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## Über dieses Buch

Affine geometry and quadrics are fascinating subjects alone, but they are also important applications of linear algebra. They give a first glimpse into the world of algebraic geometry yet they are equally relevant to a wide range of disciplines such as engineering.

This text discusses and classifies affinities and Euclidean motions culminating in classification results for quadrics. A high level of detail and generality is a key feature unmatched by other books available. Such intricacy makes this a particularly accessible teaching resource as it requires no extra time in deconstructing the author’s reasoning. The provision of a large number of exercises with hints will help students to develop their problem solving skills and will also be a useful resource for lecturers when setting work for independent study.

Affinities, Euclidean Motions and Quadrics takes rudimentary, and often taken-for-granted, knowledge and presents it in a new, comprehensive form. Standard and non-standard examples are demonstrated throughout and an appendix provides the reader with a summary of advanced linear algebra facts for quick reference to the text. All factors combined, this is a self-contained book ideal for self-study that is not only foundational but unique in its approach.’

This text will be of use to lecturers in linear algebra and its applications to geometry as well as advanced undergraduate and beginning graduate students.

## Inhaltsverzeichnis

### 1. Affine Spaces

In this chapter we introduce the most fundamental concept of these notes: the affine space. It is a natural generalization of the concept of vector space but with a clear distinction between points and vectors. Often this distinction is not made, and it is frequent not to distinguish, for instance, between the point (1,2)∈ℝ

2

and the vector

v

=(1,2)∈ℝ

2

. The problem is that ℝ

2

is at the same time a set of points and a vector space.

In the study of vector spaces, the vector subspaces and the relations among them (Grassmann's formula) play a central role. In the same way, in the study of affine spaces, the affine subspaces and the relations among them (affine Grassmann's formula) play also an important role.

The most simple figure that we can form with points and straight lines is the triangle. In this chapter we shall see two important results that refer to triangles and the incidence relation: the theorems of Menelaus and Ceva.

In the Exercises at the end of the chapter we verify Axioms 1, 2 and 3 of Affine Geometry given in the Introduction.

The subsections are

1.1

Introduction

1.2

Definition of affine space

1.3

Examples

1.4

Dimension of an affine space

1.5

First properties

1.6

Linear varieties

1.7

Examples of straight lines

1.8

Linear variety generated by points

1.9

Affine Grassmann's formulas

1.10

Affine frame

1.11

Equations of a linear variety

1.12

Barycenter

1.13

Simple ratio

1.14

Theorems of Thales, Menelaus and Ceva

Exercises

Agustí Reventós Tarrida

### 2. Affinities

In this chapter we introduce

affinities

, the most natural maps to consider between affine spaces. The definition of affine map, or affinity, is so natural that we shall see that affinities are simply those maps that take collinear points to collinear points.

We shall also see that there are enough affine maps. In fact, in an affine space of dimension

n

, given two subsets of

n

+1 points, there exists an affine map such that takes the points of the first subset to the points of the second.

In the Exercises at the end of the chapter we verify the axioms 4 and 5 of Affine Geometry given in the Introduction.

The subsections are

2.1

Introduction

2.2

Definition of affinity

2.3

First properties

2.4

The affine group

2.5

Affinities and linear varieties

2.6

Equations of affinities

2.7

Invariant varieties

2.8

Examples of affinities

2.9

Characterization of affinities of the line

2.10

Fundamental Theorem of Affine Geometry

Exercises

Agustí Reventós Tarrida

### 3. Classification of Affinities

In this and in the following chapter we answer the natural question of how many affine maps there are. To do so we first define an equivalence relation between affine maps and study all the equivalence classes that appear. In low dimensions the problem is not too hard and is solved explicitly in this chapter. The idea is that the classification of affinities is given by the classification of endomorphisms plus a geometrical property: the invariance level.

We shall also give a geometric interpretation of the affinities of the real affine plane

The subsections are

3.1

Introduction

3.2

Similar endomorphisms

3.3

Similar affinities

3.4

Computations in coordinates

3.5

Invariance level

3.6

Classification of affinities of the line

3.7

Classification of affinities of the real plane

3.8

Invariance level in the real plane

3.9

Geometrical interpretation

3.10

Decomposition of affinities in the real plane

Exercises

Agustí Reventós Tarrida

### 4. Classification of Affinities in Arbitrary Dimension

In arbitrary dimensions the problem of the classification of the affine maps is rather involved, since it depends on the classification of endomorphisms, and in particular on the Jordan normal form of a matrix. In this chapter we give all the details of this classification, since we were unable to find it in specialist literature. We prove essentially the same result that in the above chapter: Two affine maps are similar if and only if the corresponding linear part are similar and the corresponding invariance level are equal.

