# Center of Mass Technique and Affine Geometry

Askold Khovanskii The work was partially supported by the Canadian Grant No.
156833-17.

###### Abstract

The notion of center of mass, which is very useful in kinematics, proves to be
very handy in geometry (see [1]–[2]). Countless applications of center of mass
to geometry go back to Archimedes. Unfortunately, the center of mass cannot be
defined for sets whose total mass equals zero. In the paper we improve this
disadvantage and assign to an $n$-dimensional affine space $L$ over any field
$\mathbf{k}$ the $(n+1)$-dimensional vector space $\widehat{M}(L)$ over
$\mathbf{k}$ of weighty points and mass dipoles in $L$. In this space the sum
of weighted points whose total mass $\lambda\neq 0$ is equal to the center of
mass of these points, equipped with mass $\lambda$. We present several
interpretations of the space $\widehat{M}(L)$ and a couple of its applications
to geometry. The paper is self-contained and is accessible for undergraduate
students.

To the memory of Sergei Duzhin

##  1 Introduction

1.1. I first met Sergei Duzhin long time ago: Sergei was an active member
famous Arnold’s mathematical seminar in Moscow which was an important part of
my life.

Sergei was an attractive and original person. He was easily learning foreign
languages, and while visiting universities in different countries he was
starting to lecture in the language of the host country. He took an accordion
with him to trips, often sang his favorite songs, his friends and colleagues
sang along with him. Sergei organized interesting mathematical seminars, first
in Pereslavl-Zalessky, and then in St. Petersburg, which attracted many
visitors. He was very friendly, and like many others I enjoyed his company.

He had a wide range of interests, for example, he took great pleasure in
studying the history of St. Petersburg. He was interested in very different
areas of mathematics, and I would be glad to discuss the center of mass
technique with him.

1.2. A classical approach to geometry of masses had to distinguish between two
cases: that of a nonzero total mass, and that of zero total mass. The
importance of keeping in mind the second option was understood long ago; it
led naturally to the concept of projective closure and, ultimately, to
development of projective geometry. On the other hand, the idea of this paper
is to treat the two cases uniformly. We will discuss several descriptions of
the vector space, in which the “calculus of masses” naturally takes place, and
how these descriptions are interrelated.

1.3. Applications of the center of mass technique to geometry go back to
Archimedes. They are based on existence of the center of mass of any weighty
set of points (i.e., of any finite set of points equipped with masses whose
total mass is not equal to zero). Center of mass satisfies some nice
properties (see Axioms 1 and 2 in Section 2.2) which suggest its applications
to geometry.

As Archimedes, one can make geometrical discoveries heuristically, believing
in the existence of the center of mass (see Section 2). Of course, it is not
hard to rigorously justify the center of mass technique (see Theorems 2.1–2.2
below).

1.4. One tempted to look for a commutative group of weighty points (i.e., of
points equipped with non zero masses) in an affine space $L$ such that the sum
of weighty points in the group is equal to the center of mass of these points
equipped with their total mass.

Such a group cannot be defined: one cannot sum a set of weighty points, whose
total mass is equal to zero, since the center of mass of such a set does not
exist.

One can improve this situation and define a vector space $\widehat{M}(L)$,
whose elements are weighty points and mass dipoles in $L$, defined up to a
shift.

Instead of the center of mass, one can assign to any weightless set, i.e., to
a set of points whose total mass is equal to zero, its mass dipole (see below)
defined up to a shift.

A mass dipole $\\{-A,B\\}$ in $L$ is an ordered pair of points $A,B$ equipped
with masses $-1,1$, correspondingly. Two mass dipoles $\\{-A,B\\}$ and
$\\{-C,D\\}$ are equivalent, if the oriented segments $AB$ and $CD$ are
parallel, equal and having the same direction (in the other words, if the
ordered pairs of points $(A,B)$ and $(C,D)$ are equal up to a shift).

In the additive group of the space $\widehat{M}(L)$ the sum of weighty points,
whose total mass is $\lambda\neq 0$, equals the center of mass of the set of
these points, equipped with mass $\lambda$.

The space $\widehat{M}(L)$ has many geometrical applications besides classical
applications (see [1]–[2]) of the center of mass. To show how it works we
present an example of such application in Section 8.

1.5. In this paper, we introduce several interpretations of the vector space
$\widehat{M}(L)$ over $\mathbf{k}$ of weighty points and mass dipoles in an
affine space $L$ over $\mathbf{k}$. Let us briefly discuss these
interpretations.

With the affine space $L$ one can associate the vector space $M(L)$ of moment-
like affine maps $P:L\to\overline{L}$ of $L$ to the space $\overline{L}$ of
free vectors in $L$ whose linear part (which is a map of the space
$\overline{L}$ to itself) is proportional to the identity map. The space
$M(L)$ contains a subspace $M_{0}(L)$ of codimension one which consists of
constant maps $P:L\to\overline{L}$ whose linear parts are equal to zero.

The space $\widehat{M}(L)$ is isomorphic to the space $M(L)$. We prove basic
properties of the space $\widehat{M}(L)$ using this isomorphism.

The space $\widehat{M}(L)$ can be defined as the factor-space $D(L)/DM(L)$ of
the infinite dimensional vector space $D(L)$ of all weighted sets in $L$,
i.e., all finite sets of points in $L$ equipped with masses
$\lambda_{i}\in\mathbf{k}$, by its subspace $DM(L)$, consisting of null sets
in $L$, whose total masses and mass dipoles are equal to zero (see Section
5.2). The isomorphism $\widehat{M}(L)\sim D(L)/DM(L)$ allows to assign to any
affine map $F:L_{1}\to L_{2}$ the corresponding linear map
$F_{*}:\widehat{M}(L_{1})\to\widehat{M}(L_{2})$.

The space $\widehat{M}(L)$ consists of weighty points and mass dipoles in $L$
defined up to a shift. One can define vector space operations (either an
addition of two vectors or a multiplication of a vector by an element of the
field $\mathbf{k}$) on the space $\widehat{M}(L)$ by listing rules which allow
to perform these operations (see Section 7.1). Such presentation of the space
$\widehat{M}(L)$ is the most convenient for geometrical applications.

If an affine space $L$ is an affine hyperplane, not passing through the origin
in a vector space $\mathbf{L}$, then the space $\widehat{M}(L)$ can be
identified with the ambient vector space $\mathbf{L}$. Such identification
provides the most visual interpretation of the space $\widehat{M}(L)$. It
works only if $L$ is an affine hyperplane in $\mathbf{L}$ not passing through
its origin.

One can canonically represent any affine space $L$ as the characteristic
hyperplane (see Section 10.3) not passing through the origin in the space
$P_{1}^{*}(L)$, dual to the space $P_{1}(L)$ of polynomials on $L$ whose
degree is $\leq 1$ (see Section 10.3). This representation implies that the
space $\widehat{M}(L)$ is canonically isomorphic to the space $P_{1}^{*}(L)$.

Let $L_{B}$ be a vector space of all polynomials on $L$ of degree $\leq 2$,
whose homogeneous parts of degree 2 are proportional to a non-degenerate
quadratic form $Q_{B}$ (whose symmetric bilinear form is a fixed form $B$).

One can show that the space $\widehat{M}(L)$ is isomorphic to the space of
differentials of polynomials from the space $L_{B}$. This observation implies
that the critical points of polynomials from the space $L_{B}$ obey the same
laws as the centers of mass of weighty points in the affine space $L$ (see
Section 10.4).

1.6. A few words about the organization of this paper.

Section 2 contains an introduction of the classical center of mass technique.
This technique has the following disadvantage: center of mass cannot be
defined for sets whose total mass is equal to zero. In Section 3, we discuss
how to improve this method. Next sections contain a detailed presentation of
such improvement.

In Section 4, we recall definitions and basic properties of affine spaces over
arbitrary fields, and of affine maps (see [3]).

Sections 5–7 contain the central definitions and results of the paper: here we
discuss weighted sets of points in affine spaces, their moments and moments
maps, moment-like maps and the vector space $\widehat{M}(L)$ of weighty points
and mass dipoles in an affine space $L$.

In Section 8, we prove classical theorems on three altitudes in a triangle and
on the Euler line in a triangle, applying mass dipoles and centers of mass.

In Section 9, we discuss relations between affine geometry of a hyperplane $L$
in a vector space $\mathbf{L}$, not passing through its origin, and geometry
of the vector space $\mathbf{L}$. We also show that any affine space can be
canonically embedded into a vector space as an affine hyperplane not passing
through the origin.

In Section 10, we show first that an affine map $F:L_{1}\to L_{2}$ induces a
linear map $F_{*}:\widehat{M}(L_{1})\to\widehat{M}(L_{2})$. Then we present
several interpretations of the space $\widehat{M}(L)$.

1.7. Vladlen Timorin made many valuable suggestions which allowed me to
improve the exposition. He also edited my English. My wife Tatiana
Belokrinitskaya helped me writing the paper. In particular, she typed and
edited it. I am grateful to both of them: without their help, I would not be
able to complete this project.

##  2 Heuristic Applications of Centers of Mass in Geometry

A set of weighed points $\widetilde{A}=\\{(A_{i},\lambda_{i})\\}$ in a real
$n$-dimensional space $L$ is a finite set of points $A_{i}\in L$ equipped with
(positive or negative) numbers $\lambda_{i}$ which are called “masses” of the
corresponding points. The total mass $\lambda=\lambda_{T}(\widetilde{A})$ of
the set $\widetilde{A}$ is the sum $\sum\lambda_{i}$ of masses of all points
in the set. The set is weighty if its total mass is not equal to zero, and is
weightless if its total mass is equal to zero.

According to kinematics, one can assign to each weighty set $\widetilde{A}$ a
point which is called the center of mass of the set $\widetilde{A}$. The map,
that sends each weighty set to its center of mass, satisfies some nice
properties called Axioms 1 and 2 (see below).

###  2.1 Intuitive Meaning of the Center of Mass

Let us discuss an intuitive meaning of the center of mass of a weighty set
$\widetilde{A}$ in a plane $L$.

One possible way of understanding what the center of mass means is to imagine
the plane $L$ as a big flat weightless tabletop. Imagine also that to each
point $A_{i}$ equipped with a mass $\lambda_{i}$ a force proportional to
$\lambda_{i}$ is applied that pulls the tabletop down if $\lambda_{i}$ is
positive, or pulls it up if $\lambda_{i}$ is negative.

It is known from numerous experiments that one can support the tabletop on
just one leg; the position of the leg depends on the weighty set
$\widetilde{A}$. This point is called the center of mass of $\widetilde{A}$.

Let us see what the center of mass is for a system of two positive point
masses $m_{1}$ and $m_{2}$ located at points $A$ and $B$.

Experiments show that if $m_{1}$ and $m_{2}$ are positive, the center of these
masses is located at the point $C$ on the line segment $AB$ such that

$m_{1}\cdot AC=m_{2}\cdot CB.$

Here $AC$, $CB$ are the oriented lengths of the corresponding segments. Since
the point $C$ is located between the points $A$, $B$, the oriented lengths
have the same sign and can be taken positive.

One can consider negative masses as well. A point with a negative mass could
be considered as a point of application of a force proportional to its mass
and acting not down but up.

The center of a weighted couple of points $A$ and $B$, equipped with (not
necessarily positive) masses $m_{1}$ and $m_{2}$, is defined by the same
formula, but one should allow the point $C$ to be located on the line $AB$,
not necessarily in the segment $AB$.

Figure 1: If the mass at $B$ is twice as big as the mass at $A$, then the
center of mass $C$ will be two times closer to $B$ than to $A$ Figure 2: If
$AC=2BC$, then equilibrium holds if $m_{B}=-2m_{A}$

The identity defining $C$ can be rewritten more symmetrically:

$m_{1}\cdot CA+m_{2}\cdot CB=0.$

If points $A$ and $B$ are equipped with masses $-m$ and $m$ whose sum is equal
to zero, then their center of mass is not defined. If $m$ is the unit mass,
$m=1$, the corresponding pair $\\{-A,B\\}$ is called mass dipole.

Two mass dipoles $\\{-A,B\\}$ and $\\{-C,D\\}$ are equivalent if the ordered
pairs of points $(A,B)$ and $(C,D)$ are equal up to a shift.

For the sake of equilibrium of the tabletop, supported at the pivot point $O$,
the effect of putting a mass dipole $\\{-A,B\\}$ and putting a mass dipole
$\\{-C,D\\}$ on the tabletop, are equal, if the mass dipoles are equivalent.

Mass dipoles together with weighty points play a key role in the extension of
the center of mass technique presented in the paper.

The center of mass of a weighty set containing more than two points can be
found inductively using Axiom 2 (see below).

Figure 3: Force proportional to $m_{1}+m_{2}$ must be placed at $C$ to keep
the table in equilibrium

###  2.2 Axiomatic Definition of the Center of Mass

According to kinematics, centers of mass of weighty sets in a real
$n$-dimensional affine space $L$ must satisfy the following axioms.

Axiom 1. The center of mass of two points $A$ and $B$ equipped with masses
$m_{1}$ and $m_{2}$ with nonzero sum $m_{1}+m_{2}$ is the unique point $C$ on
the line $AB$ such that

$m_{1}\cdot AC=m_{2}\cdot CB\,\,\mbox{or }\,\,m_{1}\cdot CA+m_{2}\cdot CB=0.$

Axiom 2. Let $\widetilde{B}$ be a weighty subset of a weighty set
$\widetilde{A}$. Let $O_{1}$ be the center of mass of $\widetilde{B}$ and let
$m_{1}$ be the total mass of $B$. Then the center of mass of $\widetilde{A}$
coincides with the center of mass of the set obtained from $\widetilde{A}$ by
replacing the subset $\widetilde{B}$ by the point $O_{1}$ equipped with mass
$m_{1}$.

