Table of Contents
Table of Contents
"Geometry"
INTRODUCTION
GEOMETRY: BASIC CONCEPTS
EUCLIDEAN PLANE GEOMETRY
EUCLIDEAN SOLID GEOMETRY
ANALYTICAL GEOMETRY IN THE PLANE
ANALYTICAL GEOMETRY IN SPACE
NON-EUCLIDEAN GEOMETRY
COMBINATORY GEOMETRY
DISCRETE GEOMETRY
FRACTAL GEOMETRY
DIFFERENTIAL GEOMETRY
"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
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
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 see that the two half-lines divide the plane into two parts.
The angles are denoted either with lowercase letters of the Greek alphabet or with the uppercase letters of the extremes, spaced out from the point of origin of the two half-lines (called vertex) with a circumflex accent above this last letter.
By convention, angles are measured counterclockwise.
In plane geometry, if the two half-lines coincide, the angle contains the whole plane and is called a round angle, the measure of which is, by definition, 360°.
Half of a round angle is called a flat angle and measures 180° and occurs when the two half-lines are adjacent.
An angle is said to be convex if it is less than 180°, concave if it is greater, as in the figure:
Half of a straight angle is called a right angle and measures 90°.
An angle between 0° and 90° is called acute , an angle between 90° and 180° is called obtuse .
Two straight lines or two semi-lines or two segments are said to be perpendicular (or orthogonal) to each other if the angle of incidence is a right angle. It goes without saying that these geometric elements are necessarily accidents.
Two angles are said to be complementary if their sum gives a right angle, supplementary if it gives a straight angle, and complementary if it gives a round angle.
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bisector is defined as the straight line or semi-line or the segment which divides the angle subtended between two segments or two semi-lines or two straight lines into two equal parts.
A peculiarity of the bisector is given by the fact that if a half-line is bisector of an angle, its extension is also the same for the angle complementary to the first.
straight line or half-line or segment that divides the length of a given segment into two equal parts, i.e. passing through the midpoint of the segment, is defined as median .
Height is defined as a line or ray or segment that is perpendicular to another geometric entity such as a line or ray or segment. The point of intersection between the height and the given line or ray or segment is called the foot .
axis is defined as a line or ray or segment which is perpendicular to a segment at its midpoint.
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Euclid's postulates
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Euclid enunciated five postulates which, if accepted, make geometry fall into the so-called Euclidean geometry . The five postulates are given by:
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1) Between any two points it is possible to draw one and only one straight line.
2) A straight line can be extended beyond the two points indefinitely.
3) Given a point and a length, it is possible to describe a circumference.
4) All right angles are equal.
5) If a straight line cuts two other straight lines, determining on the same side internal angles whose sum is less than that of two right angles, extending the two straight lines indefinitely, they will meet on the side where the sum of the two angles is less than two right angles .
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Euclid's third postulate introduces the concept of circumference which we have not yet introduced. We define circumference as the locus of points on the plane equidistant from a point called center (the constant distance is called radius).
Some consequences of the first four postulates are as follows:
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An infinite number of straight lines pass through each point on the plane.
One and only one straight line passes through two distinct points on the plane.
Infinite planes pass through a straight line in space.
Only one plane passes through three non-aligned points in space.
Only one circle passes through three non-aligned points in the plane.
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Obviously the plan and the space take the qualifying adjective of Euclidean.
Three or more points in space are aligned if they are contained in a straight line, four or more points in space are coplanar if they are contained in a plane.
Of all Euclid's postulates, it is the fifth postulate that determines Euclidean geometry.
This postulate is also called the parallel postulate and its non-acceptance gives rise to non-Euclidean geometries, which we will not deal with in this manual which, instead, is entirely centered on Euclidean geometry.
The name of parallel postulate derives from the equivalent, and better known, version given to this postulate:
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Given any straight line r and a point P not belonging to it, it is possible to draw for P one and only one straight line parallel to the given straight line r .
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Therefore two straight lines are intersecting if they have a point of intersection, while if they do not they are said to be parallel.
In Euclidean geometry, two parallel lines always maintain the same distance between them.
In Euclidean geometry, the minimum distance between two parallel lines, or between a point outside a line and the line itself, is given by the height.
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Other definitions
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We define translation as a transformation of the Euclidean plane or space that moves
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 19.04.2023
ISBN: 978-3-7554-3942-4
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