(c_{33}c_{14}^{\;2}+c_{44}c_{13}^{\;2})$,

$R_{1}:=(-\sqrt{w}+r+2(c_{13}c_{23}c_{22}c_{44}+c_{14}c_{24}c_{22}c_{33})P_{3}$
$+\;(+\sqrt{w}+r-2(c_{13}c_{23}c_{11}c_{44}+c_{14}c_{24}c_{11}c_{33})P_{4}$,
$R_{2}:=(+\sqrt{w}+r+2(c_{13}c_{23}c_{22}c_{44}+c_{14}c_{24}c_{22}c_{33})P_{3}$
$+\;(-\sqrt{w}+r-2(c_{13}c_{23}c_{11}c_{44}+c_{14}c_{24}c_{11}c_{33})P_{4}$,

with
$r:=c_{33}(c_{11}c_{24}^{\;2}+c_{22}c_{14}^{\;2})-c_{44}(c_{11}c_{23}^{\;2}+c_{22}c_{13}^{\;2})$.

The mapping
$(A_{1}{\smallsetminus}{\smallsetminus}\,\mathcal{Q})\times(A_{2}{\smallsetminus}{\smallsetminus}\,\mathcal{Q})\to\mathbb{C},\,(S_{1},S_{2})\,\mapsto\,S_{1}{\scriptstyle{[\mathfrak{A}]}}S_{2}\,$,
is stationary at $(Q_{1},R_{1})$ and at $(Q_{2},R_{2})$,
and $Q_{1}\sqcup R_{1}$ and $Q_{2}\sqcup R_{2}$ are the two lines that meet
each of the sets $A_{1},A_{2},A_{1}^{\prime},A_{2}^{\prime}$. Thus,
$Q_{1}{=\,}\textrm{ped}(R_{1},P_{1}\sqcup P_{2})$,
$R_{1}{=\,}\textrm{ped}(Q_{1};P_{3}\sqcup P_{4})$,
$Q_{2}{=\,}\textrm{ped}(R_{2};P_{1}\sqcup P_{2})$,
$R_{2}{=\,}\textrm{ped}(Q_{2};P_{3}\sqcup P_{4}).$
$d_{0}(Q_{1},Q_{2})=d_{0}(R_{1},R_{2})=d_{0}(Q_{1},R_{2})=d_{0}(R_{1},Q_{2})=\frac{1}{2}\pi\mathbf{i}$.
Since $A_{1}\cap A_{2}=\emptyset$, we therefore get $Q_{2}\sqcup
R_{2}\leq(Q_{1}\sqcup R_{1})^{\mathfrak{A}}.$
Moreover, $\;\zeta_{0}(Q_{1}\sqcup Q_{2},R_{1}\sqcup R_{2})$
$\displaystyle=\frac{\big{(}\det\\!{\left(\begin{array}[]{cc}\\!\\!\\!\scriptstyle{Q_{1}^{\circ}}{\scriptscriptstyle{[\mathfrak{A}]}}\scriptstyle{R_{1}^{\circ}}&\\!\\!\\!\scriptstyle{0}\\!\\!\\!\\\
\\!\\!\\!\scriptstyle{0}&\\!\\!\\!\scriptstyle{Q_{2}^{\circ}}{\scriptscriptstyle{[\mathfrak{A}]}}\scriptstyle{R_{2}^{\circ}}\\!\\!\\!\\\
\end{array}\right)}\big{)}^{2}}{{\det\\!{\left(\begin{array}[]{cc}\\!\\!\\!\scriptstyle{Q_{1}^{\circ}}{\scriptscriptstyle{[\mathfrak{A}]}}\scriptstyle{Q_{1}^{\circ}}&\\!\\!\\!\scriptstyle{0}\\!\\!\\!\\\
\\!\\!\\!\scriptstyle{0}&\\!\\!\\!\scriptstyle{Q_{2}^{\circ}}{\scriptscriptstyle{[\mathfrak{A}]}}\scriptstyle{Q_{2}^{\circ}}\\!\\!\\!\\\
\end{array}\right)}}{\det\\!{\left(\begin{array}[]{cc}\\!\\!\\!\scriptstyle{R_{1}^{\circ}}{\scriptscriptstyle{[\mathfrak{A}]}}\scriptstyle{R_{1}^{\circ}}&\\!\\!\\!\scriptstyle{0}\\!\\!\\!\\\
\\!\\!\\!\scriptstyle{0}&\\!\\!\\!\scriptstyle{R_{2}^{\circ}}{\scriptscriptstyle{[\mathfrak{A}]}}\scriptstyle{R_{2}^{\circ}}\\!\\!\\!\\\
\end{array}\right)}}}$
=
$\big{(}\cosh(d_{0}(Q_{1},R_{1}))\big{)}^{2}\big{(}\cosh(d_{0}(Q_{2},R_{2}))\big{)}^{2}$.
Let us now assume that $m>1$ and that the statement holds for $m-1$.
First of all, there exists (at least) one line $L$ which meets the planes
$A_{1},A_{2},A_{1}^{\prime},A_{2}^{\prime}$. This is a consequence of a lemma
in enumerative geometry (Schubert calculus) which states that, if four
$m$-planes in a projective space of dimension $2m+1$ are in a ”general
position”, then there exist $m+1$ lines which meet all these planes, see [23]
for a proof.
If $L$ meets $A_{1}$ at $Q_{1}$, then there exist anisotropic points
$Q_{2},\dots,Q_{m+1}\in A_{1}$ such that
$d_{0}(Q_{i},Q_{j})=\frac{1}{2}\pi\mathbf{i},1{\leq}i{<}j{\leq\,}m{+}1$, and
if $L$ meets $A_{2}$ at $R_{1}$, then there are anisotropic points
$R_{2},\dots,R_{m+1}\in A_{2}$ with
$d_{0}(R_{i},R_{j})=\frac{1}{2}\pi\mathbf{i},1{\leq}i{<}j{\leq\,}m{+}1$.
The proof for $m=1$ shows that $d_{0}(Q_{i},R_{j})=\frac{1}{2}\pi\mathbf{i}$
for $\,i\neq j$. $U_{1}{:=\,}\textrm{span}(Q_{2},\dots,Q_{m+1})$ and
$U_{2}{:=\,}\textrm{span}(R_{2},\dots,R_{m+1})$ are disjoint $(m{-}1)$-planes
in $(P_{1}\sqcup Q_{1})^{\mathfrak{A}}$. We apply formula $(\ddagger)$ getting
$\zeta_{0}(A_{1},A_{2})=\big{(}\cosh(d_{0}(P_{1},Q_{1}))\big{)}^{2}\,\zeta_{0}(U_{1},U_{2})$.
We complete the proof for disjoint planes $A_{1}$ and $A_{2}$ by including the
induction assumption.
We now assume that $A_{1}$ and $A_{2}$ intersect in a non-empty subspace $U$
of dimension $s$. We can find points $Q_{1},\dots,Q_{m+1}$ which span $A_{1}$
such that $\textrm{span}(Q_{1},\dots,Q_{s+1})=U$ and
$\textrm{cosh}(d_{0}(Q_{i},Q_{j}))=0$ for
${1{\,\leq\,}i{\,<\,}j{\leq\,}}m{+}1$. Put
$R_{i}=Q_{i},\,1\leq\\!i\leq\\!s{+}1$. We can find points
$R_{s+2},\dots,R_{m+1}$ such that $\textrm{cosh}(d_{0}(R_{i},R_{j}))=0$ for
${1{\,\leq\,}i{\,<\,}j{\leq\,}}m{+}1$. The intersections
$A_{1}\sqcap\,U^{\mathfrak{A}}$ and $A_{2}\sqcap\,U^{\mathfrak{A}}$ are two
disjoint subspaces of equal dimension in $U^{\mathfrak{A}}$. With formula
$(\ddagger)$ we get $\zeta_{0}(A_{1},A_{2})=\zeta_{0}(A_{1}\sqcap
U^{\mathfrak{A}},A_{2}\sqcap U^{\mathfrak{A}}\,)$. We can apply the theorem to
the subspaces $A_{1}\sqcap U^{\mathfrak{A}}$ and $A_{2}\sqcap
U^{\mathfrak{A}}$ and finish the proof. $\scriptstyle{\Box}$

### 2.11.

The function $\psi$ and staudtian of a finite set of points.
We introduce a function $\psi$ which is defined on the powerset of
$\textrm{P}\boldsymbol{V}$ and has values in $\mathbb{R}$ by:
$\psi(\mathcal{S})=0$, unless $\mathcal{S}$ consists of finitely many
anisotropic points. If $\mathcal{S}$ consists of $s\in\mathbb{N}^{+}$
anisotropic points ${P_{1},\dots,P_{s}}$, then
$\displaystyle\psi(\mathcal{S})={\det((P_{i}^{\circ}{\scriptstyle{[\mathfrak{A}]\,}}P_{j}^{\circ})_{1\leq\,i,j\leq\,s}})\,$.
If $\psi(\mathcal{S})\neq 0$, we call the number
$\frac{1}{(s{-}1)!}\sqrt{|\psi(\mathcal{S})|}$ the staudtian of $\mathcal{S}$
(cf. [13, 7] for the case $n{\,=\,}2$).
Given two anisotropic points $P_{1},P_{2}$, then
$\psi(\\{P_{1},P_{2}\\})=\xi_{0}(P_{1},P_{2})=-\sinh^{2}(d_{0}(P_{1},P_{2}))$.
Let $P_{1}{,\dots,}P_{s}$ be independent anisotropic points and
$\mathcal{S}{=}\\{P_{1}{,\dots,}P_{s}\\}$. If the plane span$(\mathcal{S})$ is
isotropic, then $\psi(\mathcal{S})=0.$

If the planes span$(\mathcal{S})$ and $P_{1}{\sqcup\dots\sqcup}P_{s-1}$ are
anisotropic, then

$-\psi(\mathcal{S}{\smallsetminus}{\smallsetminus}\\{P_{s}\\})\sinh^{2}(d_{0}(P_{s},\textrm{span}(\mathcal{S}{\smallsetminus}{\smallsetminus}\\{P_{s}\\}))){\,=\,}\,\psi(\mathcal{S})$.

If ${P_{1},\dots,P_{s}}$ are anisotropic and
$Q=q_{1}P_{1}{\,+\dots+\,}q_{s}P_{s}$ with
$q_{1}q_{2}{\cdots}q_{s}{\,\neq\,}0$ is another anisotropic point, dependent
on $P_{1}{\,,\dots,\,}P_{s}$, then for $0<i,j\leq s$

$\displaystyle{\psi(\mathcal{S}{\smallsetminus}{\smallsetminus}\\{P_{i}\\}\cup\\{Q\\})}{\,q_{j}^{\,2}}{\,=\,}{\psi(\mathcal{S}{\smallsetminus}{\smallsetminus}\\{P_{j}\\}\cup\\{Q\\})}{\,q_{i}^{\;2}}\,$.