######  Theorem 2.1.

There is a unique way of assigning the center of mass to any weighty set of
points in a real $n$-dimensional affine space $L$ so that Axioms 1 and 2 hold.

We will not prove Theorem 2.1 now. Its classical proof can be described in the
following way (see proof of Theorem 2.1, presented right after Lemma 7.2).

One can define the center of mass by an explicit formula and check that the
center of mass thus defined satisfies Axioms 1, 2. Uniqueness of the center of
mass (see Theorem 2.2 below) implies that the explicit formula provides the
unique possible definition of the center of mass and proves its existence.

######  Theorem 2.2.

Assuming that there is a way of assigning a center of mass to any weighty set
of points in a real $n$-dimensional affine space $L$ so that Axioms 1 and 2
hold, one can compute the center of mass of any weighty set (in many different
ways) using an inductive procedure presented in the proof of Theorem.

######  Remark 2.1.

Theorem does not imply the existence of a center of mass satisfying Axioms 1,
2: different ways of computation of the center of mass could provide different
answers.

######  Remark 2.2.

Theorems 2.1, 2.2 hold for affine spaces over arbitrary field $\mathbf{k}$
whose characteristic is not equal to two.These theorems are not applicable to
affine spaces over fields of characteristic two (the proof of Lemma 2.1
implicitly uses division by two and it does not work over fields of
characteristic two). Our extension of the center of mass technique works for
affine spaces over arbitrary fields.

######  Lemma 2.1.

If a weighty set $\widetilde{A}$ contains at least two points, then it
contains a weighty subset of two points.

###### Proof.

If all subsets of $\widetilde{A}$ containing two points are weightless, then
weights of all points are equal up to sign. A triple of such weighty points
contains a pair of weighty points equipped with equal masses. This pair of
points is a weighty subset of $\widetilde{A}$. ∎

###### Proof of Theorem 2.2.

Using Axioms 1, 2 one can compute the center of mass of any weighty set of
points. Indeed:

  * •

Axiom 1 allows to compute the center of mass for any weighty set containing
two points;

  * •

by Lemma 2.1, a weighty set containing at least two points, contains a weighty
pair of points. By Axiom 2, one can replace this pair of points by its center
of mass equipped with its total mass;

  * •

the above properties allow to find the center of mass of a weighty set
recursively, by reducing it to computation of the center of mass of weighty
sets, containing smaller number of points;

∎

Theorems 2.1, 2.2 have countless applications in geometry: Theorem 2.1 implies
that all the different ways of finding of the center of mass of a given
weighty set, suggested by Theorem 2.2, have to give the same answer.

Classical applications of centers of mass in geometry are based on the above
statement.

Let us present the simplest classical application of centers of mass in
geometry which goes back to Archimedes.

###  2.3 Three Medians in a Triangle (Archimedes)

Let us prove the theorem on three medians in a triangle using the center of
masses technique. A similar proof was discovered by Archimedes who invented
the center of mass technique in geometry.

Suppose we start with a triangle $ABC$ and three unit mass at its vertices.
How can we find the center of mass $O$ of the resulting system?

One way is to combine masses at $A$ and $B$ first. This will give us mass 2
placed at the midpoint of $AB$.

Now, combine the resulting mass with the unit mass at $C$. By doing so, we
obtain that the center of mass $O$ lies on the segment connecting the vertex
$C$ to the midpoint of $AB$ and divides it in the proportion $2:1$ (the part
of it adjacent to the vertex being the longer one).

Figure 4: Center of 3 unit masses and 3 medians

But we could proceed differently: first, combine the masses at $A$ and $C$,
and then combine the result with the mass at $B$. In this way we obtain that
the center of mass $O$ of the system lies on the median from vertex $B$ and
divides it in the ratio $2:1$.

By combining the masses in the third possible order ($B$ and $C$ first and
adding $A$ afterwards), we see that it also lies on the median from the vertex
$A$ and divides it in the ratio $2:1$.

Thus, the center of mass $O$ lies on all three medians of triangle $ABC$ and
divides them in the ratio $2:1$. We proved that all three medians of the
triangle $ABC$ pass through a point $O$ which divides them in the ratio $2:1$.

##  3 How to Improve the Center of Mass Technique?

The classical center of mass technique has the following disadvantage: the
center of mass is not defined for weightless sets.

One is tempted to look for a commutative group of weighty points in $L$ with
the following addition: the sum of weighty points is equal to the center of
mass of the set of these points, equipped with its total mass.

Unfortunately, such a group does not exist: the sum of weighted points whose
total mass is equal to zero, is not defined.

###  3.1 The Group $\widehat{M}(L)$ of Weighty Points and Mass Dipoles

It is possible to improve the center of mass technique and to define a
commutative group $\widehat{M}(L)$ whose elements are weighty points in $L$
and mass dipoles in $L$, defined up to a shift.

Instead of the center of mass, one can assign to a any weightless set its mass
dipole, defined up to a shift (one can identify a mass dipole $\\{-O,B\\}$,
defined up to a shift, with the free vector, represented by an ordered pair of
points $(O,B)$, defined up to a shift).

Addition in the group $\widehat{M}(L)$ satisfies the following condition: the
sum of points, contained in a set $\widetilde{A}$, whose total mass is
$\lambda\neq 0$, equals to the center of mass of $\widetilde{A}$, equipped
with the mass $\lambda$.

Addition in the commutative group $\widehat{M}(L)$ could be defined
axiomatically. Instead of one axiom, describing the center of mass of a
weighty pair of points, one has to introduce four axioms, describing the
following four different types of addition in $\widehat{M}(L)$:

  * •

addition of two weighty points whose total mass is not equal to zero;

  * •

addition of two weighty points whose total mass is equal to zero;

  * •

addition of a weighty point and a mass dipole defined up to a shift;

  * •

addition of two mass dipoles defined up to a shift.

Numerous geometric applications of the group $\widehat{M}(L)$ are based on its
associative property:

addition of elements of the group $\widehat{M}(L)$ performed in different
orders gives the same answer.

Along with classical applications dealing only with the centers of mass of
weighty sets, there are many natural applications, that use weightless sets
and their mass dipoles as well. To show how it works, we describe one simple
geometric application of the group $\widehat{M}(L)$ (see Section 8).

Below we define the group $\widehat{M}(L)$ for any affine space $L$ over an
arbitrary field $\mathbf{k}$. Moreover, we show that the group
$\widehat{M}(L)$ admits a structure of a vector space over the field
$\mathbf{k}$.

The vector space $\widehat{M}(L)$ can be naturally identified with the space
of moment-like maps $P:L\to\overline{L}$ of the affine space $L$ to the space
of free vectors $\overline{L}$ in $L$. We deduce properties of the group
$\widehat{M}(L)$ from the corresponding properties of the additive group of
$M(L)$.

The definition of moment-like maps is motivated by kinematics. In the next
section, we discuss a heuristic equilibrium criterion which suggests an
explicit formula for the center of mass of a weighted set of points. It allows
to justify the classical center of mass technique. It also leads to
definitions of moment maps of weighted sets, of moment-like maps and of group
$\widehat{M}(L)$.

###  3.2 Moments of Weighted Sets about Pivot Points and Moment Maps

Imagine a weightless tabletop $L$ with a weighted set of points
$\widetilde{A}$ on it. Assume that the tabletop is supported by one leg at a
point $O$.

######  Definition 3.1.

The moment $P(O)=\Delta_{\widetilde{A}}(O)$ of a weighted set
$\widetilde{A}=\\{(A_{i},\lambda_{i})\\}$ about a pivot point $O$ is a linear
combination of oriented segments $OA_{i}$ (which we consider as free vectors
in $L$) with coefficients $\lambda_{i}$.

$P(O)=\Delta_{\widetilde{A}}(O)=\sum\lambda_{i}OA_{i}.$ (1)

Kinematics suggests a simple criterion of stability of the tabletop with the
set $\widetilde{A}$ on it, supported at a point $O$.

Kinematical Criterion. The tabletop is in equilibrium if and only if the
moment $P(O)=\Delta_{\widetilde{A}}(O)$ of $\widetilde{A}$ about the pivot
point $O$ equals zero.

Only the moment of the set $\widetilde{A}$ about the pivot point $O$ is
important for the equilibrium of a tabletop supported at $O$: for the sake of
equilibrium of the tabletop supported at t $O$, the effects of putting a
weighted set $\widetilde{A}$ and putting a weighted set $\widetilde{B}$ on the
tabletop are equal if the moments of $\widetilde{A}$ and $\widetilde{B}$ about
the point $O$ are equal.

######  Definition 3.2.

The moment map $P:L\to\overline{L}$ corresponding to a weighted set
$\widetilde{A}$ is the map $P=\Delta_{\widetilde{A}}$, whose value $P(O)$ at a
point $O\in L$ is equal to the moment $\Delta_{\widetilde{A}}(O)$ of the set
$\widetilde{A}$ about the pivot point $O$.

######  Theorem 3.1 (Change of pivot point).

Let $P=\Delta_{\widetilde{A}}$ be the moment map, corresponding to a weighted
set $\widetilde{A}$ whose total mass is equal to
$\lambda=\lambda_{T}(\widetilde{A})$. Then, for any two points $O_{1},O_{2}\in
L$, the map $P$ satisfies the following characteristic relation:

$P(O_{1})-P(O_{2})=-\lambda O_{2}O_{1}.$ (2)

###### Proof.

By (1), we have:

$P(O_{1})=\sum\lambda_{i}O_{1}A_{i}\quad{\text{and}}\quad
P(O_{2})=\sum\lambda_{i}O_{2}A_{i}.$

Subtracting these identities, we obtain

$P(O_{1})-P(O_{2})=-\lambda O_{2}O_{1}.$

∎

######  Definition 3.3.

A moment-like map $P:L\to\overline{L}$ is a map of the affine space $L$ to the
vector space $\overline{L}$ of free vectors on $L$ that satisfies the relation
(2) for some parameter $\lambda\in\mathbf{k}$. The parameter $\lambda$, which
depends on the map $P$, is called the total mass of the map $P\in M(L)$.

It is easy to see that the set $M(L)$ of all moment-like maps has a natural
structure of a vector space, and the total mass $\lambda$ is a linear function
on $M(L)$.

Applying Theorem 6.3, one can check that, if the total mass $\lambda$ of a map
$P\in M(L)$ does not equal to zero, then $P$ vanishes at a unique point $O$
called the center of mass of the map $P\in M(L)$. Such a map is uniquely
determined by its center of mass $O$ and its total mass $\lambda$ (see Theorem
6.4).

Moreover, the map $P$ is equal to the moment map $P=\Delta(\widetilde{A})$ of
the weighty set $\widetilde{A}$ consisting of one point $O$ equipped with the
mass $\lambda$.

Equation (2) implies that, if the total mass $\lambda$ of a map $P\in M(L)$
equals zero, then $P$ is a constant map, i.e., $P\equiv v$, where
$v\in\overline{L}$ is a free vector. The map $P$ is determined by the free
vector $v$.

######  Lemma 3.1.

The moment map $P=\Delta_{\widetilde{A}}$ of a mass dipole
$\widetilde{A}=\\{-O,B\\}$ is a constant map $P\equiv v$, where $v$ is the
free vector represented by an ordered pair of points $(O,B)$.

###### Proof.

The moment of $\widetilde{A}$ about a pivot point $O_{1}$ is equal to the
difference $O_{1}B-O_{1}O$ of oriented segments $O_{1}B$ and $O_{1}O$,
considered as free vectors. This difference is equal to the oriented segment
$OB$ which can be considered as the free vector $v$, represented by an ordered
pair of points $(O,B)$. ∎

Let us introduce the following identifications:

  * •

a moment-like map $P$ with total mass $\lambda\neq 0$ is identified with the
weighty point $\\{(O,\lambda)\\}$, where $O$ is the center of mass of $P$;

  * •

a constant moment-like map $P\equiv v$ is identified with a mass dipole
$\\{-O,B\\}$, defined up to a shift, the ordered pair of points $(O,B)$
represents the free vector $v$.

By this identification, we obtain a vector space $\widehat{M}(L)$ whose
elements are weighty points and mass dipoles, defined up to a shift. The
additive group of the space $\widehat{M}(L)$ can be defined by the axioms
which define the addition in the set $\widehat{M}(L)$.

Let us implement this program in more details.

##  4 Affine Spaces and Affine Maps

Below, we recall that Euclidean spaces, vector spaces over an arbitrary field
$\mathbf{k}$ and shifted vector subspaces in such spaces can be considered as
affine spaces.

###  4.1 Vector Space of Free Vectors in a Euclidean Space

Classical Euclidean geometry, in particular, deals with points, lines and
planes embedded in the three dimensional space. These objects are Euclidean
spaces of dimensions zero, one, two and three.

A Euclidean space $E$ does not have a structure of a real vector space: an
operation of multiplication of a point $A\in E$ by a real number
$\lambda\in\mathbb{R}$ and an operation of addition of two points $A,B\in E$
are not defined.

Nevertheless, to any Euclidean space $E$ one can assign a real vector space
$\overline{E}$ of free vectors in $E$ and define an action of the additive
group of free vectors on the space $E$.

Let us define free vectors in a Euclidean space $E$.

