Geometry

Geometry

SIMONE MALACRIDA

All topics concerning geometry are presented in this book:

Euclidean plane geometry

euclidean solid geometry

analytic geometry in the plane

projective geometry

analytic geometry in space

non-Euclidean geometries

combinatorial geometry

discrete geometry

fractal geometry

differential geometry

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – GEOMETRY: BASIC CONCEPTS

Definitions

Euclid's postulates

Other definitions

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II – EUCLIDEAN PLANE GEOMETRY

Definitions

Circumference

Ellipse

Parable

Polygons: definitions

Triangle

Quadrilaterals

More polygons

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III - EUCLIDEAN SOLID GEOMETRY

Definitions

Sphere

Cone

Cylinder

Polyhedra: definitions

Pyramid

Prism

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IV - ANALYTICAL GEOMETRY IN THE PLANE

Definitions

Translation and distance

Practical applications

The straight line in the Cartesian plane

Properties of the straight line in the Cartesian plane

The parabola in the Cartesian plane

Circumference

Ellipse

Hyperbole

General considerations on conics

Generalization of analytic geometry in the plane

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V - ANALYTICAL GEOMETRY IN SPACE

The plane in space

The straight line in space

Surfaces in space

The quadrics

Other surfaces

Projective geometry

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VI - NON-EUCLIDEAN GEOMETRIES

Introduction

Elliptical geometry

Spherical geometry

Hyperbolic geometry

Projective geometry

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VII - COMBINATORY GEOMETRY

Introduction

Graphs

Trees

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VIII - DISCRETE GEOMETRY

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IX - FRACTAL GEOMETRY

Introduction

Types of fractals

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X - DIFFERENTIAL GEOMETRY

Introduction

Operations

INTRODUCTION

Geometry is certainly one of the most important fields of mathematics and this has been known since ancient times.

The geometric study has always supported and supported the mathematical one, generating a series of reciprocal influences that have lasted up to the present day.

It is useless to recall the enormous applications of geometry not only at a scientific and technological level, but in everyday life.

This book deals with all aspects of geometry, from the elementary one that is taught from the first years of school, up to the most advanced knowledge at the university level.

The first three chapters introduce the geometric discourse, substantially as already known by the Greeks, exposing the elementary concepts and implications of plane geometry and solid geometry within the Euclidean vision.

The fourth and fifth chapters instead draw inspiration from Descartes' studies on analytical geometry and extend the concepts of high school up to university-level knowledge, going to study analytical geometry in space and on the plane through increasingly sophisticated formalisms.

The sixth chapter is dedicated to the introduction of non-Euclidean geometries and to the fundamental study that emerged for two centuries at the mathematical level.

The last four chapters make us understand how the role of geometry in modern society has evolved exponentially.

Geometry has connections with algebra and combinatorics, with logic and with analysis.

There are geometries of different types, among which we mention the discrete, the combinatorial and the fractal.

Particularly important for its physical and mathematical consequences is differential geometry, presented in the tenth and final chapter.

This book therefore wants to be a summa of geometry in every possible mathematical application.

I

GEOMETRY: BASIC CONCEPTS

Definitions

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Geometry is that branch of mathematics that deals with shapes and figures in a given setting.

Below we give the foundations of elementary geometry, largely developed already in ancient Greece.

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The primitive concept of geometry is the point, conceived as a dimensionless and indivisible entity, which characterizes the position and is characterized by it .

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An infinite and successive set of points is called a segment , if this set is delimited by two points called extremes.

Two segments are consecutive if they have an end point in common, while they are external if they have no point in common.

Two segments are said to be incident if they have only one point in common, called the point of intersection , which however is not an extreme.

The midpoint of a segment is the point that exactly divides the segment in half.

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An infinite and successive set of points is called a straight line , if this set is not bounded by any end point, while it is called a semi-line if there is only one end point.

A segment can therefore be seen as part of a straight line.

Two consecutive segments are adjacent if they belong to the same line.

Lines, segments and semi-lines are characterized by a single dimension called length.

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The geometric entity characterized by two dimensions, called length and height, is the plane , while the one characterized by three dimensions (in addition to those mentioned there is the width) is called space . Plane geometry deals with the study of the two-dimensional case, solid geometry with the three-dimensional case.

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Two straight lines or two segments are said to be coplanar if they lie in the same plane, otherwise they are called skew .

In geometry, points are indicated with capital letters, segments with capital letters of the two extremes barred at the top by a line, while straight lines and semi-lines with small letters.

Furthermore, all geometric dimensions are, by definition, positive.

Two segments, two straight lines or two semi-lines are said to coincide if and only if all the points present in the first geometric element are exactly the same as in the second geometric element.

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In plane geometry, in the case of two half-lines having a common end point, the concept of angle can be defined .

In fact we