### 2.12.

Trigonometry.
Three independent points $P_{1},P_{2},P_{3}$ in $\textrm{P}\boldsymbol{V}$
form a triangle. We assume that the plane ${P_{1}\sqcup P_{2}\sqcup P_{3}}$
and the three sidelines $P_{2}\sqcup P_{3},P_{3}\sqcup P_{1},P_{1}\sqcup
P_{2}$ of this triangle are anisotropic. The sidelengths are
$a_{1}:=\mu([P_{2},P_{3}]_{+}),a_{2}:=\mu([P_{3},P_{1}]_{+}),a_{3}:=\mu([P_{1},P_{2}]_{+})$;
the measures of the interior angles are $\alpha_{1}:=\mu([P_{1}\sqcup
P_{2},P_{1}\sqcup P_{3}]_{+}),\alpha_{2}:=\mu([P_{2}\sqcup P_{3},P_{2}\sqcup
P_{1}]_{+}),\alpha_{3}:=\mu([P_{3}\sqcup P_{1},P_{1}\sqcup P_{2}]_{+})$. We
mention that

$\hskip 3.0pt\sinh(a_{1})=\sinh(d_{0}(P_{2},P_{3})),\dots,\hskip
20.0pt\sinh(\alpha_{1})\;=\sinh(d_{0}(P_{1}\sqcup P_{2},P_{1}\sqcup
P_{3})),\dots,$

$\;\,\cosh^{2}(a_{1})=\cosh^{2}(d_{0}(P_{2},P_{3})),\dots,\hskip
10.0pt\cosh^{2}(\alpha_{1})=\cosh^{2}(d_{0}(P_{1}\sqcup P_{2},P_{1}\sqcup
P_{3})),\dots$.

The law of sines:

$\displaystyle\sinh^{2}(a_{1}):\sinh^{2}(\alpha_{1})\;=\;\sinh^{2}(a_{2}):\sinh^{2}(\alpha_{2})\;=\;\sinh^{2}(a_{3}):\sinh^{2}(\alpha_{3})\,$.

Two laws of cosines:

$\displaystyle\cosh^{2}(\alpha_{1})=\big{(}\frac{\cosh(a_{1})-\cosh(a_{2})\cosh(a_{3})}{\sinh(a_{2})\sinh(a_{3})}\big{)}^{2}$

$\hskip
6.0pt\displaystyle\cosh^{2}(a_{1})=\big{(}\frac{\cosh(\alpha_{1})+\cosh(\alpha_{2})\cosh(\alpha_{3})}{\sinh(\alpha_{2})\sinh(\alpha_{3})}\big{)}^{2}\,.$

Figure 1. All figures were created with the software program GeoGebra [33].

Two generalized laws of sines:
Let $U\leq\textrm{P}\boldsymbol{V}$ be a anisotropic plane of dimension $s\geq
2$ and let $S{\,:=\,}\\{P_{1},\dots,P_{s+1}\\}$ be a set of independent
anisotropic points in $U$. Hence $S$ is the set of vertices of an $s$-simplex
in $U$ and the sets
$S_{i}{\,:=\,}S{\smallsetminus}{\smallsetminus}\\{P_{i}\\}$,
$i{\,=\,}1,\dots,s{+}1$, are the sets of vertices of its $(s{-}1)$-faces.
Define for $i\neq j$ points $P_{ij}:=(P_{i}\sqcup P_{j})\sqcap
P_{i}^{\mathfrak{A}}$ and put $\alpha_{i}:=\big{\\{}P_{ij}|1\leq j\leq
s{+}1\;\textrm{and}\;j\neq i\big{\\}}$. $\alpha_{i}$ can be interpreted as the
interior angle of the simplex at the vertex $P_{i}$. Then, cf. [5]:
(1) If $\textrm{span}(S_{1}),\dots,\textrm{span}(S_{s+1})$ are anisotropic and
$i\in\\{1,\dots,s{+}1\\}$,

$\displaystyle\frac{\psi(\alpha_{i})}{\psi(S_{i})}=\frac{\psi^{s-1}(S)}{\prod_{j=1}^{s{+}1}\psi(S_{j})}\;.$

(2) If for distinct numbers $i,j\in\\{1,\dots,s{+}1\\}$ the three planes
generated by $S_{i},S_{j}$ and
$\hskip 18.49428ptS{\smallsetminus}{\smallsetminus}\\{P_{i},P_{j}\\})$ are
anisotropic, then

$\xi_{0}(\textrm{span}(S_{i}),\textrm{span}(S_{j})\big{)}=-\displaystyle{\sinh^{2}}{\big{(}d\big{(}\textrm{span}(S_{i}),\textrm{span}(S_{j})\big{)}\big{)}}=\frac{\psi(\mathcal{S})\,\psi(\mathcal{S}{\smallsetminus}{\smallsetminus}\\{P_{i},P_{j}\\})}{\psi({S_{i}})\,\psi({S_{j}})}$

### 2.13.

Quadrics and spheres. Let $(P_{1},\dots,P_{s+1})$ be a tuple of anisotropic
points generating an anisotropic $s$-plane $A$. Let
$\mathfrak{N}{\,=\,}(\mathfrak{n}_{ij}){\,\in\,}\mathbb{R}^{(s{+}1)\times(s{+}1)}$
be an indefinite, regular symmetric matrix. Then
$\mathcal{Q}_{\mathfrak{N}}{\,=\,}\mathcal{Q}(\mathfrak{N},(P_{1}{,\dots,}P_{\\!{s+1}})){\,:=\,}\\{r_{1}P_{1}{+\dots+\,}r_{\\!{s+1}}P_{s+1}|\\!\displaystyle\sum_{1\leq
i,j\leq s{+}1}\\!\\!\\!\mathfrak{n}_{ij}r_{i}r_{j}{=\,}0\\}$ is a non-empty
quadric in $A$. The polar of a point $R{\,=}\sum_{j}r_{j}P_{j}$ with respect
to $\mathcal{Q}_{\mathfrak{N}}$ , polar$(R,\mathcal{Q}_{\mathfrak{N}})$
$:=\\{\sum_{j}t_{j}P_{j}\,|\\!\displaystyle\sum_{1\leq i,j\leq
s{+}1}\\!\\!\\!\mathfrak{n}_{ij}t_{i}r_{j}{\,=\,}0\\}$, is a hyperplane of A.
An anisotropic point $R$ is a symmetry point of $\mathcal{Q}_{\mathfrak{N}}$
iff polar$(R,\mathcal{Q}_{\mathfrak{N}})=R^{\mathfrak{A}}\sqcap A$.
Proof: Take any line through $R$ that meets $\mathcal{Q}_{\mathfrak{N}}$ in
two points $T_{1}$ and $T_{2}$. This line meets the
polar$(R,\mathcal{Q}_{\mathfrak{N}})$ at the harmonic conjugate $R^{\prime}$
of $R$ with respect to $T_{1}$ and $T_{2}$. Precisely when $R$ is an
anisotropic midpoint of $\\{T_{1},T_{2}\\}$, the point $R^{\prime}$ is a
(second) anisotropic midpoint of $\\{T_{1},T_{1}\\}$ and lies on
$R^{\mathfrak{A}}.\;\;\scriptstyle{\Box}$

Figure 2. The blue conic is the absolute in a hyperbolic plane. The green
conics are 1-spheres (circles) with center $M$. The numbers indicate their
quadrances.

$\mathcal{Q}_{\mathfrak{N}}$ is a hypersphere of $A$ if there exists an
anisotropic hyperplane $H$ of $A$ whose anisotropic points are symmetry points
of $\mathcal{Q}_{\mathfrak{N}}$. In this case, the point
$H^{\mathfrak{A}}\sqcap A$ is called the center of the hypersphere.
Let us assume that $\mathcal{Q}_{\mathfrak{N}}$ is a hypersphere of $A$ and
$M$ is its center. Then $M^{\mathfrak{A}}\cap\mathcal{Q}_{\mathfrak{N}}$ is
the set of isotropic points of this sphere. A sphere can have real isotropic
points; but these form a thin set within this sphere.
Suppose the center $M$ of $\mathcal{Q}_{\mathfrak{N}}$ is anisotropic. Then
$M$ is also a symmetry point of $\mathcal{Q}_{\mathfrak{N}}$. If $T_{1}$ and
$T_{2}$ are two distinct anisotropic points on $\mathcal{Q}_{\mathfrak{N}}$,
then $d_{0}(T_{1},M)$ $=d_{0}(T_{2},M)$, because: Let $R$ be the intersection
of the line $T_{1}\sqcup T_{2}$ with the hyperplane $M^{\mathfrak{A}}$ of
$\textrm{P}\boldsymbol{V}$, then the triple $(T_{1},M,T_{2})$ is mapped onto
the triple $(T_{2},M,T_{1})$ by the (distance preserving) reflection in the
point ${R}$.
If $M$ is anisotropic, we can define the radius $r$ of a hypersphere as the
distance between its center and any of its anisotropic points, and we call
$\xi_{0}(\mathcal{Q}_{\mathfrak{N}}){\,:=\,}1-\cosh^{2}(r)$ the quadrance of
$\mathcal{Q}_{\mathfrak{N}}$.
If the center $M$ of $\mathcal{Q}_{\mathfrak{N}}$ is isotropic, $M$ is a
touchpoint of the two quadrics $\mathcal{Q}_{\mathfrak{N}}$ and
$\mathcal{Q}_{\mathfrak{A}}$, and $\mathcal{Q}_{\mathfrak{N}}$ is called a
horosphere.
If $\mathcal{Q}_{\mathfrak{N}}$ has several (more than one) isotropic points,
its center is a point outside that sphere and the lines connecting an
isotropic point of $\mathcal{Q}_{\mathfrak{N}}$ with the center are tangents
of the sphere. The union of these tangents through $M$ is called the tangent-
cone of $\mathcal{Q}_{\mathfrak{N}}$ with vertex $M$.

Suppose $\mathcal{Q}_{\mathfrak{N}}$ has an anisotropic center
$M=m_{1}P_{1}{+\dots+}m_{s+1}P_{s+1}$ and radius $r\in\mathbb{D}$, then

$\mathcal{Q}_{\mathfrak{N}}{=}\big{\\{}r_{1}\\!P_{1}{+\dots+\,}r_{s+1}\\!P_{s+1}|\displaystyle(\sum_{1\leq
i,j\leq
s{+}1}\\!\\!\\!\\!\mathfrak{n}_{ij}r_{i}m_{j})^{2}{=}\cosh^{2}(r)\,(\sum_{1\leq
i,j\leq s{+}1}\\!\\!\\!\mathfrak{n}_{ij}m_{i}m_{j})(\sum_{1\leq i,j\leq
s{+}1}\\!\\!\\!\mathfrak{n}_{ij}r_{i}r_{j})\big{\\}}.$

If $\mathcal{Q}_{\mathfrak{N}}$ passes through a point $T=t_{1}P_{1}{+\dots}$
$+\,t_{s+1}P_{s+1}$, $\mathcal{Q}_{\mathfrak{N}}$ consists of all points
${Q}\,=r_{1}\\!P_{1}{+\dots+\,}r_{s+1}\\!P_{s+1}$ satisfying the equation

$\displaystyle(\sum_{1\leq i,j\leq
s{+}1}\\!\\!\\!\mathfrak{n}_{ij}r_{i}m_{j})^{2}(\sum_{1\leq i,j\leq
s{+}1}\\!\\!\\!\mathfrak{n}_{ij}t_{i}t_{j}){\;=\;}(\sum_{1\leq i,j\leq
s{+}1}\\!\\!\\!\mathfrak{n}_{ij}t_{i}m_{j})^{2}(\sum_{1\leq i,j\leq
s{+}1}\\!\\!\\!\mathfrak{n}_{ij}r_{i}r_{j}).$

Let $H_{1}$ and $H_{2}$ be two hyperplanes of $A$ touching
$\mathcal{Q}_{\mathfrak{N}}$ in anisotropic points $Q_{1},Q_{2}$,
respectively. If the center $M$ of $\mathcal{Q}_{\mathfrak{N}}$ is
anisotropic, then
$\zeta_{0}(H_{1},M^{\mathfrak{A}})=\zeta_{0}(H_{2},M^{\mathfrak{A}})$.