######  Definition 4.1.

Two ordered pairs of points $(A,B)$ and $(C,D)$ from the space $E$ are
equivalent $(A,B)\sim(C,D)$, or are equal up to a Sift, if the oriented
segments $AB$ and $CD$ are parallel, equal in length and point in the same
direction.

######  Definition 4.2.

A free vector in $E$ is an ordered pair of points $(A,B)$ in $E$, defined up
to the equivalence $\sim$.

One can multiply an ordered pair of points $(O,A)$ by any real number
$\lambda$. By definition, $\lambda(O,A)$ is an ordered couple of points
$(O,B)$ such that the points $O,A,B$ belong to the same line, and ratio
$OB:OA$ of oriented length of the segments $OB$ and $OA$ equals $\lambda$.

It is easy to check that if $(A_{1},B_{1})\sim(A_{2},B_{3})$, then
$\lambda(A_{1},B_{1})\sim\lambda(A_{2},B_{3})$.

######  Definition 4.3.

Let $(A,B)$ be an ordered pair of points that represents a free vector
$v\in\overline{E}$. Then, by definition, $\lambda v\in\overline{L}$ is the
free vector represented by the ordered pair $\lambda(A,B)$.

One can add any two ordered pairs of points $(O,A)$ and $(O,B)$. By
definition, $(O,A)+(O,B)$ is an ordered pair of points $(O,C)$ such that the
oriented segments $AC$ and $OB$ are parallel, equal to each other and point to
the same direction.

Easy to check that if $(A_{1},B_{1})\sim(A_{2},B_{2})$ and
$(A_{1},C_{1})\sim(A_{2},C_{2})$, then
$(A_{1},B_{1})+(A_{1},C_{1})\sim(A_{2},B_{2})+(A_{2},C_{2})$.

######  Definition 4.4.

Let $(A,B)$, $(A,C)$ be ordered pair of points which represent free vectors
$V_{1},V_{2}\in\overline{E}$. Then, by definition,
$V_{1}+V_{2}\in\overline{L}$ is the free vector represented by the oriented
pair $(A,B)+(A,C)$.

One can verify that if $E$ in an $n$-dimensional Euclidean space
($n=0,1,2,3$), then the space $\overline{E}$ of free vectors in $E$ is an
$n$-dimensional real vector space.

There is a natural operation of addition for a free vector and a point in the
space $E$. The result of such addition is a point in the space $E$.

One can add a vector to a point.

######  Definition 4.5.

By definition, a point $P\in E$ is the sum $O+v$ of a point $O\in E$ and a
free vector $v\in\overline{E}$ if the ordered pair of points $(O,P)$
represents the free vector $v\in\overline{E}$.

It is easy to check the following lemma.

######  Lemma 4.1.

Let $O$ be any point in $E$. Then

  1. 1.

the identity $O+v=O$ holds if and only if $v$ is the zero vector in the space
of free vectors $\overline{E}$;

  2. 2.

for $v_{1},v_{2}\in\overline{E}$ the identity
$O+(v_{1}+v_{2})=(O+v_{1})+v_{2}$ holds.

  3. 3.

for any point $P\in E$, there is a unique free vector $v\in\overline{E}$ such
that $P=O+v$.

A free vector $v$ defines the shift $Sh_{v}$ of the space $E$ that maps a
point $O\in E$ to the point $Sh_{v}(O)=O+v$.

Lemma 4.1 implies that the shifts $Sh_{v}$ constitute the action of the
additive group of free vectors on the space $E$. Moreover, this action is
transitive and free.

######  Remark 4.1.

Euclidean geometry equips the vector space $\overline{E}$ of free vectors in
$E$ with the inner product $\langle x,y\rangle$ of vectors
$x,y\in\overline{E}$. The inner product is a symmetric bilinear form in the
vector space $\overline{E}$ that is positively defined, i.e., $\langle
x,x\rangle\geq 0$ and $\langle x,x\rangle=0$ only if $x=0$. The inner product
allows to define distances between points and angles between lines in a
Euclidean space $E$.

###  4.2 Vector Spaces as Affine Spaces

Let $\mathbf{L}$ be a vector space over an arbitrary field $\mathbf{k}$. In
this section, we define the structure of an affine space on $\mathbf{L}$,
i.e., we define the vector space $\overline{\mathbf{L}}$ of free vectors in
$\mathbf{L}$ and the action of the additive group of free vectors on the space
$\mathbf{L}$.

######  Definition 4.6.

A shift of the space $\mathbf{L}$ by a vector $v\in\mathbf{L}$ is the map
$Sh_{v}:\mathbf{L}\to\mathbf{L}$ which sends a point $O\in\mathbf{L}$ to the
point $O+v$.

The additive group of the vector space $\mathbf{L}$ acts on the space
$\mathbf{L}$ by shifts: a vector $v\in\mathbf{k}$ sends a point
$O\in\mathbf{L}$ to the point $Sh_{v}(O)$.

######  Definition 4.7.

A free vector in $\mathbf{L}$ is an ordered pair of points $(A,B)$ defined up
to a shift (i.e., for any $v\in\mathbf{L}$ the ordered pair of points
$(A+v,B+v)$ defines the same free vector as the ordered pair of points
$(A,B)$).

One can identify a free vector $f\in\overline{\mathbf{L}}$, represented by a
pair $(A,B)$, with the vector $v=B-A\in\mathbf{L}$. By definition, the vector
$v$ is well defined, i.e., is independent of a choice of a pair $(A,B)$, that
represents the free vector $f$.

######  Definition 4.8.

By definition, a vector space operation on the space $\overline{\mathbf{L}}$
of free vectors is induced from the corresponding vector space operation on
the vector space $\mathbf{L}$ under the identification of
$\overline{\mathbf{L}}$ with the vector space $\mathbf{L}$.

Thus, we defined the vector space $\overline{\mathbf{L}}$ of free vectors in
$\mathbf{L}$. The action of the additive group of $\overline{\mathbf{L}}$ on
$\mathbf{L}$ is defined as follows: a free vector
$(A,B)\in\overline{\mathbf{L}}$ sends a point $O\in\mathbf{L}$ to the point
$Sh_{B-A}(O)=O+B-A\in\mathbf{L}$.

###  4.3 Shifted Vector Subspaces as Affine Spaces

An affine subspace $L\subset\mathbf{L}$ is a vector subspace
$\widetilde{L}\subset\mathbf{L}$ shifted by a vector $O\in\mathbf{L}$. In
general, if $O$ does not belong to $\widetilde{L}$, the space
$L\subset\mathbf{L}$ is not a vector space. Instead, it can be naturally
equipped with a structure of an affine space, i.e., with a vector space
$\overline{L}$ of free vectors on $L$ and with an action of the additive group
of free vectors $\overline{L}$ on the space $L$.

We define an affine structure on $L$ using vector space operations in the
ambient space $\mathbf{L}$.

######  Definition 4.9.

A set $L\subset\mathbf{L}$ is a shifted subspace
$\widetilde{L}\subset\mathbf{L}$ by a vector $O\in\mathbf{L}$ if
$L=Sh_{O}(\widetilde{L})$, i.e., if each point $A\in L$ is representable in
the form $A=\widetilde{A}+O$, where $\widetilde{A}\in\widetilde{L}$. (In other
words, the set $L$ is a coset of the additive group of the vector subspace
$\widetilde{L}$, containing the point $O$.)

######  Definition 4.10.

Two ordered pairs of points $(A,B)$ and $(C,D)$ in an affine space $L$ are
equivalent if $B-A=D-C$. (In other words, the pairs are equivalent if they
define the same free vector in the ambient space $\mathbf{L}$.)

A free vector in an affine space $L$ is an equivalence class of ordered pairs
of points in $L$.

There is a natural embedding of the set $\overline{L}$ to the vector space
$\overline{\mathbf{L}}$ which sends a free vector in $\overline{L}$,
represented by an ordered pair of points $(A,B)$ from $L$, to the free vector
in $\overline{\mathbf{L}}$ represented by the same ordered pair of points
$(A,B)$.

######  Lemma 4.2.

Each free vector $(A,B)\in\overline{L}$ as a free vector in
$\overline{\mathbf{L}}$ has a unique representative of the form
$(\mathbf{O},C)$, where $\mathbf{O}$ is the origin of the vector space
$\mathbf{L}$, and $C$ belongs to the vector space $\widetilde{L}$.

###### Proof.

Indeed, a free vector $(A,B)$ in $\overline{\mathbf{L}}$ is equivalent to a
unique ordered pair of points $(\mathbf{O},B-A)$ of the needed form. ∎

One can identify a free vector $f\in\overline{L}$, represented by a pair
$(A,B)$ with the vector $v=B-A\in\widetilde{L}$. The vector $v$ is well
defined, i.e., is independent of a choice of a pair $(A,B)$ which represent
the free vector $f$.

######  Definition 4.11.

By definition, a vector space operation on the space $\overline{L}$ of free
vectors in $L$ is induces from the corresponding vector space operation on the
space $\widetilde{L}$ under the identification of $\overline{L}$ with the
vector space $\widetilde{L}$.

Thus, we defined the vector space $\overline{L}$. The action of the additive
group of $\overline{L}$ on $L$ is defined as follows: a free vector
$(A,B)\in\overline{L}$ sends a point $O\in L$ to the point
$Sh_{B-A}(O)=O+B-A\in L$. The vector space $\overline{L}$ is naturally
embedded in the of the vector space $\overline{\mathbf{L}}$ and the actions of
the additive groups of these spaces on $L$ agree with this embedding.

Let us choose any point $O$ in an affine space $L$ and identify a point $A\in
L$ with the free vector $\tau_{O}(A)$, such that $O+\tau_{O}(A)=A$ (in other
words, one identifies a point $A\in L$ with the free vector $\tau_{O}(A)$,
represented by the ordered pair of points $(O,A)$.

This identification provides an affine space $L$ with the structure of a
vector space $L_{O}$ whose origin is the point $O$. Different choice of the
origin $O\in L$ provide $L$ with different structures of vector spaces
$L_{O}$.

For any point $O\in L$, the affine space $L$ can be considered as an affine
subspace in the vector space $L_{O}$ (witch as a set coincides with $L_{O}$).
The affine structures induced in $L$ as a subset of $L_{O}$ are the same for
different choices of the point $O$:

  1. 1.

vector spaces of free vectors in different spaces $L_{O}$ can be naturally
identified with each other;

  2. 2.

identified free vectors from different spaces $L=L(O)$ realize the same action
on the space $L$.

The center of mass technique deals with affine spaces over arbitrary fields.
In particular, it is applicable to $n$-dimensional Euclidean spaces which are
real $n$ dimensional affine spaces equipped with an extra structure.

In the next section, we briefly recall the definition of multidimensional
Euclidean spaces. We will not use this definition later.

###  4.4 Multidimensional Euclidean Spaces

One can assign the structure of an affine space $L$ to any vector space
$\mathbf{L}$: as a set, the affine space $L$ coincides with the vector space
$\mathbf{L}$, the space $\overline{L}$ of free vectors in the affine space $L$
and the action of the additive group of free vector on the affine space $L$
have been defined above.

The real $n$-dimensional affine space $R^{n}$ is, by definition, the affine
space associated with the real $n$-dimensional vector space $\mathbb{R}^{n}$.

An $n$-dimensional Euclidean space is a real $n$-dimensional affine space $L$
equipped with a positive definite quadratic form on the vector space of free
vectors on $L$.

Using a positive definite quadratic form on $\overline{L}$, one can define
distances between points, angles between lines in $L$, and develop geometry
based on this notions in $L$.

For $n=1,2,3$ this geometry can be identified with the classical Euclidean
geometries of corresponding dimensions.

Euclidean geometry in the spaces of dimension $n>3$ is a natural and rich
generalization of classical Euclidean geometry.

###  4.5 Affine Maps

Let $L_{1}$ and $L_{2}$ be affine spaces over a field $\mathbf{k}$. A map
$F:L_{1}\to L_{2}$ sends an ordered pair of points $(A,B)$ in $L_{1}$ to the
ordered pair of points $(F(A),F(B))$ in $L_{2}$.

######  Definition 4.12.

A map $F:L_{1}\to L_{2}$ is an affine map if:

  * •

the induced map $\overline{F}:\overline{L}_{1}\to\overline{L}_{2}$ is well
defined, i.e., if the pairs $(A,B)$ and $(C,D)$ define the same free vector in
$\overline{L}_{1}$, then the pairs $(F(A),F(B))$ and $F((C),F(D))$ define the
same free vector in $\overline{L}_{2}$;

  * •

the induced map $\overline{F}:\overline{L}_{1}\to\overline{L}_{2}$ is a linear
map.

Let us choose some points $O_{1}\in L_{1}$ and $O_{2}\in L_{2}$ and consider
affine spaces $L_{1}$ and $L_{2}$ as the vector spaces $(L_{1})_{O_{1}}=L_{1}$
and $(L_{2})_{O_{2}}=L_{2}$.

######  Definition 4.13.

A map $F:(L_{1})_{O_{1}}\to(L_{2})_{O_{2}}$ is a shifted linear map if it can
be represented in the form

$F(x)=Ax+b,$

where $A:(L_{1})_{O_{1}}\to(L_{2})_{O_{2}}$ is an arbitrary linear map (which
is called the linear part of $F$), $b\in(L_{2})_{O_{2}}$ is an arbitrary
vector, and $x$ is any point in the vector space $(L_{1})_{O_{1}}$.

