Geometry: A Comprehensive Course

Dan Pedoe

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486131733
• Topic: Mathématiques
• Subtopic: Géométrie

"A lucid and masterly survey." — Mathematics Gazette Professor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to help students enjoy geometry. Among the topics discussed: the use of vectors and their products in work on Desargues' and Pappus' theorem and the nine-point circle; circles and coaxal systems; the representation of circles by points in three dimensions; mappings of the Euclidean plane, similitudes, isometries, mappings of the inversive plane, and Moebius transformations; projective geometry of the plane, space, and n dimensions; the projective generation of conics and quadrics; Moebius tetrahedra; the tetrahedral complex; the twisted cubic curve; the cubic surface; oriented circles; and introduction to algebraic geometry. In addition, three appendices deal with Euclidean definitions, postulates, and propositions; the Grassmann-Pluecker coordinates of lines in S3, and the group of circular transformations. Among the outstanding features of this book are its many worked examples and over 500 exercises to test geometrical understanding.

I

VECTORS

1.1 Vectors in the Euclidean (and affine) plane

The concept of a vector as a directed segment is a familiar one, and it can be placed on a firm mathematical foundation. Much of what follows will apply not only to vectors in the real Euclidean plane, but also to vectors in the real affine plane (§0.2). This is indicated by choosing a system of cartesian axes which are not necessarily orthogonal

Fig. 1.1

Fig. 1.2

(Fig.1.1). Let P = (x₁, x₂) and Q = (y₁, y₂) be two points in the plane. Then the vector

is defined to be the ordered pair of real numbers:

Two vectors

and

are thought of as being equal, in ordinary language, when PQ is parallel to RS, when the length of PQ = the length of RS, and when the direction of motion from P to Q is the same as that from R to S. All this is expressed, in our language, thus: if R = (z₁, z₂) and S = (t₁, t₂) then

if and only if the ordered pair of real numbers (y₁ – x₁, y₂ – x₂) is the same as the ordered pair of real numbers (t₁ – z₁, t₂ – z₂) (See Fig.1.2).

This definition of equality between vectors sets up an equivalence relation between ordered pairs of real numbers (§0.10, and 1.1). In particular, in the equivalence class which contains a given vector there is always one vector with its first number pair at the origin, (0, 0). We take this as the vector representative of the class.

Such a vector joins the origin to a point P = (a₁, a₂), say, and we call this vector the position–vector of the point P (Fig.1.3). The remaining vectors of the equivalence class represented by this vector are called free vectors. The representative of the class is called a bound vector.

Fig. 1.3

1.2 Addition of bound vectors

Our operations on vectors will be confined to bound vectors. Since the origin is kept fixed, a bound vector is defined by a pair of real numbers, which define the end point of the vector. Thus if P = (x₁, x₂), the vector

is defined by the ordered number–pair (x₁, x₂), and we may use the symbol P, by itself, to denote the position–vector

. We therefore write

P = (x₁, x₂),

and regard P as a vector. If, similarly, Q = (y...

Table of contents

• Cover
• Title Page
• Dedication
• Copyright Page
• Contents
• Preface
• Chapter 0: Preliminary Notions
• Chapter I: Vectors
• Chapter II: Circles
• Chapter III: Coaxal Systems of Circles
• Chapter IV: The Representation of Circles by Points in Space ofThree Dimensions
• Chapter V: Mappings of the Euclidean Plane
• Chapter VI: Mappings of the Inversive Plane
• Chapter VII: The Projective Plane and Projective Space
• Chapter VIII: The Projective Geometry of n Dimensions
• Chapter IX: The Projective Generation of Conics and Quadrics
• Chapter X: Prelude to Algebraic Geometry
• Appendix I
• Appendix II
• Appendix III
• Bibliography and References
• Index