Here are some examples of hyperspheres: (1) Suppose $A$ is an anisotropic
line. Then two distinct anisotropic points $Q_{1},Q_{2}\in A$ form a
hypersphere iff they have a midpoint. A single anisotropic point $Q\in A$
(when counted with multiplicity $2$) is a hypersphere in $A$ with radius $0$.
(2) A (non-empty) tangent-cone of a hypersphere in an anisotropic plane $A$
with dim$(A)>1$ is a hypersphere in $A$ with radius $r=0$. (3) If $M$ is an
anisotropic point in an anisotropic plane $A$, then $A\cap M^{\mathfrak{A}}$
is (doubly counted) a hypersphere in $A$; $M$ is its center and its radius is
$\frac{1}{2}\pi{\mathbf{i}}$.

### 2.14.

Simplices and their classical centers444) For a definition of the term
triangle center see [14]. See also [6, 8, 31, 32] .). Let
$(P_{1},\dots,P_{s+1})$ be a tuple of anisotropic points generating an
anisotropic $s$-plane $A$. We denote a point
$R{\,=\,}r_{1}P_{1}{\,+\dots+\,}r_{s+1}P_{s+1}\in A$ by
$[r_{1}:\dots:r_{s+1}]$. The set
$\\{[r_{1}:\dots:r_{s+1}]\,|\,r_{1}r_{2}{\cdots}r_{s+1}$ $\neq 0\,\\}$
consists of $2^{s}$ connected components. The closure of each component is a
simplex with vertices $P_{1},\dots,P_{s+1}$. We will especially focus on the
simplex
$\Sigma=\Sigma_{0}{\,:=\,}\\{[r_{1}{\,:\dots:\,}r_{s+1}]|\,p_{1}{,\dots,}\,p_{s+1}\geq
0\,\\}$; it will be our reference simplex. The other simplices with these
vertices will be called the mates of $\Sigma$.
A plane of dimension $k\leq s$ through $k{+}1$ vertices of $\Sigma$ is called
a $k$-sideplane, the intersection of $\Sigma$ with one of its $k$-sideplanes a
$k$-face of $\,\Sigma$. Instead of $1$-sideplanes we usually speak of
sidelines, and the $1$-faces are also called edges, the $(n-1)\\!$ -faces
facets of $\Sigma$. Even the vertices $P_{i}$ are accepted as sideplanes of
$\Sigma$.
The classical centers of an $n$-simplex are the centroid $G$, the circumcenter
$O$ and the incenter $I$.
We want all the edges of $\Sigma$ to have midpoints and demand that all
sidelines of $\Sigma$ are anisotropic and all vertices $P_{1},\dots,P_{s+1}$
have the same sign. In this case, the centroid of $\Sigma$ is the point
$G=[1:\dots:1]$ and, more generally, the centroid of a $k$-face
$\Sigma\cap(P_{i_{1}}\sqcup\cdots\sqcup P_{i_{k+1}})$ is the point
$P_{i_{1}}+\dots+P_{i_{k+1}}$. The line joining the centroid of a $k$-face of
$\Sigma$ with the centroid of the opposite $(s{-}k{-}1)$-face passes through
$G$. The centroids of the mates of $\Sigma$ are the points
$R{\,=\,}[r_{1}:\dots:r_{s+1}]$ with $P\neq G$ and $r_{j}\in\\{-1,1\\}$ for
$1\leq j\leq s{+}1$. Without loss of generality we can put $r_{1}$ to $1$. We
label the centroids of $\Sigma$ and its mates by integers from $0$ to
$2^{s}-1$ as follows. Given any centroid $[1:r_{2}:\dots:r_{s+1}]$, define
numbers $r_{j}^{\star}\in\\{0,1\\}$ by $r_{j}^{\star}=1$ iff $r_{s+1-j}=-1$.
Then $G_{k}:=[1:r_{2}:\cdots:r_{s+1}]$ iff $k=\sum_{\begin{subarray}{c}0\,\leq
j\leq{s}\end{subarray}}r_{j}^{\star}2^{j}$. Here are two examples: $G_{0}=G$
and $G_{5}=[1:\cdots:1:-1:1:-1]$.
The simplices with vertices $P_{1},\dots,P_{s+1}$ will now be labeled such
that $\Sigma_{k}$ is the simplex with $G_{k}\in\Sigma_{k}$ for
$k=0,\dots,2^{s}{-}1$. We accept each point $G_{k},0\leq k\leq 2^{s}{-}1$, as
a centroid of each of the simplices $\Sigma_{j},0\leq j\leq 2^{s}{-}1$, but
define $G_{k}$ to be the proper centroid of $\Sigma_{j}$ iff $k=j$.
We specify the edge lengths of simplex $\Sigma_{k}$. The edge of $\Sigma_{k}$,
$k=\sum_{\begin{subarray}{c}0\,\leq
j\leq{s}\end{subarray}}r_{j}^{\star}2^{j}$, with vertices $P_{j}$ and $P_{l}$
has the length

$\mu_{k;jl}\,=\begin{cases}\mu_{0;jl}=\mu([P_{j},P_{l}]_{+})\,,&\textrm{if}\;r^{\star}_{j}r^{\star}_{l}=1,\\\
\mu([P_{j},P_{l}]_{-})\,,&\textrm{if}\;r^{\star}_{j}r^{\star}_{l}=-1.\\\
\end{cases}$

A circumsphere of $\Sigma$ is a hypersphere in $A$ passing through all
vertices of $\Sigma$. There are $2^{s}$ circumspheres
$\mathcal{S}_{0^{\,}},\dots,\mathcal{S}_{2^{s}-1}$ of $\Sigma$, and all of
them are also hyperspheres of the mates of $\Sigma$. But as for the centroids,
there is a bijective correspondence between these hyperspheres and simplices
in such a way that we say that $\mathcal{S}_{k}$ is the proper circumsphere of
$\Sigma_{k}$. All inner points of $\Sigma_{k}$ are points inside the proper
circumsphere of $\Sigma_{k}$, but the center of this circumsphere need not lie
inside $\Sigma_{k}$.
The proper circumsphere of $\Sigma$ is
$\mathcal{S}_{0}=\big{\\{}[r_{1}:\dots:r_{s+1}]\,|\\!\displaystyle\sum_{1\leq
i,j\leq
s{+}1}\\!\\!\\!\sinh^{2}(\frac{1}{2}d_{0}(P_{i},P_{j}))\,r_{i}r_{j}{\,=\,}0\big{\\}}$.
Proof: It is obvious that if we intersect a circumsphere $\mathcal{S}_{i}$
with $k$-sideplanes of $\Sigma$, we get spheres of dimension $k-1$, each
passing through $k+1$ of the vertices of $\Sigma$. For $k=2$ we get
circumcircles of triangles. The validity of the equation is proven for these
one-dimensional subsets of $S$ (see [7, 8]). From this follows the correctness
of the equation. $\scriptstyle{\Box}$

Define $\mathfrak{C}=(\mathfrak{c}_{ij})\in\mathbb{R}^{(s{+}1)\times(s{+}1)}$
by
$\mathfrak{c}_{ij}:=P_{i}^{\,\circ}{\scriptstyle{[\mathfrak{A}]}}P_{j}^{\,\circ}$.
The proper circumcenter of $\Sigma_{0}$ is $O=O_{0}=[r_{1}:\dots:r_{s+1}]$
with $(r_{1},\dots,r_{s+1})\,\mathfrak{C}=(1,1,\dots,1)$, the proper
circumcenter of $\Sigma_{1}$ is $O_{1}=[r_{1}:\dots:r_{s+1}]$ with
$(r_{1},\dots,r_{s+1})\,\mathfrak{C}=(1,\dots,1,-1)$ etc., cf. [13, 8].
Proof: We give a proof for $O=O_{0}$: $O$ is a point on all bisectors of
segments $[P_{i},P_{j}],0\leq i{\,<\,}j{\,\leq\,}s{+}1$. Therefore,
$\xi_{0}(O,P_{i}-P_{j})=1\;\,(0{\leq}i{<}j{\leq}s{+}1)$ and
$O^{\mathfrak{A}}\sqcap A=\\{[x_{1}\dots
x_{n+1}]\,|\,x_{1}+\dots+x_{n+1}=0\\}.\;\Box$
Remark: If $Q=[q_{1}:\dots,q_{s+1}]$ is a point in $A$ with $q_{1}q_{2}\dots
q_{s+1}\neq 0$, we call the $(s{-}1)$-plane $\displaystyle
Q^{\Sigma}{:=\,}\big{\\{}[x_{1}{\,:}\dots:\,x_{s+1}]\,|\,\frac{x_{1}}{q_{1}}+\dots+\frac{x_{s+1}}{q_{s+1}}=0\big{\\}}$
the $\Sigma$-polar of $Q$. For $i=0,\dots,2^{s}{-}1$ the equation
$O_{i}:=A\sqcap(G_{i}^{\,\Sigma})^{\mathfrak{A}}$ applies.
An equation for the radius $r$ of $\mathcal{S}_{0^{\,}}$ is

$\tanh^{2}(r)=\tanh^{2}(d_{0}(O,P_{1}))=1-\textrm{sgn}(P_{1})\big{(}(1,\dots,1){\scriptstyle{[\mathfrak{C}^{-1}]}}(1,\dots,1)\big{)}$.

Example: Let $P_{1},P_{2},P_{3}$ be three distinct points in the elliptic
plane $(\textrm{P}\mathbb{R}^{3},\mathfrak{A}=\textrm{diag}(1,1,1))$. Put
$c_{ij}:=\cosh(d_{0}(P_{i},P_{j}))\leq 1$. Then
$\mathfrak{C}=\left(\begin{array}[]{rrr}1_{\;\,}&c_{12}&c_{13}\\\
c_{12}&1_{\;\,}&c_{23}\\\ c_{13}&c_{23}&1_{\;\,}\end{array}\right)$.
We calculate $O$ and $\tanh^{2}(r)$:
$O=(1{\,-\,}c_{23})(1{+}c_{23}{-}c_{12}{-}c_{13})\,P_{1}$${\,+\,}(1{\,-\,}c_{13})(1{+}c_{13}{-}c_{12}{-}c_{23})\,P_{2}$
${\,+\,}(1{\,-\,}c_{12})(1{+}c_{12}{-}c_{13}{-}c_{23})\,P_{3}$,
$\displaystyle\tanh^{2}(r)=\frac{2(c_{12}-1)(c_{13}-1)(c_{23}-1)}{\det(\mathfrak{C})}=\frac{2(c_{12}-1)(c_{13}-1)(c_{23}-1)}{2c_{12}c_{13}c_{23}-c_{12}^{2}-c_{13}^{2}-c_{23}^{2}+1}$.
We denote the $(s{-}1)$-sideplane of $\Sigma$ opposite the vertex $P_{k}$ by
$H_{k}$ and put ${Q_{k}{:=\,}H_{k}^{\,\mathfrak{A}}{\,\sqcap\,}\,A}$. There
are $2^{s}$ simplices $\Sigma^{0},\dots,\Sigma^{2^{s}-1}$ with vertices
$Q_{1},\dots,Q_{s+1}$ and there is a $1{:}1$-corres-pondence
$\Sigma_{k}\mapsto\Sigma^{k}$ given by
$\Sigma^{k}:=A\,{\smallsetminus}{\smallsetminus}\bigcup\limits_{R\,\textrm{inner
point of}\,\Sigma_{k}}R^{\mathfrak{A}}$. We call $\Sigma^{k}$ the dual of
$\Sigma_{k}$.