######  Theorem 4.1.

The map $F:L_{1}\to L_{2}$ is an affine map if and only if, for any choice of
points $O_{1}\in L_{1},O_{2}\in L_{2}$, it gives rise to a shifted linear map
of $(L_{1})_{O_{1}}$ to $(L_{2})_{O_{2}}$.

Moreover, the linear part $A$ of the map $F$ can be naturally identified with
the linear map $\overline{F}:\overline{L}_{1}\to\overline{L}_{2}$, which is
independent of choice of points $O_{1}$ and $O_{2}$.

###### Proof.

Assume that $F:L_{1}\to L_{2}$ is an affine map. By definition, $F$ induces a
linear map $\overline{F}:\overline{L}_{1}\to\overline{L}_{2}$. Let us identify
the space $\overline{L}_{1}$ with the space $(L_{1})_{O_{1}}$ and the space
$\overline{L}_{2}$ with the space $(L_{2})_{F(O_{1})}$. Under this
identification, the map $\overline{F}$ can be considered as a linear map
$A:(L_{1})_{O_{1}}\to(L_{2})_{F(O_{1})}$. Let $F(O_{1})$ be a point
$b\in(L_{2})_{O_{2}}$. Then the map $F:L_{1}\to L_{2}$ can be identified with
the map $F(x)=A(x)+b$ from $(L_{1})_{O_{1}}$ to $(L_{2})_{O_{2}}.$

Conversely, any map $F(x)=A(x)+b$ of $(L_{1})_{O_{1}}$ to $(L_{2})_{O_{2}}$,
where $A$ is a linear map and $b$ is an arbitrary vector in $(L_{2})_{O_{2}}$,
obviously, gives rise to an affine map of $L_{1}$ to $L_{2}$. ∎

######  Corollary 4.1.

The space of polynomials of degree $\leq 1$ on $L$ coincides with the space of
affine maps of an affine space $L$ to the one dimensional affine space
$\mathbf{k}^{1}$.

###### Proof.

In the vector space $L_{O}=L$ with the origin at a point $O\in L$, a function
$F:L\to\mathbf{k}$ is a polynomial of degree $\leq 1$ if it can be represented
as $P(x)=L(x)+b$, where $L(x)$ is a linear function, and $b\in\mathbf{k}$ is a
constant. Thus, $F$ is a polynomial of degree $\leq 1$ if and only if it is a
shifted linear map of $L_{O}$ to $\mathbf{k}$. ∎

One can automatically verify the following two lemmas.

######  Lemma 4.3.

Let $L_{1}$ be an affine space over $\mathbf{k}$, and let $L_{2}$ be a vector
space over $\mathbf{k}$. Then the space of affine maps $F:L_{1}\to L_{2}$ is a
vector space over $\mathbf{k}$, i.e., the sum $F_{1}+F_{2}$ of affine maps
$F_{1},F_{2}$ and the product $\lambda F$ of affine map $F$ on
$\lambda\in\mathbf{k}$ are affine maps.

######  Lemma 4.4.

Let $L_{1},L_{2},L_{3}$ be affine spaces over $\mathbf{k}$ and let
$F_{1}:L_{1}\to L_{2}$ and $F_{2}:L_{2}\to L_{3}$ be affine maps, then:

  1. 1.

the composition $F_{3}=F_{2}\circ F_{1}:L_{1}\to L_{2}$ is an affine map;

  2. 2.

the linear part $\overline{F}_{3}$ of the composition is the composition
$\overline{F}_{2}\circ\overline{F}_{1}$ of the linear parts $\overline{F}_{2}$
and $\overline{F}_{1}$ of the maps $F_{2}$ and $F_{1}$.

It is easy to verify the following theorem.

######  Theorem 4.2.

An affine map $F:L_{1}\to L_{2}$ is invertible if and only if its linear part
$\overline{F};\overline{L}_{1}\to\overline{L}_{2}$ is invertible.

All invertible affine maps $F:L\to L$ of an affine space to itself form a
group under composition.

######  Definition 4.14.

The group of affine transformations of an affine space $L$ is the group of all
invertible affine maps $F:L\to L$.

######  Remark 4.2.

According to Felix Klein’s Erlangen program, in affine geometry of an affine
space $L$, two subsets $X$, $Y$ are considered as equal sets if there is an
affine transformation of $L$ which sends the set $X$ to the set $Y$.

Affine geometry studies properties of sets, which are preserved under affine
transformations (i.e., which are the same for equal sets).

##  5 Weighted Sets and their Moment Maps

In this section, we discuss weighted sets in an affine space $L$, their
moments about a pivot point and their moment maps.

###  5.1 The Vector Space of Weighted Sets

In this section, we define the vector space of finite sets of points in $L$
equipped with weights from the field $\mathbf{k}$. The definition does not
rely on the affine structure of $L$ and can be applied to any set $L$.

Let $L$ be an affine space over an arbitrary field $\mathbf{k}$.

######  Definition 5.1.

A weighted set $\widetilde{A}=\\{(A_{i},\lambda_{i})\\}$ in $L$ is a finite
set $A=\\{A_{i}\\}$ of points in $L$ equipped with elements
$\\{\lambda_{i}\\}$ of the field $\mathbf{k}$. The element
$\lambda_{i}\in\mathbf{k}$ is the mass assigned to the point $A_{i}$. The
total mass of a weighted set $\widetilde{A}$ is the element
$\lambda\in\mathbf{k}$ equal to the sum $\sum\lambda_{i}$ of masses
$\lambda_{i}$ of all points $A_{i}$ in the set.

The set $D(L)$ off all weighted sets in $L$ has a natural structure of a
vector space over the field $\mathbf{k}$. Indeed, one can interpret a weighted
set $\widetilde{A}$ as a function $f_{\widetilde{A}}:L\to\mathbf{k}$ which
vanishes on $L\setminus A$ and whose value at each point $A_{i}\in A$ is equal
to $\lambda_{i}$.

######  Definition 5.2.

The space $\widetilde{D}(L)$ is the vector space over $\mathbf{k}$ of all
functions on $L$, taking values in the field $\mathbf{k}$ and equal to zero
everywhere but on a finite set of points. A weighted set $\widetilde{A}\in
D(L)$ can be identified with the function
$f_{\widetilde{A}}\in\widetilde{D}(L)$. This identification provides the set
$D(L)$ with the structure of a vector space over $\mathbf{k}$.

######  Example 5.1.

A partition of a weighted set is a representation of the set as a union of its
disjoint subsets equipped with the masses induced from the weighted set.

If weighted subsets $\widetilde{B}$ and $\widetilde{C}$ of a weighted set
$\widetilde{A}$ realize its partition, then the weighted set $\widetilde{A}$
is the sum of the weighted sets $\widetilde{B}$ and $\widetilde{C}$.

A function $\lambda_{T}:D(L)\to\mathbf{k}$, which assigns to a weighted set
$\widetilde{A}$ its total mass, is a linear function on the space $D(L)$.

######  Definition 5.3.

A weighted set $\widetilde{A}$ in $L$ with total mass
$\lambda_{T}\in\mathbf{k}$ is called:

  1. 1.

a weighty set if its total mass $\lambda_{T}$ is not equal to zero;

  2. 2.

a weightless set if its total mass $\lambda_{T}$ is equal to zero.

######  Definition 5.4.

Let us denote by $D_{0}(L)$ the subset of the set $D(L)$ that consists of all
weightless sets $\widetilde{A}$ in $L$.

Since the total mass of a weighted set is a linear function on the vector
space $D(L)$, the set of weightless sets is a vector subspace of codimension
one in $D(L)$.

###  5.2 The Moment of a Weighted Set

One can repeat almost verbatim definitions of the moment about a pivot point
and of the moment maps of of weighty sets in an affine space over an arbitrary
field $\mathbf{k}$ (see below).

######  Definition 5.5.

The moment $P(O)=\Delta_{\widetilde{A}}(O)$ of a weighted set
$\widetilde{A}=\\{(A_{i},\lambda_{i})\\}$ about a pivot point $O$ is a linear
combination of the free vectors $(O,A_{i})$ with coefficients $\lambda_{i}$:

$P(O)=\sum\lambda_{i}(O,A_{i}).$ (3)

The following Theorem is obvious:

######  Theorem 5.1 (Linearity of moment).

For each pivot point $O$, the moment $P(O)=\Delta_{\widetilde{A}}(O)$ of a
weighted set $\widetilde{A}$ about the pivot point $O$ gives a linear map of
the space $D(L)$ to the space $\overline{L}$ of free vectors in $L$.

###### Proof.

To prove the theorem one has to check that:

  1. 1.

if $\widetilde{A}=\widetilde{B}+\widetilde{C}$, then, for every point $O\in
L$, the identity
$\Delta_{\widetilde{A}}(O)=\Delta_{\widetilde{B}}(O)+\Delta_{\widetilde{c}}(O)$
holds;

  2. 2.

if $\widetilde{A}=\tau\widetilde{B}$ for $\tau\in\mathbf{K}$, then, for every
point $O\in L$, one has
$\Delta_{\widetilde{A}}(O)=\tau\Delta_{\widetilde{B}}(O).$

One can verify easily each of these statements. ∎

######  Corollary 5.1.

For any partition of a weighted set into a union of weighted subsets, the
moment of the set about any pivot point is equal to the sum of the moments of
the subsets about the same pivot point.

######  Definition 5.6.

Two weighted sets $\widetilde{A}$ and $\widetilde{B}$ are equivalent, i.e.,
$\widetilde{A}\sim\widetilde{B}$ if their moments about any pivot point are
equal. A weighty set in $L$ is a null set if its moment about any pivot point
is equal to zero. Denote the set of all null sets in $L$ by $DM(L)\subset
D(L)$.

By definition, two sets $\widetilde{A}$ and $\widetilde{B}$ are equivalent if
$\widetilde{A}-\widetilde{B}$ is a null set.

Theorem 5.1 implies the following corollary.

######  Corollary 5.2.

The set $DM(L)$ of all null sets in $L$ is a vector subspace of the vector
space $D(L)$. The equivalence of weighted sets respects the linear operations
on weighted sets, i.e., the following relations hold:

  1. 1.

if $\widetilde{A}\sim\widetilde{B}$, then
$\mu\widetilde{A}\sim\mu\widetilde{B}$ for any $\mu\in\mathbf{k}$;

  2. 2.

if $\widetilde{A}_{1}\sim\widetilde{A}_{2}$ are
$\widetilde{B}_{1}\sim\widetilde{B}_{2}$, then
$\widetilde{A}_{1}+\widetilde{B}_{1}\sim\widetilde{A}_{2}+\widetilde{B}_{2}$.

######  Definition 5.7.

The moment map $\Delta=\Delta_{\widetilde{A}}$ of a weighted set
$\widetilde{A}$ is the map $\Delta:L\to\overline{L}$ which sends a point $O\in
L$ to the moment $\Delta_{\widetilde{A}}(O)$ of the set $\widetilde{A}$ about
the pivot point $O$.

The following theorem repeats Theorem 5.2 almost verbatim and can be checked
in the same way.

######  Theorem 5.2 (Change of a pivot point).

Let $P=\Delta_{\widetilde{A}}$ be the moment map corresponding to a weighted
set $\widetilde{A}$, whose total mass is equal to
$\lambda=\lambda_{T}(\widetilde{A})$. Then, for any two points $O_{1},O_{2}\in
L$, the map $P$ satisfies the following relation:

$P(O_{1})-P(O_{2})=-\lambda(O_{2}O_{1}).$ (4)

##  6 The Space of Moment-like Maps and the Space of Weighty Points and Mass
Dipoles

In this section, we study the space of moment-like maps and define the space
of weighty points and mass dipoles in an affine space $L$ over an arbitrary
field $\mathbf{k}$.

###  6.1 Moment-like Maps

In this section, we will study a special class of affine maps which plays a
key role for the center of mass technique.

######  Definition 6.1.

An affine map $P$ of an affine space $L$ over a field $\mathbf{k}$ to the
vector space $\overline{L}$ of free vectors in $L$ is a moment-like map if its
linear part $\overline{P}:\overline{L}\to\overline{L}$ is equal to $-\lambda
I$, where $I:\overline{L}\to\overline{L}$ is the identity map and
$\lambda\in\mathbf{k}$ is a parameter called the total mass of $P$ and denoted
by $\lambda_{T}(P)$. Denote by $M(L)$ the collection of all moment-like maps
of an affine space $L$ to the vector space $\overline{L}$ of free vectors on
$L$.

The definition implies that a map $P:L\to\overline{L}$ is a moment-like map
with total mass $\lambda_{T}(P)$ if and only if, for any two points
$O_{1},O_{2}\in L$, the following characteristic identity holds:

$P(O_{2})-P(O_{1})=-\lambda_{T}(P)(O_{2}-O_{1}).$ (5)

######  Lemma 6.1.

The set $M(L)$ of all moment-like maps of an affine space $L$ over a field
$\mathbf{k}$ is a vector space over $\mathbf{k}$, i.e., the sum of two moment-
like maps of $L$ is a moment-like map of $L$; the product of a moment-like map
of $L$ on an element $\lambda\in\mathbf{k}$ is a moment-like map of $L$.

The function $\lambda_{T}:M(L)\to\mathbf{k}$, which assigns to a map $P\in
M(L)$ its total mass $\lambda_{T}(P)$, is a linear function on $M(L)$.

###### Proof.

For each fixed pair of points $O_{1},O_{2}\in L$ the characteristic identity
can be considered as a linear homogeneous equation on a map
$P:L\to\overline{L}$. Solutions of any system of linear homogeneous equations
form a vector space.