The subsections are

4.1

Introduction

4.2

Jordan matrices

4.3

Similar endomorphisms and canonical decomposition

4.4

Clarifying examples

4.5

Classification of affinities in arbitrary dimension

Exercises

Agustí Reventós Tarrida

### 5. Euclidean Affine Spaces

In this chapter we consider affine spaces on which a distance has been defined. Thus we have a model of classical Euclidean Geometry, where, for instance, Pythagoras' Theorem works well. We give a short method to compute the distance between two varieties of arbitrary dimension.

The subsections are

5.1

Introduction

5.2

Definition of Euclidean affine space. Pythagoras' Theorem

5.3

Distance between two varieties

5.4

Common perpendicular

Exercises

Agustí Reventós Tarrida

### 6. Euclidean Motions

In this chapter we study distance preserving maps, that is, the

Euclidean motions

. Since there are fewer Euclidean motions than affine maps, the classification is simpler. We also introduce a natural equivalence relation among Euclidean motions, similar to that for affine maps, and we characterize each equivalence class by a sequence of numbers (the coefficients of a polynomial and a

metric invariant

).

We associate a vector, the

glide vector

, to each Euclidean motion

f

. This vector, and in particular its module

τ

(

f

), plays an important role in the study and classification of Euclidean motions. In fact we have that

$$\tau(f)=\inf_{P\in \mathbb{A}} d(P,f(P)).$$

The subsections are

6.1

Introduction

6.2

Definition of Euclidean motion

6.3

Examples of Euclidean motions

6.4

Similar Euclidean motions

6.5

Calculations in coordinates

6.6

Glide vector

6.7

Classification of Euclidean motions

6.8

Invariance of the glide module

Exercises

Agustí Reventós Tarrida

### 7. Euclidean Motions of the Line, the Plane and of Space

In this chapter we study Euclidean motions in dimension 1, 2 and 3. For instance, in dimension three there are only three types of Euclidean motions:

helicoidals

(that include rotations, translations and the identity),

glide reflections

(that include mirror symmetries) and

anti-rotations

.

The subsections are

7.1

Introduction

7.2

Classification of Euclidean motions of the line

7.3

Classification of Euclidean motions of the plane

7.4

Geometrical interpretation

7.5

Classification of Euclidean motions of the space

7.6

Geometrical interpretation

7.7

Composition of rotations in dimension three

Exercises

Agustí Reventós Tarrida

### 8. Affine Classification of Real Quadrics

In this chapter we study

considering equivalent two quadrics when there is an affinity that takes one onto the other. Quadrics are the zeros of quadratic polynomials, and therefore they are the most natural objects to consider after straight lines (zeros of linear polynomials). From this point of view, there are essentially only three quadrics (

conics

) in the plane: ellipse, hyperbola and parabola. We also give the complete list of affine quadrics in dimension three.

We study some properties of a given quadric, such as the tangent cone, the polar hyperplane, the center, etc.

The subsections are

8.1

Introduction

8.2

8.3

8.4

Change of affine frame

8.5

Image of a quadric under an affinity

8.6

8.7

Invariants

8.8

Canonical representatives without linear part

8.9

Canonical representatives with linear part

8.10

Number of equivalence classes of quadratic polynomials

8.11

Regular zeros

8.12

8.13

Affine classification of conics

8.14

Affine classification of quadrics in dimension three

8.15

8.16

8.17

Tangent cone

Exercises

Agustí Reventós Tarrida

### 9. Orthogonal Classification of Quadrics

In this chapter we study quadrics but unlike the previous chapter considering equivalent two quadrics when there is a Euclidean motion that takes one onto the other. From this point of view there are infinitely many quadrics (conics) in the plane, since ellipses, parabolas or hyperbolas of different size are respectively different to each other. Nevertheless, we give the classification in dimensions two and three and find lists of real numbers which represent them. For instance, in the plane, there are as many ellipses as pairs (

a

,

b

) of real numbers, with 0

a

b

and as many hyperbolas as pairs (

a

,

b

) of real numbers, with 0

a

, 0

b

. We give a faithful list of all quadrics in arbitrary dimensions. For this we shall need to introduce an adequate definition of

good order

among various real numbers. Most textbooks are not concerned with the faithfulness of this list: that is, that each quadric appears in the list once and only once; for this reason this concept of good order is, as far as we know, new in this context.

We also study the symmetries of a given quadric. The subsections are

9.1

Introduction

9.2

9.3

Criterion for ordering

9.4

Canonical representatives without linear part

9.5

Canonical representatives with linear part

9.6

Invariants

9.7

9.8

Orthogonal classification of conics

9.9

Orthogonal classification of quadrics in dimension three

9.10

Exercises

Agustí Reventós Tarrida

### Backmatter

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