An insphere of $\Sigma$ is a hypersphere in $A$ which touches each
$(s{-}1)$-dimensional sideplane $H_{k}$ of $\Sigma$. Thus, its center has the
same distance from all these planes and hence the same distance from the
vertices $Q_{k}$ of $\Sigma^{0}$. In other words, the incenters of $\Sigma$
are the circumcenters of $\Sigma^{0}$. Since circumcenters of a simplex exist
only if all its vertices have the same sign, a necessary condition for the
existence of incenters of $\Sigma$ is that all vertices of $\Sigma^{0}$ have
the same sign. We assume now, that $\Sigma_{0}$ has got an insphere and that
the facet of $\Sigma_{0}$ opposite vertex $P_{i}$ has the staudtian $s_{i}$.
Then $s_{i}/s_{j}\in\mathbb{R}^{+}$ for $1{\,\leq\,}i,j{\,\leq\,}n{+}1$. The
proper incenter of $\Sigma_{0}$ is $I_{0}=[s_{1}:{\dots}:s_{n+1}]$
$=[1/\sinh(d_{0}(P_{1},H_{1})):\dots:1/\sinh(d_{0}(P_{s+1},H_{s+1}))]$.
$I_{0}$ can also be written as a function of the interior angles $\alpha_{i}$
at the vertices of $\Sigma_{0}$:
$I_{0}=[\sqrt{\psi(\alpha_{1})}:\dots:\sqrt{\psi(\alpha_{s+1})}\,]$. The
proper circumcenter of $\Sigma^{k}$ is the proper incenter of $\Sigma_{k}$,
and the proper incenter of $\Sigma^{k}$ is the proper circumcenter of
$\Sigma_{k}$. We denote the proper incenter of $\Sigma_{k}$ by $I_{k}$. This
incenter $I_{k}$ is always a point inside the simplex $\Sigma_{k}$, but all
the points inside this insphere may lie outside the simplex, see [8]. Since
$I_{k}\in\Sigma_{k}$, $I_{k}$ is the barycentric product of $I_{0}$ with the
centroid $G_{k}$. For example, if $s=3$ and $k=2$, then
$I_{k}=[s_{1}:s_{2}:-s_{3}:s_{4}]$.
Let $H_{i}$, as before, be the $(s-1)$-sideplane of $\Sigma$ opposite vertex
$P_{i}$. The points $R_{1},\dots,R_{s+1}$, $R_{i}:=(P_{i}\sqcup
H_{i}^{\,\mathfrak{A}})\sqcap H_{i}$, are called the pedals of $\Sigma$. The
tuples $P_{1},\dots,P_{s+1}$ and $R_{1},\dots,R_{s+1}$ are, in general, not
perspective. If they are, the perspector is called orthocenter of $\Sigma$.

### 2.15.

Radical centers of spheres. Let $(P_{1},\dots,P_{s+1})$ be a tuple of
anisotropic points generating an anisotropic $s$-plane $A$. We assume that all
these points have the same sign and that all lines $P_{i}\sqcup P_{j}$ are
anisotropc lines of the same type, either all elliptic or all hyperbolic. As
in the last section, we regard $(P_{1},\dots,P_{s+1})$ as vertices of $2^{s}$
simplices $\Sigma_{i}$ and we define the matrix $\mathfrak{C}$ by
$\mathfrak{c}_{ij}:=P_{i}^{\,\circ}{\scriptstyle{[\mathfrak{A}]}}P_{j}^{\,\circ}$.
Around each point $P_{i}$ we put an $(s{-}1)$-sphere $\mathcal{S}_{i}$ (a
hypersphere in $A$) in such a way that all anisotropic points on these spheres
have the same sign. We denote the radius of $\mathcal{S}_{i}$ by
$\rho_{i}\,(\neq\frac{1}{2}\pi\textrm{i})$ and put
$r_{i}{\,:=\,}\cosh(\rho_{i}),1\leq i\leq s{+}1$.

Figure 3. A 3-simplex (triangle) $(P_{1},P_{2},P_{3})$ is situated in an
elliptic plane. The blue dotted circle
$\\{(q_{1}{:}q_{2}{:}q_{3})\,|\,q_{1}^{\,2}+q_{2}^{\,2}=q_{3}^{\,2}\\}$ is
used as a construction aid and serves for orientation. The points
$Z,Z_{1},Z_{2},Z_{3}$ are the radical centers of the black circles around the
vertices of the triangle. The red and orange lines are the radical axes. The
red circle is the radical circle with center $Z$.

If $Z$ is outside one of these spheres, then it is outside of all these
spheres and a radical $(s{-}1)$-sphere can be drawn around $Z$, a hypersphere
in $A$ that meets all the spheres $\mathcal{S}_{1},\dots,\mathcal{S}_{n+1}$
orthogonally.
The radius $\rho$ of this radical sphere satisfies the equations
$\displaystyle\cosh(\rho)=\frac{\cosh(d_{0}(Z,P_{k}))}{\cosh(\rho_{k})))}$,
$\,k=1,\dots,s{+}1$.
$Z$ is not the only radical center of these spheres; the other radical centers
are the points $Z^{\prime}{\,=\,}(z^{\prime}_{1}{:\dots:}z^{\prime}_{s+1})\neq
Z$, with
$(z^{\prime}_{1}{,\dots,}z^{\prime}_{s+1})\,\mathfrak{C}\in\\{(t_{1}r_{1},\dots,t_{s+1}r_{s+1})\,|\,t_{1},\dots,t_{s+1}=\pm
1\\}$.

Figure 4. $(P_{1}P_{2}P_{3})$ is a triangle in an elliptic plane. The point
$T$ is the inner center of similitude of the black circles around the vertices
of this triangle.

If $Z$ is outside one of these spheres, then it is outside of all these
spheres and a radical $(s{-}1)$-sphere can be drawn around $Z$, a hypersphere
in $A$ that meets all the spheres $\mathcal{S}_{1},\dots,\mathcal{S}_{n+1}$
orthogonally.
The radius $\rho$ of this radical sphere satisfies the equations
$\displaystyle\cosh(\rho)=\frac{\cosh(d_{0}(Z,P_{k}))}{\cosh(\rho_{k})))}$,
$\,k=1,\dots,s{+}1$.
$Z$ is not the only radical center of these spheres; the other radical centers
are the points $Z^{\prime}{\,=\,}(z^{\prime}_{1}{:\dots:}z^{\prime}_{s+1})\neq
Z$, with
$(z^{\prime}_{1}{,\dots,}z^{\prime}_{s+1})\,\mathfrak{C}\in\\{(t_{1}r_{1},\dots,t_{s+1}r_{s+1})\,|\,t_{1},\dots,t_{s+1}=\pm
1\\}$.

A special case: If all the radii $\rho_{k}$ agree, then the radical centers
are the circumcenters of $\Sigma$ and $Z$ is the proper circumcenter $O$ of
$\Sigma_{0}$.

Let us consider the plane $H_{12}:=Z\sqcup(P_{1}\sqcup P_{2})^{\mathfrak{A}}$.
$Z$ is not a point on $(P_{1}\sqcup P_{2})^{\mathfrak{A}}\sqcap A$, therefore
$H_{12}$ is a hyperplane in $A$ and all points in $H_{12}$ have the same power
with respect to $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$. But there is another
hyperplane $\tilde{H}_{12}$ in $A$ consisting also of points with the same
power with respect to both hyperspheres. Both planes are radical planes of
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ in $A$. $P_{1}^{\,\mathfrak{A}}\sqcap
A,{H}_{12},P_{2}^{\,\mathfrak{A}}\sqcap A,\tilde{H}_{12}$ form a harmonic
pencil, see Figure 3.
To each $\mathcal{S}_{k}$ is assigned a dual $(s-1)$-sphere
$\mathcal{S}_{k}^{\delta}=\\{T^{\mathfrak{A}}\sqcap A|\;T\,\textrm{is a
tangent hyperplane}$ $\textrm{of}\,\mathcal{S}_{k}\textrm{in\,}A\\}$. Let us
look at the two spheres $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$. Their duals
have together two radical hyperplanes, $H^{\prime}_{12}$ and
$\tilde{H}^{\prime}_{12}$. The duals in $A$ of these hyperplanes are the
points $R_{12}^{\,\pm}=\sinh(\rho_{2})P_{1}\pm\sinh(\rho_{1})P_{2}$ on the
line $P_{1}\sqcup P_{2}$. These two points are called centers of similitude of
$S_{1}$ and $S_{2}$. Together with $P_{1}$ and $P_{2}$ they form a harmonic
range $P_{1},R_{12}^{\,+},P_{2},R_{12}^{\,-}$. If the spheres $S_{1}$, $S_{2}$
have a common tangent plane, it passes either through $R_{12}^{\,+}$ or
through $R_{12}^{\,-}$.
We call the point $[1/\sinh(\rho_{1}):\cdots:1/\sinh(\rho_{n+1})]$ the inner
center of similitude of the collection $\\{S_{k}|1\leq k\leq s+1\\}$, and it
is obvious that subsets of this collection have their own inner center of
similitude. The barycentric product of an inner center of similitude with a
centroid of $\Sigma$ gives a center of similitude.
Two special cases: (1) If all radii $\rho_{k}$ are the same, then the inner
center of similitude is $G_{0}$. (2) If $\rho_{k}$ equals the distance between
the vertex $P_{k}$ and its opposite $(n{-}1)$-sideplane $H_{k}$, $1\leq k\leq
s{+}1$, then $I_{0}$ is the inner center of similitude.

##  3\. Geometry on Cayley-Klein spaces

### 3.1.

Cayley-Klein spaces. A Cayley-Klein space with underlying projective space
$\textrm{P}\boldsymbol{V}$ is a nested sequence of finitely many, say
$\rho+1$, semi CK spaces $(A_{i},{{\beta_{i}}})$, such that
$A_{0}=\textrm{P}\boldsymbol{V}$, $A_{i+1}$ is a nonempty radical
rad(${\beta_{i}}$), $0\leq i\leq\rho-1$, and
$A_{\rho+1}{:=\textrm{rad}(\beta_{\rho})}=\emptyset.$ Since the subspaces
$A_{i}$ can be read off from the functions $\beta_{i-1}$, we denote this CK-
space by
$\big{(}\textrm{P}\boldsymbol{V};{\beta_{0}},{\beta_{1}},\dots,{\beta_{\rho}}\big{)}$.
If $\beta_{0}$ is nondegenerate (i.e. $\rho=0$), then
$(\textrm{P}\boldsymbol{V},{\beta_{0}})$ is a CK-space.
We call a point $P\in\textrm{P}\boldsymbol{V}$ anisotropic of degree
$k\in\\{0,\dots,\rho\\}$ if $P\in A_{k}$
$\textrm{\,and\,}P\not\in\mathcal{Q}_{\beta_{k}}$, and we call a point $P$
isotropic of degree $k$ (or shorter $k$-isotropic) if
$P\in\mathcal{Q}_{\beta_{k}}\\!{\smallsetminus}{\smallsetminus}\,A_{k+1}$. A
point $P$ is called isotropic if it is $k$-isotropic for some
$k\in\\{0,\dots,\rho\\}$, otherwise it is called anisotropic.
A mapping $\phi\in\textrm{Aut}(\textrm{P}\boldsymbol{V})$ is an automorphism
on
$\big{(}\textrm{P}\boldsymbol{V};{\beta_{0}},{\beta_{1}},\dots,{\beta_{\rho}}\big{)}$
if the restriction of $\phi$ to $A_{i}$ is an automorphism on
$(A_{i},\beta_{i}),0\leq i<\rho$. See [10] for a more detailed study of these
mappings.