Linearity of the function $\lambda_{T}:M(L)\to\mathbf{k}$ is also a
straightforward consequence of definitions. ∎

The characteristic identity can be rewritten in the following form:

$P(O_{2})=P(O_{1})-\lambda(O_{2}-O_{1}).$ (6)

Formula (6) immediately implies the following theorem.

######  Theorem 6.1.

The total mass of a map $P\in M(L)$ is equal to zero if and only if $P$ is a
constant map $P\equiv v$, where $v\in\overline{L}$. The space $M_{0}(L)\subset
M(L)$ of all maps $P\in M(L)$, whose total mass equals to zero, is a vector
subspace in $M(L)$ of codimension one. The map $\tau:M_{0}(L)\to\overline{L}$,
that sends $P\equiv v$ to the free vector $v$, is a natural isomorphism
between the space $M_{0}(L)$ and the space $\overline{L}$.

###### Proof.

According (6), a map $P\in M(L)$ is a constant map if and only if its total
mass is equal to zero. The total mass $\lambda_{T}:M(L)\to\mathbf{k}$ is a
linear function on $M(L)$. Thus, the equation $\lambda_{T}(P)=0$ defines a
vector subspace $M_{0}(L)$ of $M(L)$ whose codimension is one.

The second statement of the theorem is obvious. ∎

######  Lemma 6.2.

A constant map $P\equiv v$ is the moment map $\Delta_{\widetilde{A}}$ of a
mass dipole $\widetilde{A}=\\{-O,B\\}$ such that the ordered pair of points
$(O,B)$ defined up to a shift, represents the free vector $v$.

###### Proof.

The lemma can be proven in almost the same way as Lemma 3.1. ∎

######  Lemma 6.3.

There is a unique map $P\in M(L)$, whose total mass is a given element
$\lambda\in\mathbf{k}$ and whose value $P(O_{1})$ at a chosen point $O_{1}\in
L$ is a given free vector $v\in\overline{L}$.

###### Proof.

According to (6), the value $P(O_{2})$ of the map $P$ at a point $O_{2}\in L$
is equal to $v-\lambda(O_{2}-O_{1})$.

Conversely, the map $P$ defined by the formula
$P(O_{2})=v-\lambda(O_{2}-O_{1})$ satisfies the assumption of the lemma. ∎

######  Corollary 6.1.

The vector space $M(L)$ is isomorphic to the direct sum
$\overline{L}\oplus\mathbf{k}^{1}$ (an isomorphism is not canonical).

###### Proof.

Fix a point $O\in L$. The map
$\tau_{O}:M(L)\to\overline{L}\oplus\mathbf{k}^{1}$ that sends a map $P\in
M(L)$ to the pair $(P(O),\lambda_{T}(P))$ for each point $O\in L$, provides an
isomorphism between $M(L)$ and $\overline{L}\oplus\mathbf{k}^{1}$. This
isomorphism depends on the point $O\in L$ and is not canonical. ∎

######  Corollary 6.2.

The dimension of the space $M(L)$ is equal to the dimension of the space $L$
plus one, i.e., the dimension $\dim_{\mathbf{k}}M(L)$ of the space $M(L)$ is
equal to $\dim_{\mathbf{k}}\overline{L}+1=\dim_{\mathbf{k}}L+1$.

######  Definition 6.2.

Let $P\in M(L)$ be a map whose mass total is not equal to zero. The center of
mass of the map $P$ is a point $O\in L$ at which the map $P$ vanishes, i.e.,
$P(O)=0$.

######  Definition 6.3.

The normalized map of a map $P\in M(L)$ whose total mass $\lambda$ is not
equal to zero, is the map $\lambda^{-1}P$.

######  Lemma 6.4.

A map $P\in M(L)$ with nonzero total mass $\lambda$ and the normalized map
$\lambda^{-1}P$ vanish at the same point. The normalized map $\lambda^{-1}P$
has the total mass 1.

###### Proof.

Two proportional vector-valued functions $P$ and $\lambda^{-1}P$ vanish at the
same points. The total mass is a linear function on the space $M(L)$. The
total mass of the normalized map is equal to
$\lambda^{-1}\cdot\lambda=1\in\mathbf{k}$. ∎

######  Theorem 6.2.

Assume that the total mass of a map $P\in M(L)$ is equal to $1\in\mathbf{k}$.
Then, for any point $Q\in L$, the free vector $P(Q)\in\overline{L}$ sends the
point $Q$ to the point $O=Q+P(Q)$ which is independent of the point $Q$.

Moreover, the point $O$ is the unique point at which the function $P$
vanishes.

###### Proof.

Indeed, for any two points $Q_{1},Q_{2}\in L$ the characteristic identity
$P(Q_{1})-P(Q_{2})=Q_{2}-Q_{1}$ can be rewritten in a following form:
$Q_{1}+P(Q_{1})=Q_{2}+P(Q_{2})$. Thus, the points $Q_{1}$ and $Q_{2}$ are
mapped by the free vectors $P(Q_{1})$ and $P(Q_{2})$ to the same point
$O=Q_{1}+P(Q_{1})=Q_{2}+P(Q_{2})$.

The point $O$ itself has to be sent to the point $O$, so that$O=O+P(O)$ which
implies that $P(O)=0$. For any other point $Q\neq O$, we have $Q+P(Q)=O\neq
Q$. Thus, $P(Q)\neq 0$. The theorem is proven. ∎

######  Theorem 6.3.

Assume that the total mass $\lambda$ of a map $P\in M(L)$ is not equal to
zero. Then $P$ has a unique center of mass $O$. For any point $Q\in L$ the
following relation holds: the free vector $\lambda^{-1}P(Q)\in\overline{L}$
sends the point $Q$ to the center of mass $O$, i.e.,

$O=Q+\lambda^{-1}P(Q).$ (7)

Moreover, if the affine space $L$ has a structure of a liner space
$\mathbf{L}$, then the vector $O\in\mathbf{L}$ satisfies the following
relation:

$O=\lambda^{-1}P(\mathbf{O}),$ (8)

where $\mathbf{O}\in\mathbf{L}$ is the zero in the space $\mathbf{L}$.

###### Proof.

The theorem follows from Theorem 6.2 applied to the normalized map
$\lambda^{-1}P$. Formula (8) can be obtained by plugging $Q=\mathbf{O}$ into
formula (7). ∎

######  Theorem 6.4.

A map $P\in M(L)$ with nonzero total mass is uniquely determined by its center
of mass $O$ and by its total mass $\lambda$.

Moreover, for any point $Q\in L$, the following identity holds:
$P(Q)=-\lambda(O-Q)$. Thus, $P$ is the moment map of the set $\widetilde{A}$
consisting of one point $O$ equipped with mass $\lambda$.

Conversely, the moment map $P=\Delta_{\widetilde{A}}$ of the set
$\widetilde{A}=\\{(O,\lambda)\\}$ has total mass $\lambda$ and center of mass
$O$.

###### Proof.

Two maps from the space $M(L)$ having the same center of mass $O$ and the same
total mass coincide, since their values at the point $O$ are equal, and their
total masses are the same.

The point $O$, obviously, is the center of mass of the map $P\in M(L)$ given
by the relation $P(Q)=-\lambda(O-Q)$, and total mass of $P$ is $\lambda$. ∎

######  Lemma 6.5.

If two maps $P_{1},P_{2}\in M(L)$ are equal at two different points
$O_{1},O_{2}\in L$, then they are equal identically.

###### Proof.

Indeed, any non-zero map $P\in M(L)$ vanishes no more than at one point. The
maps $P_{1}-P_{2}$ vanishes at two different points $O_{1}$ and $O_{2}$. Thus,
$P_{1}-P_{2}$ is identically equal to zero. ∎

###  6.2 The Space $M(L)$ and the Space $D(L)$ of Weighted Sets in $L$

In this section, we show that the space $M(L)$ of moment-like maps is
isomorphic to space of equivalence classes of weighted sets, i.e., is
isomorphic to the space $D(L)/DM(L)$ (see Corollary 5.2).

######  Definition 6.4.

The moment correspondence is the map $\Delta:D(L)\to M(L)$ which sends each
weighted set $\widetilde{A}\in D(L)$ to its moment map
$\Delta_{\widetilde{A}}\in M(L)$.

The kernel of the moment correspondence is the space $DM(L)\subset D(L)$ of
null sets in $L$. It is easy to see that a weighted set $\widetilde{A}$ is a
null set if and only if its total mass and its mass dipole are equal to zero.

######  Theorem 6.5.

The moment correspondence is a surjective map, i.e., any moment-like map is a
moment map of some weighted set $\widetilde{A}$. The space $M(L)$ is
isomorphic to a factor-space $D(L)/DM(L)$ of the space $D(L)$ by its subspace
$DM(L)$.

###### Proof.

Assume that the total mass $\lambda$ of a moment-like map $P\in M(L)$ is not
equal to zero, and its center of mass is a point $O$. Then, by Theorem 6.4,
$P=\Delta_{\widetilde{A}}$, where $\widetilde{A}$ consists of one point $O$
equipped with mass $\lambda$.

Assume that the total mass $\lambda$ of a moment-like map $P\in M(L)$ is equal
to zero, that is $P$ is a constant map $P\equiv v$. Then, by Lemma 6.2,
$P=\Delta_{\widetilde{A}}$, where $\widetilde{A}$ is a mass dipole
$\\{-O,B\\}$, where $(O,B)$ is an ordered pair of points that represents the
free vector $v$.

Thus, the correspondence map $\Delta$ is onto. Since its kernel is the space
$DM(L)$ of null sets in $L$, the space $M(L)$ is isomorphic to the factor-
space $D(L)/DM(L)$. ∎

##  7 The Space $\widehat{M}(L)$ of weighty Points and Mass Dipoles in $L$

Let $L$ be an affine space over an arbitrary field $\mathbf{k}$. In this
section, we define the vector space $\widehat{M}(L)$ over $\mathbf{k}$ of
weighty points and mass dipoles in $L$. The additive group of that vector
space has many applications in geometry.

The space $\widehat{M}(L)$ can be interpreted as the factor-space $D(L)/DM(L)$
of the space $D(L)$ of weighted points in $L$ by its subspace $DM(L)$ of all
null sets in $L$. Our presentation is based on the isomorphism between the
spaces $D(L)/DM(L)$ and $M(L)$ discussed above and on the properties of the
space $M(L)$.

######  Definition 7.1.

A weighty point $\\{(O,\lambda)\\}$ in $L$ is the point $O\in L$ equipped with
a nonzero mass $\lambda\in\mathbf{k}$.

######  Definition 7.2.

A mass dipole $\\{-A,B\\}$ in $L$ is an ordered pair of points $(A,B)$ in $L$
equipped with masses $-1,1$ respectively. Two mass dipoles $\\{-A,B\\}$ and
$\\{-C,D\\}$ are called equal if the ordered pairs of points $(A,B)$ and
$(C,D)$ are equal up to a shift.

By definition, a mass dipole $\\{-A,B\\}$ can be identified with a free
vector, represented by the ordered pair of points $(A,B)$ in $L$. The class of
the empty set can in $D(L)/DM(L)$ be considered as a mass dipole $\\{-A,A\\}$
or as the zero free vector.

######  Definition 7.3.

The set $\widehat{M}(L)$ defined as the set of all weighty points in $L$ and
all mass dipoles in $L$ defined up to a shift.

There is a natural one-to-one correspondence $\tau:\widehat{M}(L)\to M(L)$
between the sets $\widehat{M}(L)$ and $M(L)$ (see Definition 7.5 below).

The correspondence $\tau:M(L)\to M(L)$ allows to identify the sets
$\widehat{M}(L)$ with the space $M(L)$ and to define a structure of a vector
space on the set $\widehat{M}(L)$.

######  Definition 7.4.

The structure of a vector space over $\mathbf{k}$ on $\widehat{M}(L)$ is the
structure induced by the identification $\tau:\widehat{M}(L)\to M(L)$ from the
$\mathbf{k}$-vector space structure on $M(L)$.

The above definition implies that the vector space operations on the set
$\widehat{M}(L)$ are defined as follows. To perform a vector space operation
on points from the set $\widehat{M}(L)$ one has to:

  * •

identify the points in $\widehat{M}(L)$ with the maps from the space $M(L)$;

  * •

perform the desired vector space operation on the maps from $M(L)$, and

  * •

identify the map, obtained as the result of this operation, with a point of
$\widehat{M}(L)$.

We will specify vector space operation on the space $\widehat{M}(L)$ in
Section 7.1. Now we will define the one-to-one correspondence
$\tau:\widehat{M}(L)\to M(L)$.

The space $M(L)$ contains maps of two types: maps of the first type whose
total mass is a nonzero element of the field $\mathbf{k}$, and maps of the
second type whose total mass is equal to zero.

Each map of the first type is uniquely determined by its center of mass $O$
and by its total mass $\lambda\neq 0$.

Each map of the second type is a constant map $P\equiv v$, where
$v\in\overline{L}$ is a free vector represented by an ordered pair of points
$(A,B)$ defined up to a shift.

We now define the map $\tau$.

######  Definition 7.5.

The map $\tau$ sends a weighty point $\\{(O,\lambda)\\}$ to the map $P\in
M(L)$ of first type whose total mass is $\lambda$ and whose center of mass is
the point $O$.

The map $\tau$ sends a mass dipole $\\{-A,B\\}$ to the map $P\equiv v$, $P\in
M(L)$ of the second type, where $v$ is a free vector, corresponding to an
ordered pair of points $(A,B)$.

Now, that vector space operations on the set $\widehat{M}(L)$ are defined, one
can describe each type of vector space operation separately, see lemmas in
Section 7.1 below. Each of these lemmas can be easily verified. The content of
the lemmas can be considered as an axiomatic definition of the space
$\widehat{M}(L)$. The additive group of the space $\widehat{M}(L)$ is
important for geometrical applications.

Since the set $\widehat{M}(L)$ contains points of very different nature,
definition of addition is a bit tricky: addition of different types of pairs
of points in $\widehat{M}(L)$ is performed by different rules. Because of
that, the associativity of addition is not obvious.

Geometrical applications of the additive group $\widehat{M}(L)$ come from the
following statement:

The sum of several elements of the group $\widehat{M}(L)$ is well defined: it
does not depend on order in which the addition of pair of elements is
performed.

###  7.1 The Vector Space Operations on $\widehat{M}(L)$

Let us describe vector space operation on the space $\widehat{M}(L)$.

######  Lemma 7.1.

The product of a weighty point $\\{(O,\lambda)\\}$ by $\mu\in\mathbf{k}$, is
the weighty point $\\{(O,\mu\lambda)\\}$ if $\mu\neq 0$, and is a zero mass
dipole $\\{-A,A\\}$ if $\mu=0$. The product of a mass dipole $\\{-A,B\\}$,
corresponding to a free vector $(A,B)$, by $\mu\in\mathbf{k}$, is the mass
dipole, corresponding to the free vector $\mu(B-A)$.

In geometrical applications, the additive group of the space $\widehat{M}(L)$
plays a key role. Let us give more detailed description of addition in
$\widehat{M}(L)$.

######  Lemma 7.2.

The sum of two weighty points $\\{(A,\mu)\\}$ and $\\{(B,\rho)\\}$ with
$\mu+\rho=\lambda\neq 0$ is the weighty point $\\{(O,\lambda)\\}$, where $O$
is a point such that the free vectors $(A,O)$ and $(O,B)$ are proportional and
their ratio $(A,O):(O,B)$ satisfies the relation

$(A,O):(O,B)=\frac{\rho}{\lambda}.$ (9)

###### Proof.

By definition, the center of mass of the weighty set defined in the statement
of the lemma, is a point $O$ such that $\mu(O,A)+\rho(O,B)=0$, or
$\mu(A,O)=\rho(O,B)$ which is equivalent to (9). ∎

If $L$ is a real affine space, then Lemma 7.2 implies that the oriented
segments $AO$ and $OB$ are proportional and

$AO:OC=\frac{\rho}{\lambda}.$

Lemma 7.2 implies that the point $O$ as a vector in a vector space
$\mathbf{L}$ containing the affine space $L$, satisfies the following
relation:

$O=\frac{\mu}{\lambda}A+\frac{\rho}{\lambda}B.$

Let us justify the classical center of mass technique.

###### Proof of Theorem 2.1.

We have to prove that there is a way of assigning the center of mass to every
weighty set that satisfies Axioms 1, 2. The uniqueness of such center of mass
is proved above (Theorem 2.2). Above, we defined the center of mass of a
weighty set $\widetilde{A}$ as the only point where the moment map
$\Delta_{\widetilde{A}}$ vanishes. Lemma 7.2 implies that the center of mass
defined in such a way satisfies Axiom 1. It satisfies Axiom 2, since the
moment map $\Delta_{\widetilde{A}}$, by Theorem 5.1, depends linearly on the
weighted set $\widetilde{A}$. ∎

The next three lemmas describing other types of addition in the group
$\widehat{M}(L)$, are also straightforward. We will nor present their proofs.

######  Lemma 7.3.

The sum of a weighty point $\\{(A,\lambda)\\}$ and a mass dipole $\\{-B,C\\}$
is the weighted point $\\{(O,\lambda)\\}$, where the point $O$ is the sum
$A+\lambda^{-1}(B,C)$ of the point $A$ and the free vector
$\lambda^{-1}(B,C)$.

Figure 5: The center of mass of a weighted set $\\{(A,2),(O,-1),(B,1)\\}$ is
the point $C$, where $2AC=OB$

If $L$ is a real affine space, then the lemma implies that the oriented
segments $AO$ and $\lambda^{-1}(B,C)$ are parallel, equal and point to the
same direction (see Figure 5).

The lemma implies that the point $O$ as a vector in a vector space
$\mathbf{L}$ containing the affine space $L$, satisfies the following
relation:

$O=\lambda^{-1}(B,C).$

######  Lemma 7.4.

The sum of two mass dipoles $\\{-A,B\\}$ and $\\{-C,D\\}$ is the mass dipole
$\\{-E,F\\}$ that corresponds to the free vector $(A,B)+(C,D)$.

If $L$ is a real affine space, then the lemma implies that the oriented
segments $AB,CD,CF$ (which we consider us free vectors defined up to shifts)
satisfy the following relation:

$EF=AB+CD.$

The lemma implies that the points $R,F$ as vectors in a vector space
$\mathbf{L}$ containing an affine space $L$, satisfy the following relation:

$F-E=(B-A)+(C-D).$

######  Lemma 7.5.

The sum of two weighted points $\\{(A,\mu)\\}$ and $\\{(B,-\mu)\\}$ is the
mass dipole $\\{-C,D\\}$ that corresponds to the free vector $\mu(A,B)$.

If $L$ is a real affine space, then the lemma implies that the oriented
segments $CD,\mu AB$, defined up to a shift, satisfy the following relation:

$CD=\mu AB.$

The lemma implies that the points $C,D$ as vectors in a vector space
$\mathbf{L}$ containing an affine space $L$, satisfy the following relation:

$D-C=\mu(B-A).$

##  8 The Orthocenter and the Euler Line

In this section, we give a simple application of the extended center of mass
technique (in other words, of the additive group $\widehat{M}(L)$). We will
prove the following classical results:

  1. 1.

the three altitudes of a triangle have a common point; it is called the
orthocenter;

  2. 2.

for any triangle, its orthocenter $H$, its barycenter $M$ (i.e., the
intersection points of its medians) and its circumcenter $O$ (i.e., the center
$O$ of a circle, passing through all the vertices of the triangle) belong to
one line (which is called the Euler line of the triangle). Moreover, the
relation $HM:OM=2:1$ holds.

###  8.1 The Three Altitudes of a Triangle and Masses

Applying the extended center of mass technique, one can show that the three
altitudes of a triangle pass through a common point (which is called the
orthocenter of the triangle). Moreover, one can see how the orthocenter is
located at each of these altitudes (see see Theorem 8.1).

Consider a triangle $ABC$. Let $O$ be the center of the circle passing through
the vertices $A,B,C$.

Consider a weighted set $\widetilde{T}$ containing the points $A,B,C,O$, in
which the points $A,B,C$ have mass $1$ and the point $O$ has mass $-2$.

Let us compute the center of mass of the set $\widetilde{T}$ in several
different ways.

Let $CC^{\prime}$ be the altitude of $ABC$ passing through the vertex $C$ and
let $O_{AB}$ be the midpoint of the side $AB$.

######  Lemma 8.1.

Denote by $H_{C}$ the point on the line $CC^{\prime}$ such that the oriented
segment $CH_{C}$ is equal to the oriented segment $OO_{A,B}$ multiplied by
two, i.e., $CH_{C}=2OO_{AB}$. Then the point $H_{C}$ is the center of mass of
weighty set $\widetilde{T}$.

Figure 6: If the masses at $A,B,C$ are equal to 1, and the mass at $O$ is
equal to $-2$, then the center of mass is the point $H$ such that
$CH=2OO_{A,B}$

###### Proof.

Partition $\widetilde{T}$ into two subsets: $\\{A,B,O\\}$ and $\\{C\\}$.

The sum $1+1-2$ of the masses of the points $A,B,O$ is equal to zero. The mass
dipole of these weighted points is equal to the sum of vectors $OA+OB$ which
is equal to the vector $2\cdot OO_{AB}$. Since the point $C$ has mass $1$, the
center of mass of the set $\widetilde{T}$ is at the point $C$ shifted by the
vector $2\cdot OO_{AB}$ (see Lemma 7.3). Thus, it is equal to the point
$H_{C}$. ∎

In the same way, one can compute the center of mass of the set $\widetilde{T}$
by its subdividing it into two weighted subsets: a weighty vertex of the
triangle and a weightless set complimentary to the vertex. Since the center of
mass is well defined, we obtain the following result.

######  Theorem 8.1 (The three altitudes theorem).

In a triangle $ABC$ the three altitudes pass through a common point $H$.
Moreover, for any vertex $V$ (where $V$ is $A$, $B$ or $C$), the vector $VH$
is equal to $2\cdot OV^{\prime}$, where $V^{\prime}$ is the midpoint of the
side opposite to the vertex $V$.

###  8.2 The Euler line of a Triangle and Masses

In the previous section, we considered the weighty set $\widetilde{T}$
consisting of the vertices $A,B,C$ of a triangle equipped with the unit masses
and the center $O$ of the circle passing through the vertices $A,B,C$ equipped
with mass $-2$. We proved that the center of mass of the set $\widetilde{T}$
is the orthocenter $H$. In the proof, we used three different subdivisions of
the set $\widetilde{T}$.

Let us subdivide $\widetilde{T}$ in a fourth way and apply this subdivision to
another computation of the center of mass of $\widetilde{T}$. It will allow us
to prove a beautiful Euler line theorem ofor the triangle $ABC$.

Let $H$ be the orthocenter of the triangle $ABC$, i.e., the point of
intersection of its three altitudes.

Let $M$ be the barycenter of the triangle $ABC$, i.e., the point of
intersection of its three medians (see Section 2.3).

Let $O$ be circumcenter of the triangle $ABC$.

######  Theorem 8.2 (The Euler line).

For any triangle $ABC$, the points $H,M,O$ are collinear. Moreover, the point
$M$ divides the segment $HO$ in proportion $2:1$, i.e.,

$HM:MO=2:1.$

###### Proof.

Let us subdivide the weighted set $\widetilde{T}$ into weighted sets
$\\{A,B,C\\}$ and $\\{O\\}$. Instead of the weighted set $\\{A,B,C\\}$, one
can take its center of mass $M$ equipped with mass $1+1+1=3$. Thus, the center
of mass $H$ of $\widetilde{T}$ is equal to the center of mass of the weighted
set $\\{M,O\\}$, where $M$ has mass $3$ and $O$ has mass $-2$.

From the properties of the center of mass, we conclude that

$3\cdot MH=-2\cdot HO.$

We also have the following relations:

$MH=-HM\quad\text{and}\quad HM+MO=HO.$

These identities imply the theorem. ∎

##  9 Affine Hyperplanes not Passing Through the Origin in a Vector Space

In this section we discuss geometry of an affine hyperplane not passing
through the origin in a vector space. We also show that any affine space can
be canonically embedded into a vector space as an affine hyperplane not
passing through the origin.

###  9.1 Affine Transformations of a Hyperplane and Linear Transformations of
the Ambient Space

In this section, we discuss a natural relation between affine geometry of
hyperplane, not passing through the origin, and linear algebra of the ambient
vector space.

Let $L$ be an affine hyperplane in a vector space $\mathbf{L}$ not passing
through the origin $\mathbf{O}$ of $\mathbf{L}$. Let $\widetilde{L}$ be a
vector subspace parallel to $L$. The space $\widetilde{L}$ can be naturally
identified with the vector space $\overline{L}$ of free vectors in the
hyperplane $L$.

######  Theorem 9.1.

Any affine map $F:L\to L$ extends to a unique linear map
$\widehat{F}:\mathbf{L}\to\mathbf{L}$. The affine space $L$ and the vector
space $\widetilde{L}$ are invariant subspaces for $\widehat{F}$.

Conversely, let $G:\mathbf{L}\to\mathbf{L}$ be a linear map such that $L$ is
invariant under $G$. Then the restriction $G|_{L}$ of $G$ to $L$ is an affine
map $G|_{L}:L\to L$.

###### Proof.

The space $\overline{L}$ of free vectors in $L$ can be identified with the
space $\widetilde{L}$. Under this identification, the linear part
$\overline{F}:\overline{L}\to\overline{L}$ of an affine map $F:L\to L$ becomes
a linear map $\widetilde{F}:\widetilde{L}\to\widetilde{L}$. Let us define
$\widehat{F}:\mathbf{L}\to\mathbf{L}$ as the linear map whose restriction to
$\widetilde{L}$ is equal to $\widetilde{F}$ and whose value $\widehat{F}(A)$
at a chosen vector $A\in\mathbf{L}$ is equal to the vector $F(A)\in
L\subset\mathbf{L}$.

It is easy to see that the linear map $\widehat{F}$, thus defined is an
extension of $F:L\to L$. The uniqueness of an extension is obvious, since any
vector in $\mathbf{L}$ can be represented as a linear combination of vectors
from $L$.

Conversely, it is easy to see that the restriction of $G$ to an invariant
subspace $L$ is an affine map $G_{L}:L\to L$. ∎

######  Definition 9.1.

Let $GL(\mathbf{L},L)$ be the subgroup of the group $GL(\mathbf{L})$
consisting of invertible linear transformations of $\mathbf{L}$ under which
the hyperplane $L$ is invariant.

######  Theorem 9.2.

The restriction of a transformation $G\in GL(\mathbf{L},V)$ to the $L$ is an
affine automorphism of $L$. Any affine automorphism of $L$ is the restriction
of a unique linear transformation $G\in GL(\mathbf{L},V)$.