### 3.2.

The $\mathbf{\varepsilon}$-quadrance as a distance-function on Cayley-Klein
spaces. Let $(n^{\downarrow},n^{+},n^{-},n^{\uparrow})$ be a quadruple of
nonnegative integers with $n^{\downarrow}+n^{+}+n^{-}+n^{\uparrow}=n{+}1$. A
matrix $\mathfrak{A}=(\mathfrak{a})_{ij}\in\mathbb{R}^{(n+1)\times(n+1)}$ we
denote by $\mathfrak{A}^{(n^{\downarrow},\,n^{+},\,n^{-},\,n^{\uparrow})}$ if
$\mathfrak{a}_{ii}{\,=\,}1$ for $n_{\downarrow}\leq i\leq
n_{\downarrow}+n_{+}$, $\mathfrak{a}_{ii}{\,=\,}{-}1$ for
$n^{\downarrow}+n^{+}{\,<\,}i\leq n^{\downarrow}+n^{+}+n^{-}$, and
$\mathfrak{a}_{ij}{\,=\,}0$ otherwise. We now refer to the CK space introduced
in the last section. We put
$\beta_{i}^{\,\downarrow}{:=\,}\sum_{k<i}\,(\beta_{k}^{\;+}+\beta_{k}^{\;-}),\beta_{i}^{\,\uparrow}{:=\,}\sum_{k>i}\,(\beta_{k}^{\;+}+\beta_{k}^{\;-})$,
$\beta_{0}^{\,\downarrow}{:=\,}0$, $\beta_{\rho}^{\,\uparrow}{:=\,}0$ and
$\mathfrak{A}_{i}:=\mathfrak{A}^{(\beta_{i}^{\;\downarrow},\,\beta_{i}^{\;+},\,\beta_{i}^{\;-},\,\beta_{i}^{\;\uparrow})}$.
Then
$\big{(}\textrm{P}\boldsymbol{V};{\beta_{0}},{\beta_{1}},\dots,{\beta_{\rho}}\big{)}$
and
$\big{(}\textrm{P}\mathbb{R}^{n+1};{\mathfrak{A}_{1}},\dots,{\mathfrak{A}_{\rho}}\big{)}$
are isomorphic CK spaces.

Let ${\mathbf{\varepsilon}}{\,\in\,}\mathbb{R}^{\ast}$ be a positive
infinitesimal. We put
$\displaystyle\mathfrak{A}_{\mathbf{\varepsilon}}{\,:=\,}{{\mathfrak{A}_{0}}}+\\!\\!\sum_{1\leq
i\leq\rho}\\!\\!{\mathfrak{A}_{i}}\mathbf{\varepsilon}^{i}\,$.
$\mathfrak{A}_{\mathbf{\varepsilon}}$ holds the complete information about the
CK space
$\big{(}\textrm{P}\mathbb{R}^{n+1};{\mathfrak{A}_{0}},\dots,{\mathfrak{A}_{\rho}}\big{)}$.
In the following we study the CK space
$\big{(}\textrm{P}\mathbb{R}^{n+1};\mathfrak{A}_{\mathbf{\varepsilon}}\big{)}$
more closely.
It can be easily checked that a point
$P=\mathbb{R}\boldsymbol{p}\in\textrm{P}\boldsymbol{V}$ is isotropic precisely
when
$\boldsymbol{p}{\;\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]}\;}\boldsymbol{p}=0.$
Let $U$ be a plane of dimension $s$ generated by points
$P_{1}=\mathbb{R}\boldsymbol{p}_{1},\dots,P_{s+1}=\mathbb{R}\boldsymbol{p}_{s+1}$.
Then $U$ is called isotropic if the matrix
$(\boldsymbol{p}_{i}{\scriptstyle{\,[\mathfrak{A}_{\mathbf{\varepsilon}}]\,}}\boldsymbol{p}_{j})_{{}_{ij}}$
is degenerate. We now assume that $U$ is anisotropic. If $\tilde{U}$ is a
second anisotropic plane of the same dimension $s$, generated by points
$\tilde{P}_{1}=\mathbb{R}\tilde{\boldsymbol{p}}_{1},\dots,\tilde{P}_{s+1}=\mathbb{R}\tilde{\boldsymbol{p}}_{s+1}$,
we define the $\mathbf{\varepsilon}$-quadrance of $U$ and $\tilde{U}$ by

$\,\displaystyle\xi_{\mathbf{\varepsilon}}(U,\tilde{U}):=\star_{\mathbf{\varepsilon}}\,\big{(}1-\frac{\big{(}\det\,({\boldsymbol{p}}_{i}\,{\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]}}\,\tilde{\boldsymbol{p}}_{j})_{{}_{ij}}\big{)}^{2}}{\big{(}\det\,({\boldsymbol{p}}_{i}\,{\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]}}\,{\boldsymbol{p}}_{j})_{{}_{ij}}\big{)}\big{(}\det\,({\tilde{\boldsymbol{p}}}_{i}\,{\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]}}\,{\tilde{\boldsymbol{p}}}_{j})_{{}_{ij}}\big{)}}\big{)}$
.

We call a plane $U{\,\leq\,}\textrm{P}\boldsymbol{V}$ regular if
$U{\,\smallsetminus}{\smallsetminus\,}A_{0}\neq\emptyset$, and we call a
regular plane $U\leq\textrm{P}\boldsymbol{V}$ a light-plane if
$\xi_{\mathbf{\varepsilon}}(Q,R)\in\mathbb{R}\,\mathbf{\varepsilon}^{2}$ for
all regular points $P,Q\in U$. (According to this definition regular points
are light-planes of dimension 0.)

### 3.3.

Total polars and polar varieties.
There are various definitions for a (total) polar of a plane in a CK-space in
the literature. We adopt the definition given in [11]. If $P$ is an
anisotropic point of degree $s<\rho$ in
$(\textrm{P}V,\mathfrak{A}_{1},\dots,\mathfrak{A}_{\rho})$, then any
hyperplane in $\textrm{P}V$ that contains $P^{\mathfrak{A}_{s+1}}$ is a total
polar of $P$. Let $U\leq\textrm{P}V$ be a $k$-plane. We can find an
independent set $B=\\{P_{1},\dots,P_{k+1}\\}$ of points that generate $U$ such
that $\\#(B\cap A_{j})=\dim(U\cap A_{j}){\,+\,}1$, $1\leq j\leq\rho{+}1$. If
$H_{i}$ is a total polar of $P_{i}$ for $1\leq i\leq s{+}1$ and
$H_{1},\dots,H_{s+1}$ are independent, the $(n{-}s)$-plane
$\bigcap\limits_{1\leq i\leq s{+}1}H_{i}$ is a total polar of U.
If $\tilde{U}$ is a total polar of $U$, then $U$ is a total polar of
$\tilde{U}$. If $P$ is a point in $U$ and $Q$ a point in a total polar of $U$,
then $\xi_{\mathbf{\varepsilon}}(P,Q)=1$.
The set of all total polars of a $k$-plane $U$ is called polar variety of $U$,
denoted by $U^{\perp}$.

### 3.4.

Examples. (1)
$\big{(}\textrm{P}\mathbb{R}^{1};{\mathfrak{A}_{\mathbf{\varepsilon}}=\textrm{diag}{(1,0)}}\big{)}$
is a CK space of dimension $0$; it consists of one point,
$E_{1}=\mathbb{R}\boldsymbol{e}_{1}$.
$\xi_{\mathbf{\varepsilon}}(E_{1},E_{1})=0$. The only total polar of $E_{1}$
is the empty set.
(In all CK spaces with underlying projective space $\textrm{P}\boldsymbol{V}$
the total polar of $0$ is $\textrm{P}\boldsymbol{V}$ and vice versa.)
(2) The isotropic points on the hyperbolic line
$\big{(}\textrm{P}\mathbb{R}^{2};{\mathfrak{A}_{\mathbf{\varepsilon}}=\textrm{diag}{(1,-1)}}\big{)}$
are the points $P_{1}=(1:1)$ and $P_{2}=(1:-1)$. (They are isotropic of degree
0.) The quadrance of two distinct points on this line is not defined if one of
these is $P_{1}=(1:1)$ or $P_{2}=(1:-1)$. The $\mathbf{\varepsilon}$-quadrance
of two anisotropic points $Q=(q_{1}{:}q_{2})$ and $R=(r_{1}{:}r_{2})$ is
$\displaystyle\xi_{\mathbf{\varepsilon}}(Q;R)=\frac{(q_{1}r_{2}-r_{1}q_{2})^{2}}{(q_{1}^{\,2}-q_{2}^{\,2})(r_{1}^{\,2}-r_{2}^{\,2})}$.
As ${\mathfrak{A}_{\mathbf{\varepsilon}}}$, $\xi_{\mathbf{\varepsilon}}(Q;R)$
is independent of $\mathbf{\varepsilon}$.
$Q^{\perp}=\\{(q_{2}{:}q_{1})\\}$, $P_{1}^{\perp}=\\{P_{1}\\}$ and
$P_{2}^{\perp}=\\{P_{2}\\}$.
(3) On the parabolic line
$\big{(}\textrm{P}\mathbb{R}^{2};{{\mathfrak{A}_{\mathbf{\varepsilon}}}}\big{)}$,
$\mathfrak{A}_{\mathbf{\varepsilon}}=\textrm{diag}(1,\mathbf{\varepsilon})$,
the point $P=(0{:}1)$ is anisotropic of degree 1, all other points are
anisotropic of degree $0$. The $\mathbf{\varepsilon}$-quadrance between two
points $Q=(1{:}q_{2})$ and $R=(1{:}r_{2})$ is
$\mathbf{\varepsilon}(q_{2}-r_{2})^{2}$, the quadrance between $P$ and $Q$ is
1. The only total polar of $Q$ is the point $P$, while
$P^{\perp}=\textrm{P}\mathbb{R}^{2}$.
(4) Consider the CK space
$(\textrm{P}\mathbb{R}^{3};\mathfrak{A}_{\mathbf{\varepsilon}})$,
$\mathfrak{A}_{\mathbf{\varepsilon}}=\textrm{diag}(1,1,\mathbf{\varepsilon})$
. It is called the co-euclidean plane or polar-euclidean plane. The point
$E_{3}$ is anisotropic of degree 1; it is often called the absolute midpoint.
All other points are anisotropic of degree 0. $E_{3}^{\;\perp}$ is the set of
all lines in $\textrm{P}\mathbb{R}^{3}$. $E_{1}^{\;\perp}$ consists of only
one element, the line $E_{2}\sqcup E_{3}$. The polar variety of the line
$E_{1}\sqcup E_{2}$ consists of one point, the point $E_{3}$, while the polar
variety of the line $E_{1}\sqcup E_{3}$ is the set (the range) of points on
the line $E_{1}\sqcup E_{2}$, thus $(E_{1}\sqcup E_{3})^{\perp}=E_{1}\sqcup
E_{2}$.
Let $Q=(q_{1}{:}q_{2}{:}q_{3})$ and $R=(r_{1}{:}r_{2}{:}r_{3})$ be anisotropic
points of degree 0. If $Q$ and $R$ are not on one line with $E_{3}$, then
$\displaystyle\xi_{\mathbf{\varepsilon}}(Q,R)=\frac{(q_{1}r_{2}-r_{1}q_{2})^{2}}{(q_{1}^{\,2}-q_{2}^{\,2})(r_{1}^{\,2}-r_{2}^{\,2})}$
. If $Q,R,E_{3}$ are collinear, then
$\displaystyle\xi_{\mathbf{\varepsilon}}(Q,R)=\mathbf{\varepsilon}\,\frac{(q_{1}r_{3}-r_{1}q_{3})^{2}+(q_{2}r_{3}-r_{2}q_{3})^{2}}{(q_{1}^{\,2}-q_{2}^{\,2})(r_{1}^{\,2}-r_{2}^{\,2})}$.
The quadrance between $E_{3}$ and an anisotropic point $P^{{\,}^{\,}}\neq
E_{3}$ is $\,\xi_{0}(E_{3},P)=\xi_{\mathbf{\varepsilon}}(E_{3},P)=1$. (The
quadrance between two points which are anisotropic of different degree takes
always the value 1.)
Let us calculate the quadrance between two lines in this plane. All lines are
anisotropic, and for all lines $L$ we find
$\xi_{\mathbf{\varepsilon}}(L,L)=0$. Let $L_{1},L_{2}$ be two distinct lines.
If $L_{1},L_{2}$ both pass through $E_{3}$, then
$\xi_{\mathbf{\varepsilon}}(L_{1},L_{2})=\xi_{\mathbf{\varepsilon}}(S,T)$,
where $S\in L_{1}$ and $T\in L_{2}$ are any two points different from $E_{3}$.
If just one of these lines passes through $E_{3}$, then
$\xi_{\mathbf{\varepsilon}}(L_{1},L_{2})=1$. We now assume that neither
$L_{1}$ nor $L_{2}$ passes through $E_{3}$ and that they meet at a point
$P=(p_{1}{:}p_{2}{:}p_{3})$. If $Q=(q_{1}{:}q_{2}{:}q_{3})\neq P$ is another
point on $L_{1}$ and $R=(r_{1}{:}r_{2}{:}r_{3})\neq P$ a point on $L_{2}$ we
calculate $\xi_{\mathbf{\varepsilon}}(L_{1},L_{2})$, getting

$\displaystyle\xi_{\mathbf{\varepsilon}}(L_{1},L_{2})=\mathbf{\varepsilon}\,\frac{(p_{1}^{2}+p_{2}^{2})(\det\left(\begin{array}[]{rrr}p_{1}&p_{2}&p_{3}\\\
q_{1}&q_{2}&q_{3}\\\
r_{1}&r_{2}&r_{3}\end{array}\right))^{2}}{(p_{1}q_{2}-p_{2}q_{1})^{2}(p_{1}r_{2}-p_{2}r_{1})^{2}}\,$.

When we choose $Q=L_{1}\sqcap P^{\mathfrak{A}_{0}}$ and $R=L_{2}\sqcap
P^{\mathfrak{A}_{0}}$, then
$\xi_{\mathbf{\varepsilon}}(L_{1},L_{2})=\xi_{\mathbf{\varepsilon}}(Q,R)$.
(5) We calculate the $\boldsymbol{\varepsilon}$-quadrance of two lines in the
euclidean space $(\textrm{P}\mathbb{R}^{4}$,
$\mathfrak{A}_{\boldsymbol{\varepsilon}}),\mathfrak{A}_{\boldsymbol{\varepsilon}}=\textrm{diag}(1,\boldsymbol{\varepsilon},\boldsymbol{\varepsilon},\boldsymbol{\varepsilon})$.
Let $P:=(1{:}p_{2}{:}p_{3}{:}p_{4})$ and $Q:=(1{:}q_{2}{:}q_{3}{:}q_{4})$ be
two regular points in $\textrm{P}\mathbb{R}^{4}$, and let
$R:=(0{:}r_{2}{:}r_{3}{:}r_{4})$, $S:=(0{:}s_{2}{:}s_{3}{:}s_{4})\in A_{1}$ be
points ”at infinity”. Put $\boldsymbol{p}:=(p_{2},p_{3},p_{4})$,
$\boldsymbol{q}:=(q_{2},q_{3},q_{4})$, $\boldsymbol{r}:=(r_{2},r_{3},r_{4})$
and $\boldsymbol{s}:=(s_{2},s_{3},s_{4})$. Without loss of generality we may
assume that
$|\boldsymbol{r}|^{2}=r_{2}^{\,2}{\,+\,}r_{3}^{\,2}{\,+\,}r_{4}^{\,2}=1=|\boldsymbol{s}|^{2}$.
If $R\neq S$, then $\xi_{\boldsymbol{\varepsilon}}(P\sqcup R,Q\sqcup
S)=|\boldsymbol{r}\times\boldsymbol{s}|^{2}=\xi_{\boldsymbol{\varepsilon}}(R,S)$.
If $R=S$, then $\xi_{\boldsymbol{\varepsilon}}(P\sqcup R,Q\sqcup
S)=\boldsymbol{\varepsilon}|\boldsymbol{r}\times(\boldsymbol{p}-\boldsymbol{q})|^{2}$.
This number can be interpreted as the euclidean squared distance of the
parallel lines $P\sqcup R$, $Q\sqcup S$. There is one total polar of $P\sqcup
R$ passing through $P$. This plane meets the line $Q\sqcup R$ at a point
$Q^{\prime}$ with
$\xi_{\boldsymbol{\varepsilon}}(P,Q^{\prime})=\boldsymbol{\varepsilon}|\boldsymbol{r}\times(\boldsymbol{p}-\boldsymbol{q})|^{2}.$
Remark: E. Study [29, p. 205] introduced an angular distance of two skew lines
which is a dual number made up of the angle and the euclidean distance of
these lines.

### 3.5.

Reflections on
$\big{(}\textrm{P}\mathbb{R}^{n+1};\mathfrak{A}_{\mathbf{\varepsilon}}\big{)}$.
Let $Q,R,S\in\textrm{P}\mathbb{R}^{n+1}$ be anisotropic of degree 0.
(1) Assume that $U$ is a plane in $\textrm{P}\mathbb{R}^{n+1}$ and $R\in U$.
The image of $Q$ under the reflection in $R$ has the same distance from $U$ as
$Q$.
(2) Let $Q^{\prime}$ be the image of $Q$ under a reflection in $R$ and
$Q^{\prime\prime}$ the image of $Q^{\prime}$ under a reflection in $S$. Put
$E:=Q\sqcup R\sqcup S$. If $\textrm{dim}(E){\,<\,}2$, then the distance of $Q$
and $Q^{\prime\prime}$ is twice distance of $R$ and $S$. Otherwise there is
uniquely determined point $P\in E$, the pole of $R\sqcup S$ in $E$, and angle
between $P\sqcup Q$ and $P\sqcup Q^{\prime\prime}$ is twice the angle between
$P\sqcup R$ and $P\sqcup S$.
Proof of (1). Put $m:=\textrm{dim}(U)$. We may assume that $Q\notin U$. We can
find points
$P_{1}=\mathbb{R}(\boldsymbol{p}_{1}),\dots,P_{m+2}=\mathbb{R}(\boldsymbol{p}_{m+2})\in\textrm{P}\mathbb{R}^{n+1}$
such that $U=P_{1}\sqcup\dots\sqcup P_{m+1}$, $Q\in U\sqcup P_{m+2}$ and
$\boldsymbol{p}_{i}{\scriptstyle{[\mathfrak{A}_{0}]\,}}\boldsymbol{p}_{j}=0$
for $1\leq i<j\leq m{+}2$. Put
$\tau_{i}{:=\,}\boldsymbol{p}_{i}{\,\scriptstyle{[\mathfrak{A}_{0}]\,}\,}\boldsymbol{p}_{i},\,i=1,\dots,n{+}2$.
We can find real numbers $q_{1},\dots,q_{m+2},r_{1},\dots,r_{m+1}$ with
$Q{\,=\,}\mathbb{R}(\sum_{1\leq i\leq
m{+}2}q_{i}\boldsymbol{p}_{i}),\,R{\,=\,}\mathbb{R}(\sum_{1\leq i\leq
m{+}1}r_{i}\boldsymbol{p}_{i})$. The image of $Q$ under the reflection in $R$
is the point

$Q^{\prime}=\mathbb{R}\big{(}q_{m+2}\boldsymbol{p}_{m+2}+\sum_{1\leq i\leq
m{+}1}(q_{i}+\lambda r_{i})\boldsymbol{p}_{i}\big{)}$ with
$\displaystyle\lambda=-2\frac{\sum_{1\leq i\leq
m{+}1}q_{i}r_{i}\tau_{i}}{\sum_{1\leq i\leq m{+}1}r_{i}^{\;2}\tau_{i}}$.

The distance between $Q$ and $U$ is the same as the distance between the
points $Q$ and $\tilde{Q}:=\mathbb{R}(\sum_{1\leq i\leq
m{+}1}q_{i}\boldsymbol{p}_{i})=(P_{m+2}\sqcup Q)\sqcap U$ and the distance
between ${Q}^{\prime}$ and $U$ agrees with the distance between $Q^{\prime}$
and $\tilde{Q}^{\prime}:=\mathbb{R}\big{(}\sum_{1\leq i\leq
m{+}1}(q_{i}+\lambda r_{i})\boldsymbol{p}_{i}\big{)}=(P_{m+2}\sqcup
Q^{\prime})\sqcap U$. So we calculate
$\xi_{\mathbf{\varepsilon}}(Q,\tilde{Q})$ and
$\xi_{\mathbf{\varepsilon}}(Q^{\prime},\tilde{Q}^{\prime})$ and get

$\displaystyle\xi_{\mathbf{\varepsilon}}(Q,\tilde{Q})=\xi_{\mathbf{\varepsilon}}(Q^{\prime},\tilde{Q}^{\prime})=\frac{q_{m+2}(\boldsymbol{p}_{m+2}{\,\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]\,}\,}\boldsymbol{p}_{m+2})}{\sum_{{1\leq
i\leq m{+}2}}q_{i}^{\,2}\tau_{i}}\;.$