###### Proof.

The theorem follows from Theorem 9.1. We only have to show that if an affine
map $F:L\to L$ is invertible, then its extension
$\widehat{F}:\mathbf{L}\to\mathbf{L}$ also is invertible.

Indeed, the linear part $\overline{F}:\overline{L}\to\overline{L}$ is
invertible. Thus, the restriction of $\widehat{F}$ to the invariant subspace
$\widetilde{L}$ is invertible. The dimension of the factor-space
$\mathbf{L}/\widetilde{L}$ is equal to one, and the image of $L$ spans the
space $\mathbf{L}/\widetilde{L}$. The map $\widehat{F}$ induces the identity
transformation the factor-space $\mathbf{L}/\widetilde{L}$, since it sends $L$
to $L$. Thus, $\widehat{F}$ is an invertible map. ∎

######  Remark 9.1.

According to Felix Klein’s Erlangen program, geometry of an affine space $L$
is determined by the group of affine transformations of the space $L$.

Theorem 9.2 provides a simple description of this group for affine hyperplanes
in a vector space. This description is applicable to any affine spaces, since
any affine space can be canonically embedded as a hyperplane in a vector space
(see Theorem 9.4 below).

######  Theorem 9.3.

Any polynomial $F$ of degree $\leq 1$ on an affine hyperplane
$L\subset\mathbf{L}$ in a vector space $\mathbf{L}$ can be uniquely extended
to linear function $\widehat{F}$ on the space $\mathbf{L}$.

Conversely, the restriction of any linear function on $\mathbf{L}$ to $L$ is a
polynomial of degree $\leq 1$.

###### Proof.

A polynomial $F$ of degree $\leq 1$ on an affine space $L$ is an affine map
$F:L\to\mathbf{k}^{1}$. The linear part
$\overline{F}:\overline{L}\to\mathbf{k}$ is a linear function on
$\overline{L}$ which can be considered as a linear function on the space
$\widetilde{L}$. Let us choose any point $A\in L\subset\mathbf{L}$. An
extension $\widehat{F}$ can be constructed as the linear function on $L$ that
coincides with $\overline{F}$ on the $\widetilde{L}$ and whose value a chosen
point $A\in L\subset\mathbf{L}$ is $F(A)$.

The uniqueness of linear extension is obvious, since any vector from
$\mathbf{L}$ can be represented as a linear combination of elements of $L$. ∎

######  Corollary 9.1.

One can naturally identify the ambient vector space $\mathbf{L}$ with the
vector space $P_{1}^{*}(L)$ dual to the space $P_{1}(L)$ of degree $\leq 1$
polynomials on $L$.

###### Proof.

Theorem 9.3 allows to canonically identify the space $P_{1}(L)$ with the space
$\mathbf{L}^{*}$ dual to the space $\mathbf{L}$. This identification
canonically identifies the space $P_{1}^{*}(L)$ dual to the space $P_{1}(L)$
with the space $\mathbf{L}$. ∎

###  9.2 Canonical Realization of an Affine Space as a Hyperplane in Vector
Space

Let us describe a canonical realization of an affine space $L$ as a
hyperplane, not passing through the origin, in a vector space.

We will need a general definition of Kodaira’s map. Let $X$ be a set, and let
$W$ be a space of functions on taking values in a field $\mathbf{k}$.

######  Definition 9.2.

Kodaira’s map $Kod:X\to W^{*}$ is the map of $X$ to the dual space $W^{*}$
that sends a point $A\in X$ to the linear function on $W$ whose value at $f\in
W$ is equal to $f(A)$.

######  Lemma 9.1.

Kodaira’s map is an embedding if $W$ separates points in $X$, i.e., for any
points $A\neq B$ in $X$, there is a function $f_{A,B}\in W$ such that
$f_{A,B}(A)\neq f_{A,B}(B)$.

###### Proof.

If $W$ separates points $A,B\in X$, then Kodaira’s map sends $A$ and $B$ to
different linear functions on $W$, since their values at the function
$f_{A,B}$ are different. ∎

######  Definition 9.3.

Assume that the space $W$ contains a constant function $f\equiv\mu$ for any
$\mu\in\mathbf{k}$. A characteristic hyperplane $H\in W^{*}$ is a hyperplane
consisting of all linear functions $f\in W^{*}$ on $W$ whose value at the
function $f\equiv 1$ is equal to $1$.

######  Lemma 9.2.

Assume that the space $W$ contains all constant functions on $X$. Then
Kodaira’s map sends $X$ to the characteristic hyperplane of the space $W^{*}$.

###### Proof.

The value of the function $f\equiv 1$ at each point $A\in X$ is equal to $1$.
∎

Set $X=L$, an affine space, and $W=P_{1}(W)$.

######  Lemma 9.3.

Kodaira’s map $L\to P_{1}^{*}(L)$ is an affine map.

###### Proof.

A polynomial $P$ of degree $\leq 1$ is an affine map $P:L\to\mathbf{k}^{1}$.
Its linear part $\overline{P}$ is a linear function on the space
$\overline{L}$ of free vectors in $L$. Thus, the difference $P(B)-P(A)$ is
invariant under all translations of the pair $(A,B)$ (i.e., if
$(A,B)\sim(C,D)$, then $P(B)-P(A)=P(D)-P(C)$) and depends linearly on the free
vector $(A,B)$. These properties imply that Kodaira’s map $L\to P_{1}(L)^{*}$
is an affine map. ∎

######  Theorem 9.4.

Kodaira’s map $L\to P_{1}^{*}(L)$ provides a canonical affine embedding of the
affine space $L$ to the vector space $P_{1}^{*}(L)$. The image of $L$ under
the embedding coincides with the characteristic hyperplane $H$ in the space
$P_{1}^{*}(L)$.

###### Proof.

By Lemma 9.1, the map $L\to P_{1}^{*}(L)$ is an embedding, since all constant
functions on $L$ are polynomial of degree $\leq 1$. By Lemma 9.3, the map
$L\to P_{1}^{*}(L)$ is affine; by Lemma 9.2, it maps $L$ to the characteristic
hyperplane $H\in P_{1}^{*}(L)$. The dimension $\dim_{\mathbf{k}}H$ of the
characteristic hyperplane is equal to the dimension $\dim_{\mathbf{k}}L$ of
$L$. Indeed,
$\dim_{\mathbf{k}}H=\dim_{\mathbf{k}}P_{1}(L)-1=\dim_{\mathbf{k}}L$. Thus, the
map $L\to P_{1}^{*}(L)$ provides an isomorphism between the affine spaces $L$
and $H$. ∎

##  10 Several Interpretations of the Space of Weighty Points and Mass Dipoles

In this section, we show first that an affine map $F:L_{1}\to L_{2}$ induces
the linear map $F_{*}:\widehat{M}(L_{1})\to\widehat{M}(L_{2})$. Then we give
three interpretations of the space $\widehat{M}(L)$.

###  10.1 A Linear Map of the Space of Weighty Points and Mass Dipoles Induced
by an Affine Map

Let $L_{1},L_{2}$ be affine spaces over a field $\mathbf{k}$, and let
$D(L_{1})$, $D(L_{2})$ be vector spaces over $\mathbf{k}$ of sets of weighted
points in $L_{1}$ and $L_{2}$ correspondingly. Consider any map $F:L_{1}\to
L_{2}$.

######  Definition 10.1.

The map $F_{*}:D(L_{1})\to D(L_{2})$ induced by the map $F:L_{1}\to L_{2}$ is
the map that sends any single point $A\in L_{1}$, equipped with mass 1, to the
single point $F(A)\in L_{2}$ equipped with mass 1, and which extends the above
map by linearity to the space $D(L)$.

The definition implies the the total mass of the set $\widetilde{A}$ in
$L_{1}$ and $F_{*}(\widetilde{A})$ in $L_{2}$ are equal. Thus, the following
inclusion holds: $F_{*}(D_{0}(L_{1})\subset D_{0}(L_{2}).$

The definition of the map $F_{*}:D(L_{1})\to D(L_{2})$ does not use affine
structures on $L_{1}$ and $L_{2}$ (and it could be applied to vector spaces of
weighted points in any sets $L_{1}$ and $L_{2}$). The following Lemma uses the
affine structures on $L_{1}$ and $L_{2}$ in a very essential way.

######  Theorem 10.1.

If $F:L_{1}\to L_{2}$ is an affine map, then $F_{*}$ takes the null sets in
$L_{1}$ to null sets in $L_{2}$.

###### Proof.

Let $\widetilde{A}$ be a set of points $A_{1},\dots,A_{N}$ equipped with
masses $\lambda_{1},\dots,\lambda_{n}$. Since $\widetilde{A}$ is a null set,
its total mass is equal to zero, and, for any point $O\in L_{1}$, the free
vector $\sum\lambda_{i}(O,A_{i})$ is equal to zero. Thus,
$\overline{F}(\sum\lambda_{i}(O,A_{i}))$ is the zero free vector in $L_{2}$.
It implies that the free vector $\sum\lambda_{i}(F(O),F(A_{i}))$ equals to
zero. Thus, the moment of the set $F_{*}(\widetilde{A})$ about the point
$F(O)$ equals to zero. Since the total mass of the set $F_{*}(\widetilde{A})$
is zero, the set $F_{*}(\widetilde{A})$ is a null set. ∎

######  Corollary 10.1.

If $F:L_{1}\to L_{2}$ is an affine map, then the map
$F_{*}:\widehat{M}(L_{1})\to\widehat{M}(L_{2})$ is a well defined linear map.

###### Proof.

By Theorem 10.1, the linear map $F_{*}:D(L_{1})\to D(L_{2})$ takes the
subspace $DM(L_{1})$ of null sets in $L_{1}$ to the subspace $DM(L_{2})$ of
null sets in $L_{2}$. Thus, $F_{*}$ induces a well defined linear map from the
factor-space $D(L_{1})/DM(L_{1})=\widehat{M}(L_{1})$ to the factor-space
$D(L_{2})/DM(L_{2})=\widehat{M}(L_{2})$. ∎

######  Corollary 10.2.

Let $F:L_{1}\to L_{2}$ be an affine map. If $\widetilde{A}$ is a weighty set
in $L_{1}$ with nonzero total mass $\lambda$ and the center of mass $O$, then
the set $F_{*}(\widetilde{A})$ is a weighty set in $L_{2}$ with nonzero total
mass $\lambda$ and the center of mass $F(O)$.

If $\widetilde{A}$ is a weightless set in $L_{1}$ whose total mass equals to
zero and whose mass dipole is $\\{-O,B\\}$, then the set
$F_{*}(\widetilde{A})$ is a weightless set in $L_{2}$ whose total mass equals
to zero and whose mass dipole is $\\{-F(O),F(B)\\}$.

###### Proof.

If two weighted sets in $L_{1}$ are equivalent in $L_{1}$, then their images
under the map $F_{*}$ are equivalent in $L_{2}$, since $F_{*}$ maps null sets
in $L_{1}$ to null sets in $L_{2}$.

If a weighty set $\widetilde{A}$ in $L_{1}$ is equivalent to a weighty point
$\\{(O,\lambda)\\}$, then the set $F_{*}(\widetilde{A})$ in $L_{2}$ is
equivalent to the weighty point $\\{(F(O),\lambda)\\}$. Thus, the first
statement oh the Corollary is proved.

If a weightless set $\widetilde{A}$ in $L_{1}$ is equivalent to a mass dipole
$\\{-O,B\\}$, then the set $F_{*}(\widetilde{A})$ in $L_{2}$ is equivalent to
the mass dipole $\\{-F(O),F(B)\\}$. Thus, the second statement of the
Corollary is proved. ∎

######  Corollary 10.3.

If $F:L_{1}\to L_{2}$ is an affine embedding, then the map
$F_{*}:\widehat{M}(L_{1})\to\widehat{M}(L_{2})$ is a linear embedding.

###### Proof.

By Corollary 10.1, the map $F_{*}$ is linear. Corollary 10.2 implies that if
$F$ is an embedding, then $F_{*}$ sends nonzero elements of the space
$\widehat{M}(L_{1})$ to nonzero elements of the space $\widehat{L}_{2}$. ∎

###  10.2 The Space $\widehat{M}(L)$ of an Affine Hyperplane $L$ in a Linear
Ambient Space $\mathbf{L}$

In this section, we will consider an affine embedding $F:L\to\mathbf{L}$,
where $L$ is an affine hyperplane in a vector space $\mathbf{L}$ not
containing the origin $\mathbf{O}$ of the space $\mathbf{L}$. We will show
that the vector space $\widehat{M}(L)$ of weighty points and mass dipoles in
$L$ is naturally isomorphic to $\mathbf{L}$.

Let $\widetilde{L}$ be the vector subspace in $\mathbf{L}$ parallel to the
affine hyperplane $L$. It is easy to verify the following lemma.

######  Lemma 10.1.

If an affine hyperplane $L\subset\mathbf{L}$ does not contain the origin
$\mathbf{O}\in\mathbf{L}$, then there is a unique linear function
$T_{L}:\mathbf{L}\to\mathbf{k}$ that is identically equal to $1\in\mathbf{k}$
on the hyperplane $L$.

Moreover, the function $T_{L}$ vanishes on the vector space
$\widetilde{L}\subset\mathbf{L}$ parallel to the affine hyperplane $L$.