Proof of (2). If $Q=R=S$, the statement is obviously true. Let us assume now
that $\textrm{dim}(E)\geq 1$. Let
$P_{1}=\mathbb{R}\boldsymbol{p}_{1},P_{2}=\mathbb{R}\boldsymbol{p}_{2},P_{3}=\mathbb{R}\boldsymbol{p}_{3}\leq\textrm{P}\mathbb{R}^{n+1}$
be independent points with $R{\,=\,}P_{1},R\,\sqcup S\leq P_{1}\sqcup
P_{2},R\,\sqcup S\sqcup Q\leq P_{1}\sqcup P_{2}\sqcup P_{3}$ and
$\boldsymbol{p}_{1}{\scriptstyle{[\mathfrak{A}_{0}]\,}}\boldsymbol{p}_{2}{\,=\,}\boldsymbol{p}_{1}{\scriptstyle{[\mathfrak{A}_{0}]\,}}\boldsymbol{p}_{3}{\,=\,}\boldsymbol{p}_{2}{\scriptstyle{[\mathfrak{A}_{0}]\,}}\boldsymbol{p}_{3}{\,=\,}0$.
(If n=1, we imagine $\textrm{P}\mathbb{R}^{2}$ embedded isometrically in a
plane spanned by $P_{1},P_{2},P_{3}$.) Put
$\tau_{i}{:=\,}\boldsymbol{p}_{i}{\,\scriptstyle{[\mathfrak{A}_{0}]\,}\,}\boldsymbol{p}_{i},\,i=1,2,3$.
Then,
$Q^{\prime}{\,=\,}\mathbb{R}(-q_{1}\boldsymbol{p}_{1}{\,+\,}q_{2}\boldsymbol{p}_{2}+q_{3}\boldsymbol{p}_{3})$
and $Q^{\prime\prime}{\,=\,}\mathbb{R}((-q_{1}+\lambda
s_{1})\boldsymbol{p}_{1}{\,+\,}(q_{2}+\lambda
s_{2})\boldsymbol{p}_{2}+q_{3}\boldsymbol{p}_{3})$ with
$\displaystyle\lambda=2\frac{q_{1}s_{1}\tau_{1}-q_{2}s_{2}\tau_{2}}{s_{1}^{\;2}\tau_{1}+s_{2}^{\;2}\tau_{2}}$.
We first consider the case $Q\in P_{1}\sqcup P_{2}$. In this case,

$\displaystyle\xi_{\mathbf{\varepsilon}}(Q,Q^{\prime\prime})=4\,\frac{s_{2}^{\;2}(\boldsymbol{p}_{2}{\,\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]\,}\,}\boldsymbol{p}_{2})}{s_{1}^{\;2}\tau_{1}}=4\,\xi_{\mathbf{\varepsilon}}(R,S)\;$
if $\tau_{2}=0$

and

$\displaystyle\xi_{\mathbf{\varepsilon}}(Q,Q^{\prime\prime})=4\,\frac{s_{1}^{\;2}s_{2}^{\;2}\tau_{1}\tau_{2}}{(s_{1}^{\;2}\tau_{1}+s_{2}^{\;2}\tau_{2})^{2}}=4\,\xi_{\mathbf{\varepsilon}}(R,S)\big{(}1-\xi_{\mathbf{\varepsilon}}(R,S)\big{)}\;$
if $\tau_{2}\neq 0$ .

This proves the statement if $Q\in P_{1}\sqcup P_{2}$. (The result is
independent of the choice of point $P_{3}$.)
We now omit the restriction $Q\in P_{1}\sqcup P_{2}$ and calculate
$\xi_{\mathbf{\varepsilon}}(P_{3}\sqcup R,P_{3}\sqcup{S})$ and
$\xi_{\mathbf{\varepsilon}}({P_{3}\sqcup Q,}$
${P_{3}\sqcup{Q}^{\prime\prime}})$. We have $P=P_{3}$ and get

$\displaystyle\xi_{\mathbf{\varepsilon}}(P\sqcup Q,P\sqcup
Q^{\prime\prime})=4\,\frac{s_{2}^{\;2}(\boldsymbol{p}_{2}{\,\scriptstyle{[\mathfrak{A}_{\mathbf{\varepsilon}}]\,}\,}\boldsymbol{p}_{2})}{s_{1}^{\;2}\tau_{1}}=4\,\xi_{\mathbf{\varepsilon}}(P\sqcup
R,P\sqcup S)\;$ if $\tau_{2}=0$

and
$\displaystyle\xi_{\mathbf{\varepsilon}}(P\sqcup Q,P\sqcup
Q^{\prime\prime})=4\,\frac{s_{1}^{\;2}s_{2}^{\;2}\tau_{1}\tau_{2}}{(s_{1}^{\;2}\tau_{1}+s_{2}^{\;2}\tau_{2})^{2}}$
= $4\,\xi_{\mathbf{\varepsilon}}(P\sqcup R,P\sqcup
S)\big{(}1-\xi_{\mathbf{\varepsilon}}(P\sqcup R,P\sqcup S)\big{)}\;$ if
$\tau_{2}\neq 0\;.\;\Box$

### 3.6.

Metric affine spaces.
We next consider a CK space
$(\textrm{P}\mathbb{R}^{n+1},\mathfrak{A}_{0},\dots,\mathfrak{A}_{\rho})$,
with $\mathfrak{A}_{0}=\textrm{diag}(1,0,\dots,0)$ and $\rho>0$. The geometry
on $\textrm{P}\mathbb{R}^{n+1}{\,\smallsetminus}{\smallsetminus\,}A_{1}$ is
called metric affine. Therefore we call
$(\textrm{P}\boldsymbol{\mathbb{R}^{n+1}};\mathfrak{A}_{0},\dots,\mathfrak{A}_{\rho})$
a metric affine CK space.
If
$Q=(q_{1}{\,:\dots:\,}q_{n+1})\in\textrm{P}\mathbb{R}^{n+1}{\,\smallsetminus}{\smallsetminus\,}A_{1}$,
we put $Q^{\circ}:=(1,{q_{2}/q_{1}{\,,\dots,\,}q_{n+1}/q_{1}})$. Given two
regular points $Q$ and $R$, then
$\xi_{\mathbf{\varepsilon}}(Q,R){\,=\,}\star_{\mathbf{\varepsilon}}\big{(}({Q^{\circ}-R^{\circ}}){\scriptstyle{\,[\mathfrak{A}_{\mathbf{\varepsilon}}]\,}}({Q^{\circ}-R^{\circ}})\big{)}$
This number can be interpreted as the squared distance of the two points.
(This squared distance can be negative.) For anisotropic points $P$ and $Q$ of
different degree we have $\xi_{\mathbf{\varepsilon}}(P,Q)=1$.
Special cases: (1) If $\rho=1$ and $(A_{1};{\mathfrak{A}_{1}})$ is an elliptic
space, the geometry on
$\textrm{P}\boldsymbol{V}{\,\smallsetminus}{\smallsetminus\,}A_{1}$ is called
euclidean. (2) If $n=4$ and $(A_{1};{\mathfrak{A}_{1}})$ is a hyperbolic
space, the geometry on
$\textrm{P}\boldsymbol{V}{\,\smallsetminus}{\smallsetminus\,}A_{1}$ is called
Minkowski-geometry (of space-time). (3) If $\rho=2$ and
$(A_{1};\mathfrak{A}_{2},\mathfrak{A}_{3})$ is a polar-euclidean space, the
geometry on
$\textrm{P}\boldsymbol{V}{\,\smallsetminus}{\smallsetminus\,}A_{1}$ is called
galilean. (4) A flag-space can be characterized by $\rho=n$.

Let $M$ be a regular point and $r\in\mathbb{R}\,\mathbf{\varepsilon}$ be a
real infinitesimal. The hypersphere with center $M$ and squared radius $r$ is
a quadric $\mathcal{S}(M,r)$ consisting of all points $Q$ which fulfill the
condition $\xi_{\varepsilon}(Q,M)=r$ if $Q$ is regular, and the condition
${Q{\scriptstyle{\,[\mathfrak{A}_{1}}]\,}}Q=0$, otherwise. Together with $M$
all points of $A_{1}$ are symmetry points of $\mathcal{S}(M,r)$, but
$\mathcal{S}(M,r)$ has more symmetry points if $\rho{\,>\,}1$. Let us denote
the set of symmetry points of $\mathcal{S}(M,r)$ by
$\mathfrak{S}(\mathcal{S}(M,r))$.
Examples: (1) We determine the sphere $\mathcal{S}(M,r)$ with regular center
$M=E_{1}$ and squared radius $r=1\mathbf{\varepsilon}{\,>\,}0$ for the spaces
listed above, together with its symmetry points.
(1a) In the euclidean space of dimension $n$ this sphere is the set
$\\{(1{:}p_{2}{:}\dots:p_{n+1})$ $|$ $\,p_{2}^{2}+\dots+p_{n+1}^{2}=1\\}$, and
$\mathfrak{S}(\mathcal{S}(M,1))=\\{\\!M\\!\\}\cup A_{1}$.
(1b) In the $4$-dimensional Minkowski space-time we have
$\mathcal{S}(M,1)=\\{(p_{1}{:}\dots:p_{5})$
$|\,p_{1}^{2}=p_{2}^{2}{+}p_{3}^{2}{+}p_{4}^{2}{-}p_{5}^{2}\\}$ and
$\mathfrak{S}(\mathcal{S}(M,r))=\\{M\\}\cup A_{1}$. (1c) In the galilean space
of dimension $n{\,>\,}1$ this sphere is $\\{(p_{1}{:}\dots:p_{n+1})$
$|\,p_{1}^{2}=p_{2}^{2}+\dots+p_{n}^{2}\\}$ and the set of its symmetry-points
is $(M\sqcup E_{n+1})\cup A_{1}$. (4) In the flag-space of dimension
$n{\,>\,}1$ we have $\mathcal{S}(M,1)=\\{(p_{1}{:}\dots:p_{n+1})$
$\,|\,p_{1}^{2}=p_{2}^{2}\\}$ and $\mathfrak{S}(\mathcal{S}(M,1))=(M\sqcup
E_{3}\sqcup\dots\sqcup E_{n+1})\cup A_{1}$.
(2) Now we consider spheres with center $M=E_{1}$ and radius $r=0$.
(2a) In the euclidean space of dimension $n$ this sphere is the set
$\\{(1{:}p_{2}{:}\dots:p_{n+1})$ $|\,p_{2}^{2}+\dots+p_{n+1}^{2}=0\\}$. So it
is the point $E_{1}$ with multiplicity $2$ (the double point $E_{1}$) and
$\mathfrak{S}(\mathcal{S}(M,1))=\\{M\\}\cup A_{1}$.
(2b) In the $4$-dimensional Minkowski space-time $\mathcal{S}(M,1)$ is the
light cone consisting of all light lines through $E_{1}$. All of these light
lines are null lines. $\mathfrak{S}(\mathcal{S}(M,0))=\\{M\\}\cup A_{1}$.
(2c) In the galilean space of dimension $n{\,>\,}1$ the sphere
$\mathcal{S}(M,0)$ is the point $E_{1}$ with multiplicity $2$. But if we
accepted the line $E_{1}\sqcup E_{n+1}$ as a null line (for example by
demanding $\mathbf{\varepsilon}^{2}=0$), then this line (with multiplicity
$2$) would be $\mathcal{S}(M,0)$. Each point on this line is a symmetry point.
(2d) In the flag-space of dimension $n{\,>\,}1$ the sphere $\mathcal{S}(M,0)$
is the point $E_{1}$ with multiplicity $2$ unless the hyperplane $E_{1}\sqcup
E_{3}\sqcup\dots\sqcup E_{n+1}$ is regarded a null plane. All points of this
hyperplane are symmetry points.

An $(n{-}1)$-dimensional quadric $\mathcal{Q}$ in a metric affine CK space
$\textrm{P}\boldsymbol{V}$ is called a horosphere if there exists a regular
plane $K$ of positive dimension such that $K$ is an axis of symmetry of $Q$
and $Q$ touches the hyperplane $A_{1}$ in a way that the points of $K\cap
A_{1}$ are the points of tangency. We regard $K\cap A_{1}$ as the center of
the horosphere.
Examples: In the galilean CK space of dimension $n$ the horospheres passing
through the point $E_{1}$ are the quadrics
$\\{(p_{1}{:}\dots:p_{n+1})$$|\,p_{1}p_{n+1}=a\,(p_{2}^{2}+\dots+p_{n}^{2})\\}$,
$a\in\mathbb{R}^{\times}$, while in the $n$-dimensional flag-space these are
the quadrics $\\{(p_{1}{:}\dots:p_{n+1})$$|\,p_{1}p_{n+1}=a\,p_{2}^{2}\\}$,
$a\in\mathbb{R}^{\times}$. In the first case $E_{n+1}$ is the center of the
horosphere, in the second $E_{3}\sqcup\dots\sqcup E_{n+1}$.

### 3.7.

($n{+}1$)-simplices in a metric affine CK space of dimension $n$.
Let
$P_{1},\dots,P_{n+1}\in\textrm{P}\boldsymbol{V}\,{\smallsetminus}{\smallsetminus}\,A_{1}$
be independent regular points in a metric affine CK space
$(\textrm{P}\boldsymbol{V},{\mathfrak{A}_{0}},\dots,\mathfrak{A}_{\rho})$ of
dimension $n>1$. Then
$\Sigma{\,:=\,}\\{[q_{1}{:}\dots:q_{n+1}]\,|\,q_{1}{,\dots,}q_{n+1}\in\mathbb{R}^{\geq
0}\\}$ is an $(n{+}1)$-simplex with vertices $P_{1}{,\dots,}P_{n+1}$. The
hyperplane at infinity $A_{1}$ consists of all points
$Q=[q_{1}{:\dots:}q_{n+1}]$ with $q_{1}{\,+\dots+\,}q_{n+1}=0$.
While all points of $\Sigma$ are regular, this is not the case for the mates
of $\Sigma$.
We assume that for each sideplane $U$ of $\Sigma$ the intersection of $U$ with
$A_{1}$ is anisotropic with respect to
${\scriptstyle{\,[\mathfrak{A}_{1}]\,}}$. (It is possible to find such a
collection of points $P_{1},\dots,P_{n+1}$.) It follows that none of the lines
$P_{i}\sqcup P_{j}$, $1\leq i<j\leq n{+}1$, is a light-line, thus
$d_{ij}{:=\,}\xi_{\mathbf{\varepsilon}}(P_{i},P_{\\!j})\not\in\mathbb{R}\,\mathbf{\varepsilon}^{2}$
for $1{\,\leq\,}i{\,<\,}j{\,\leq\,}n{+}1$.
If $Q=(q_{1}{:}\dots:q_{n+1})\in\textrm{P}\boldsymbol{V}$ is a regular point,
then the vector $\displaystyle
Q^{\circ}=(1,\frac{q_{2}}{q_{1}},\dots,\frac{q_{n+1}}{q_{1}})\in\boldsymbol{V}$
can be assigned to $Q$. If $Q$ is regular and
$Q^{\circ}=q_{1}P_{1}^{\;\circ}+\dots+q_{n+1}P_{\\!n+1}^{\;\,\circ}$, we write
$Q=[q_{1},\dots,q_{n+1}]$. Given two regular points $Q=[q_{1},\dots\,q_{n+1}]$
and $R=[r_{1},\dots\,r_{n+1}]$, their squared distance can be calculated by
$\,\xi_{\varepsilon}(Q,R)=-\sum_{1\leq i<j\leq
n{+}1}d_{ij}(q_{i}-r_{i})(q_{j}-r_{j})$. (For a proof see [4].)
Altidudes of $\Sigma$: Put ${\mathcal{S}}:=\\{P_{1},\dots,P_{n+1}\\}$, and for
$i\in\\{1,\dots,n{+}1\\}$ put
$S_{i}{\,:=\,}\textrm{span}({\mathcal{S}}{\smallsetminus}{\smallsetminus}\\{P_{i}\\})$.
The absolute polar of $S_{i}$ is an anisotropic point $R_{i}$ in $A_{\rho}$.
The line $L_{i}:=P_{i}\sqcup R_{i}$ is called the altitude for $S_{i}$. The
squared distance of two regular points on an altitude is an infinitesimal
number in $\mathbb{R}\,\mathbf{\varepsilon}^{\rho}$. Thus, the altitudes are
light lines iff $\rho>1$.

Let $Q$ be the intersection of the hyperplane $S_{n+1}=P_{1}\sqcup\dots\sqcup
P_{n}$ with its altidude $L_{n+1}$. Then
$\psi({\mathcal{S}})=\psi(\\{P_{1}{,\dots,}P_{n}\\})\,\psi(P_{n+1},Q)\in\mathbb{R}\,\mathbf{\varepsilon}^{n+\rho-1}$.
If $\rho{\,=\,}1$,
$\psi({\mathcal{S}})=\star_{\mathbf{\varepsilon}}(\det(1-\frac{1}{2}d_{ij})).$
Remark: If $\rho{\,=\,}1$, $\displaystyle\psi({\mathcal{S}})$ can also be
calculated using the Cayley-Menger determinant:

$\psi({\mathcal{S}})={(-1)^{n+1}2^{-n}\det(\left(\begin{array}[]{rr}0&\boldsymbol{1\;\;\;\;\;\;\;\;}\\\
\boldsymbol{1}&(d_{ij})_{1\leq i,j\leq n{+}1}\end{array}\right))}$ .

The measure of dihedral angles:

$\xi_{\mathbf{\varepsilon}}(S_{i},S_{j})\,\psi({\mathcal{S}}{\smallsetminus}{\smallsetminus}\\{P_{i}\\})\,\psi({\mathcal{S}}{\smallsetminus}{\smallsetminus}\\{P_{j}\\})=\psi({\mathcal{S}}{\smallsetminus}{\smallsetminus}\\{P_{i},P_{j}\\})\,\psi({\mathcal{S}}),\;1\leq
i<j\leq n{+}1$.

Important centers of $\Sigma$.
The proper centroid $G$ of $\Sigma$ is the regular point $[1:\dots:1]$. The
line through a proper centroid $Q$ of a $k$-face of $\Sigma$ and the centroid
$R$ of its opposite $(n{\,-\,}k{\,-\,}1)$-face passes through $G$, as was
mentioned before; but in addition we can state that

$(n-k)^{2}\xi_{\varepsilon}(G,Q)=(k+1)^{2}\xi_{\varepsilon}(G,R)$.

The circumcenter. A regular point $O=(1:o_{2}:\dots:o_{n+1})$ is called a
circumcenter of $\Sigma$ if

$(O^{\circ}-P_{1}^{\;\circ}){\scriptstyle{\,[\mathfrak{A}_{0}{\,+\,}\mathfrak{A}_{1}]\,}}(O^{\circ}-P_{1}^{\;\circ})=(O^{\circ}-P_{i}^{\;\circ}){\scriptstyle{\,[\mathfrak{A}_{0}{\,+\,}\mathfrak{A}_{1}]\,}}(O^{\circ}-P_{i}^{\;\circ}),\,2{\,\leq\,}i{\,\leq\,}n{+}1\;.$

$\Sigma$ has exactly one regular circumcenter iff $\rho{\,=\,}1$.
Proof. We rearrange the last system of equations:

$(P_{i}^{\;\circ}-P_{\\!1}^{\;\circ}){\scriptstyle{\,[\mathfrak{A}_{0}{\,+\,}\mathfrak{A}_{1}]\,}}O^{\circ}=(P_{i}^{\;\circ}-P_{\\!1}^{\;\circ}){\scriptstyle{\,[\mathfrak{A}_{0}{\,+\,}\mathfrak{A}_{1}]}}(\frac{1}{2}(P_{i}^{\;\circ}+P_{\\!1}^{\;\circ})),\,2{\,\leq\,}i{\,\leq\,}n{+}1\;.$

This system consists of $n{+}1$ equations. The first of these equations is
trivially fulfilled. The remaining $n$ equations form a system of maximal rank
and there is a single solution for the n-tuple $(o_{2},\dots,o_{n+1})$
precisely when $\rho=1$. We now assume that $\rho=1$. Then the circumcenter
$O$ of $\Sigma$ is a regular point. We want to determine an $(n{+}1)$-tuple
$(o_{1}{:}\dots:o_{n+1})$ of real numbers with $O=[o_{1}{:}\dots:o_{n+1}]$ and
define an $(n{+}1)\times(n{+}1)$-matrix $\tilde{\mathfrak{D}}$ by
$\displaystyle\tilde{\mathfrak{d}}_{ij}=1{\,-\,}\frac{1}{2}d_{ij}$. We can
find a tuple $(o_{1},\dots,o_{n+1})$ such that:

$(o_{1},\dots,o_{n+1})\tilde{\mathfrak{D}}=(1,1,\dots,1)\hskip
7.96677pt(\star).$

Using Cramer’s rule, we get
$O=[\det\tilde{\mathfrak{D}}^{[1]}{\,:\,}\det\tilde{\mathfrak{D}}^{[2]}{\,:\,}\dots\,:\,\det\tilde{\mathfrak{D}}^{[n+1]}]$,
where $\tilde{\mathfrak{D}}^{[i]}$ is the matrix formed by replacing the
$i$-th row of $\tilde{\mathfrak{D}}$ by the row $(1,\dots,1)$.
The circumsphere consists of all points $X=[x_{1},\dots,x_{n+1}]$ satisfying
the equation

$\sum\limits_{1{\leq}i{<}j{\leq}{n{+}1}}d_{ij}x_{i}x_{j}=0\hskip
7.96677pt(\star\star)\,.$

The squared radius of the circumsphere is $\displaystyle
r=-\frac{{\det({d}}_{ij})}{2\det(\left(\begin{array}[]{rr}0&\boldsymbol{1\;\;\;\;\;\;\;\;}\\\
\boldsymbol{1}&(d_{ij})_{1\leq i,j\leq n{+}1}\end{array}\right))}.$
Examples: (1) In a 2-dimensional CK space with $\rho=1$ we get

$O=[d_{23}(d_{12}+d_{13}-d_{23}),d_{13}(d_{12}-d_{13}+d_{23}),d_{12}(-d_{12}+d_{13}+d_{23})]$.

(2) The two above formulae $(\star),(\star\star)$ lead to the correct result
also for the 2-dimensional CK space with $\rho=2$ (the galilean plane): A
point $[q_{1}{:}q_{2}{:}q_{3}]$ in a metric affine CK space of dimension $2$
is a point on the line $A_{1}$ iff $q_{1}{+}q_{2}{+}q_{3}$ $=0$. It follows
that the circumcenter $O=$
$[d_{23}(d_{12}{+}d_{13}{-}d_{23}):d_{13}(d_{12}{-}d_{13}{+}d_{23}):d_{12}({-}d_{12}{+}d_{13}{+}d_{23})]$
is a point on $A_{1}$ iff
${-\,}d_{12}^{2}{-}d_{13}^{2}{-}d_{23}^{2}{+}2(d_{12}d_{13}{+}d_{12}d_{23}{+}d_{13}d_{23})=0$.
But this equation is a necessary and sufficient condition for the plane to be
a galilean plane, cf [6].
We will except
$\\{[x_{1}:x_{2}:x_{3}]\,|\,\sum\limits_{1{\leq}i{<}j{\leq}{3}}d_{ij}x_{i}x_{j}=0\,\\}$
as a circumcircle of the triangle $(P_{1},P_{2},P_{3})$ even though we know
that it is a horocircle.

We give up the restriction $\rho=1$. A plane $M\leq A_{1}$ is called the
circumcenter of $\Sigma$ if there exists a horosphere with center $M$ passing
through all vertices of $\Sigma$.