To provide an isomorphism between the vector spaces
$\widehat{M}(L)=D(L)/DM(L)$ and $\mathbf{L}$ we define a linear map $\Psi$ of
the space $D(L)$ to the ambient vector space $\mathbf{L}$ whose kernel is the
space $DM(L)$ of null sets in $L$.

######  Definition 10.2.

Let $D(\mathbf{L})$ be the space of weighted points in a vector space
$\mathbf{L}$. Then the evaluation map $\Phi:D(\mathbf{L})\to\mathbf{L}$ is the
map, which sends a weighty set $\widetilde{A}=\\{(A_{i},\lambda_{i})\\}$ in
$\mathbf{L}$ to the vector $\sum\lambda_{i}A_{i}\in\mathbf{L}$.

######  Lemma 10.2.

The evaluation map $\Phi:D(\mathbf{L})\to\mathbf{L}$ sends all null sets in
$D(\mathbf{L})$ to zero.

###### Proof.

Indeed, by definition, the vector $\Phi(\widetilde{A})$ is equal to the moment
of the set $\widetilde{A}$ about the origin $\mathbf{O}\in\mathbf{L}$. ∎

######  Definition 10.3.

Let $D(L)$ be the space of weighted points in an affine hyperplane
$L\subset\mathbf{L}$. Then the $\Psi$-map is the map $\Psi:D(L)\to\mathbf{L}$
which is the composition $\Psi=\Phi\circ F_{*}$ of the map $F_{*}:D(L)\to
D(\mathbf{L})$, induced by the embedding $F:L\to\mathbf{L}$ and the evaluation
map $\Phi:D(\mathbf{L})\to\mathbf{L}$.

######  Lemma 10.3.

Let $T_{L}$ be the linear function on $\mathbf{L}$ that takes value $1$ on the
affine hyperplane $L$. Then the total mass of a set $\widetilde{A}\in D(L)$ is
equal to $T_{L}(\Psi(\widetilde{A}))$.

###### Proof.

Indeed, if $\widetilde{A}$ is a weighty point $A_{1}$ of mass $1$, then
$T_{L}(\widetilde{A})=T_{L}(\Phi\circ F_{*}(\widetilde{A}))=T_{L}(A_{1})=1$.
This relation can be extended by linearity to any set
$\widetilde{A}=\\{(A_{i},\lambda_{i})\\}$:

$T_{L}(\Psi(\widetilde{A}))=T_{L}(\sum\lambda_{i}F(A_{i}))=\sum\lambda_{i}T_{L}(A_{i})=\sum\lambda_{i}.$

The lemma is proven. ∎

######  Lemma 10.4.

If $\Psi(\widetilde{A})=0$, then $\widetilde{A}$ is a null set.

###### Proof.

Let $\widetilde{A}$ be a weighted set $\\{(A_{i},\lambda_{i})\\}$ in $D(L)$.
By definition, its image $\Psi(\widetilde{A})=\sum\lambda_{i}A_{i}$ is the
moment of the set $F_{*}(\widetilde{A})\in D(\mathbf{L})$ about the origin
$\mathbf{O}\in\mathbf{L}$. Thus, if $\Psi(\widetilde{A})=0$, then the weighted
set $\widetilde{A}\in D(\mathbf{L})$ has zero moment about the pivot point
$\mathbf{O}$. The total mass of the set $\widetilde{A}$ is also equal to zero,
since $I_{L}(\Psi(\widetilde{A}))=T_{L}(0)=0$. Thus, the weighty set
$\widetilde{A}$ is a null set in $\mathbf{L}$. The map $F:L\to\mathbf{L}$ is
an embedding. Thus, the set $\widetilde{A}$ as a weighted set in $L$ also is a
null set. ∎

######  Theorem 10.2.

The map $\Psi:D(L)\to\mathbf{L}$ is an isomorphism between the space
$\widehat{M}(L)$ of weighty points and mass dipoles of an affine hyperplane
$L\subset\mathbf{L}$ not passing through the origin $\mathbf{O}\in\mathbf{L}$
and the ambient vector space $\mathbf{L}$.

###### Proof.

By Lemma 10.4, the induced map $\Psi:\widehat{M}(L)\to\mathbf{L}$ is an
embedding. The image of $\widehat{M}(L)$ under the map $\Psi$, obviously,
contains the hyperplane $L$. The smallest vector space that contains this
hyperplane is the space $\mathbf{L}$. Thus, the induced map
$\Phi:\widehat{M}(L)\to\mathbf{L}$ is an embedding and a surjective map. Thus,
$\Psi$ is an isomorphism between the spaces $\widehat{M}(L)$ and $\mathbf{L}$.
∎

Consider an affine hyperplane $L$ in a vector space $\mathbf{L}$ not passing
through the origin. Let $\widetilde{A}$ be a set of points
$\\{A_{1},\dots,A_{N}\\}$ in $L$ equipped with masses
$\lambda_{1},\dots,\lambda_{N}$. Let $\lambda$ be the total mass of the set
$\widetilde{A}$. Theorem 10.2 implies the following corollary.

######  Corollary 10.4.

If $\lambda\neq 0$, then $\sum\lambda_{i}A_{i}=\lambda O,$ where $O$ is the
center of mass of the set $\widetilde{A}$.

If $\lambda=0$, then $\sum\lambda_{i}A_{i}=B-O\in\widetilde{L}$, where
$\\{-O,B\\}$ is a mass dipole of the set $\widetilde{A}$, and $B-O$ is a well
defined vector in the vector space $\widetilde{L}$ parallel to the affine
space $L$.

###  10.3 The Spaces $\widehat{M}(L)$ and $P_{1}^{*}(L)$ are Canonically
Isomorphic

Let $L$ be an affine space over a field $\mathbf{k}$, and let $P_{1}(L)$ be a
vector space of degree $\leq 1$ polynomials on $L$.

Recall that the Kodaira’s map $Kod:L\to P_{1}^{*}(L)$ sends a point $x\in L$
to the linear function on $P_{1}(L)$ which assigns to a polynomial $f\in
P_{1}(L)$ its value $f(x)\in\mathbf{k}$ at the point $x$.

In Section 9.2, we showed that the map $Kod$ is an affine embedding of $L$ to
$P_{1}^{*}(L)$, whose image is the characteristic hyperplane (see Section 9.2)
in $P_{1}^{*}(L)$.

Thus, the Kodaira’s map provides a canonical representation of an affine space
$L$ as the characteristic hyperplane in the vector space $P_{1}^{*}(L)$.

######  Corollary 10.5.

There is a natural identification between the vector space $\widehat{M}(L)$ of
weighty points and mass dipoles in an affine space $L$ and the dual space
$P_{1}^{*}$ to the space $P_{1}(L)$ of polynomials of degree $\leq 1$ on $L$.

Let $\widetilde{A}$ be a set of points $\\{A_{1},\dots,A_{N}\\}$ in $L$
equipped with masses $\lambda_{1},\dots,\lambda_{N}$, and let $\lambda$ be the
total mass of the set $\widetilde{A}$. Theorem 10.2 and Corollary 10.5 imply
the following theorem.

######  Theorem 10.3.

Consider a linear function $Kod(\widetilde{A})$ on the space $P_{1}(L)$ which
sends a polynomial $f\in P_{1}$ to $\sum\lambda_{i}f(A_{i})\in\mathbf{k}$. If
$\lambda\neq 0$, then $\sum\lambda_{i}f(A_{i})=\lambda f(O)$, where $O$ is the
center of mass of the set $\widetilde{A}$ and $f$ is any polynomial from
$P_{1}(L)$. If $\lambda=0$, then
$\sum\lambda_{i}f(A_{i})=f(B)-f(O)\in\widetilde{L}$, where $\\{-O,B\\}$ is a
mass dipole of the set $\widetilde{A}$, and $f$ is any polynomial from the
space $\in P_{1}(L)$.

###  10.4 Differentials of Quadratic Polynomials and the Space of Weighty
Points and Mass Dipoles

Consider a vector space $\mathbf{L}$ over a field $\mathbf{k}$ as an affine
space $L$. Let $B$ be a non-degenerate symmetric bilinear form on $\mathbf{L}$
and let $L_{B}$ be a space of polynomials $T$ on $\mathbf{L}$, representable
in the form $P=\lambda Q_{B}+l+c$, where $Q_{B}(x)=B(x,x)$ is the quadratic
form, associated with $B$, $l\in\mathbf{L}^{*}$ is an arbitrary linear
function, and $\lambda,c\in\mathbf{k}$ are arbitrary constants.

The differential $DT_{x}$ of a polynomial $T$ on $\mathbf{L}$ give rise to the
map $DT:\mathbf{L}\to\mathbf{L}^{*}$ which assigns to a point $x\in\mathbf{L}$
the linear function $DT_{x}(y)$ of $y\in\mathbf{L}$.

With the space $L_{B}$, one can associate the vector space $DL_{B}$ of
differentials of all polynomials $T\in L_{B}$.

######  Theorem 10.4.

The vector space $DL_{B}$ of differentials of all polynomials from the space
$L_{B}$ is isomorphic to the space $\widehat{M}(L)$ of weighted points and
mass dipoles in $L$. Under this isomorphism:

  1. 1.

a weighty point $O\in L$ of a nonzero mass $\lambda\in\mathbf{k}$ corresponds
to the differential of a polynomial $T=\lambda Q_{B}+l+c$ whose critical point
is the point $O$, i.e., $T=\lambda Q_{B}(x-O)+c_{0}$, where
$c_{0}\in\mathbf{k}$ is an arbitrary constant;

  2. 2.

a mass dipole $\\{-C,D\\}$ corresponds to the differential of a polynomial
$P=l+c$ of degree $\leq 1$, where $l$ is the linear function defined by
relation $l(x)=-2B(D-C,x)$.

Consider a polynomial $T\in L_{B}$ representable in the form

$T(x)=\sum\lambda_{i}Q_{B}(x-A_{i}).$

Denote by $\widetilde{A}$ the set of points $\\{A_{1},\dots,A_{N}\\}$ equipped
with masses $\lambda_{1},\dots,\lambda_{N}$. Let $\lambda$ be the total mass
of the set $\widetilde{A}$. The theorem implies the following corollary.

######  Corollary 10.6.

If $\lambda\neq 0$, then $T(x)=\lambda Q_{B}(x-O)+c_{0}$, where $O$ is the
center of mass of the set $\widetilde{A}$ and $c_{0}\in\mathbf{k}$ is some
constant.

If $\lambda=0$, then $T(x)=l(x)+c_{0}$, where $l$ is the linear function
defined by relation $l(x)=-2B(D-C,x)$, where $\\{-C,D\\}$ is the mass dipole
of the set $\widetilde{A}$, and $c_{0}\in\mathbf{k}$ is some constant.

The differential $DT_{x}$ of a polynomial $T$ provides a map from the space
$\mathbf{L}$ to the space $\mathbf{L}^{*}$ of linear functions on
$\mathbf{L}$.

One can identify the vector space $\mathbf{L}$ with the space $\mathbf{L}^{*}$
by choosing a non-degenerate symmetric bilinear form $F$ on $\mathbf{L}$.

######  Definition 10.4.

The linear function $F$-dual to a vector $X\in\overline{L}$ is defined as the
linear function $l=x^{*}_{F}\in\mathbf{L}^{*}$, satisfying the following
identity $l(y)\equiv F(x,y)$.

A vector $F$-dual to a linear function $l\in\mathbf{L}^{*}$ is defined as the
vector $x=l^{*}_{F}\in\mathbf{L}$ such that $l=x^{*}_{F}$.

######  Definition 10.5.

The$F$-gradient $\nabla_{F}T(x)$ of a polynomial $T$ at a point $x\in L$ is
the vector that is $F$-dual to the differential $DT_{x}$ of $T$ at $x\in L$,
i.e., the following identity holds:

$F(\nabla_{F}T(x),y)\equiv DT_{x}(y).$

One can check that the differential $D_{x}T(y)$ of a polynomial $T=\lambda
Q_{B}+l+c$ at a point $x\in\mathbf{L}$ is equal to $\lambda 2B(x,y)+l(y)$ (as
a function of $y$.

We will identify the space $\mathbf{L}$ with the dual space $\mathbf{L}^{*}$
using a symmetric bilinear form $F=-2B$.

With this choice of a bilinear form $F$, the $F$-gradient $\nabla_{F}T(x)$
satisfies the following relation:

$\nabla_{F}T(x)=-\lambda x+b,$ (10)

where $b=l^{*}_{F}$.

###### Proof of Theorem 10.4.

The space of differentials of polynomials $T\in L_{B}$ is isomorphic to the
space of $F$-gradients of polynomials $T$. Formula (10) implies that the
$F$-gradient of a polynomial $T\in L_{B}$ is a moment-like map whose total
mass equals $\lambda$. If $\lambda\neq 0$, then the $F$-gradient vanishes at a
single point which is the center of mass of the moment-like map and at the
same time the critical point of the polynomial $T$. If $\lambda=0$, then
$F$-gradient is identically equal to the constant $b$ defined above.

The space $\widehat{M}(L)$ is isomorphic to the space $M(L)$ of moment-like
maps. It is easy to see that the isomorphism between $\widehat{M}(L)$ and
$M(L)$ agrees with the statement of the theorem. ∎

## References

  * [1] M. B. Balk, Geometric applications of the concept of the center of gravity M., Fizmatgiz, 1959.
  * [2] M. B. Balk and V. G. Boltyansky, Mass geometry. M. Nauka. Phys.-Math. lit., 1987.
  * [3] Marcel Berger, Geometry I. Springer-Verlag Berlin Heidelberg, 1987.