ACADEMIC PAPERBACKS* EDITED BY Henry Booker, D. Allan Bromley, Nicholas DeClaris, W. Magnus, Alvin Nason, and A. Shenitzer BIOLOGY Design and Function at the Threshold of Life: The Viruses HEINZ FRAENKEL-CONRAT The Evolution of Genetics ARNOLD W. RAVIN Isotopes in Biology GEORGE WOLF Life: Its Nature, Origin, and Development A. I. OPARIN Time, Cells, and Aging BERNARD L. STREHLER ENGINEERING A Vector Approach to Oscillations HENRY BOOKER Dynamic Programming and Modern Control Theory RICHARD BELLMAN and ROBERT KALABA MATHEMATICS Finite Permutation Groups HELMUT WIELANDT Elements of Abstract Harmonic Analysis GEORGE BACHMAN The Method of Averaging Functional Corrections: Theory and Applications A. Yu. LUCHKA Geometric Transformations (in two volumes) P. S. MODENOV and A. S. PARKHOMENKO Representation Theory of Finite Groups MARTIN BURROW Introduction to p-Adic Numbers and Valuation Theory GEORGE BACHMAN Linear Operators in Hubert Space WERNER SCHMEIDLER Noneuclidean Geometry HERBERT MESCHKOWSKI Quadratic Forms and Matrices N. V. YEFIMOV PHYSICS Crystals: Their Role in Nature and in Science CHARLES BUNN Elementary Dynamics of Particles H. W. HARKNESS Elementary Plane Rigid Dynamics H. W. HARKNESS Mossbauer Effect: Principles and Applications GÜNTHER K. WERTHEIM Potential Barriers in Semiconductors B. R. GOSSICK Principles of Vector Analysis JERRY B. MARION *Most of these volumes are also available in a cloth bound edition. Geometrie Transformations P. S. MODENOV and A. S. PARKHOMENKO VOLUME 1 Euclidean and Affine Transformations Translated and adapted from the first Russian edition by MICHAEL B. P. SLATER Published in cooperation with the SURVEY OF RECENT EAST EUROPEAN MATHEMATICAL LITERATURE A project conducted by ALFRED L. PUTNAM AND IZAAK WIRSZUP Department of Mathematics, The University of Chicago, under a grant from the National Science Foundation ACADEMIC PRESS New York and London First published in the Russian language under the title Geometricheskie Preobrazovaniya in 1961 by IzdateFstvo Moskovskogo Universitet, Moscow, U.S.S.R. COPYRIGHT © 1965, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l LIBRARY OF CONGRESS CATALOG CARD NUMBER: 65-25004 PRINTED IN THE UNITED STATES OF AMERICA Preface to Volume 1 of the English Edition This is the first volume of a two-volume translation of the Russian book Geometric Transformations, by Modenov and Parkhomenko. This volume embraces Chapters I—IV; Vol- ume 2 (Projective Transformations) contains the translation of the original Chapters V and VI. The treatment is elementary, and should be accessible to the high school senior. The prerequisites amount to some famil- iarity with Euclidean geometry, including the use of coordi- nates, elementary trigonometry, and linear equations (up to determinants). A little knowledge of vectors and conies might also be helpful. However, the material covered or referred to ranges much further, and should be of interest to a very broad spectrum of readers, from high school senior to college teacher. This book is not designed to be a standard text. As will be seen from the introduction, the material covered is not usually included in the curriculum, and its style is more suitable for browsing than for systematic class study. The purpose of the book is rather to introduce the reader to a fascinating and not at all difficult area of geometry, at the same time acquainting him painlessly with some of the simpler methods and concepts of advanced mathematics. Since the topic is one for which everyone will have some intuitive feeling, and the exposition is consistently straightforward, there is no danger that the reader will find himself suddenly out of his depth. The Russian authors suggest that this book can best serve as extracurricular material for geometry seminars in universi- ties and teacher-training colleges, as extra background mate- rial for school mathematics clubs (under a teacher's guidance). This translation might well serve similar purposes in American schools and colleges. Chicago, Illinois M. SLATER 1965 y Translator's Note The translation is quite free. Although it retains all of the original text, it recasts many passages and in several sections includes additional background discussion and motivation. Apart from the Appendix to Chapter II, however, the section headings are the same, and in the same order, as in the origi- nal. Wherever I have added to or changed the original, I have tried to remain consistently within its spirit. The burden of responsibility for all deviations from the original must rest entirely on me. M.S. VI Preface to the Russian Edition This book is intended for use in geometry seminars in universities and teacher-training colleges. It may also be used as supplementary reading by high school teachers who wish to extend their range of knowledge. Finally, many sections may be used as source material for school mathematics clubs under the guidance of a teacher. The subject matter is those transformations of the plane that preserve the fundamental figures of geometry: straight lines and circles. In particular, we discuss orthogonal, affine, pro- jective, and similarity transformations, and inversions. The treatment is elementary, though in a number of in- stances (where a synthetic treatment seems more cumbersome) coordinate methods are used. A little use is also made of vec- tor algebra, but the text here is self-contained. In order to clarify a number of points, we give some elemen- tary facts from projective geometry; also, in the addendum to Chapter I of Volume 2 (the topology of the projective plane), the structure of the projective plane is examined in greater detail. The authors feel obliged to express their thanks to Professor V. G. Boltyanskii, who carefully read the manuscript and made a number of valuable suggestions. They also wish to thank Miss V. S. Kapustina for editing the manuscript and re- moving many inadequacies of presentation. They would like also to say that Chapter II of Volume 2 was written with the help of an article on inversion written by V. V. Kucherenko, a second-year physics student. It is to him that we owe the elegant proof of the fundamental theorem that any circle transformation can be represented as the product of an in- version and a similarity transformation, and also as the product of an inversion and a rotation (or a reflection). Moscow THE AUTHORS January 1961 vii Introduction In the study of a number of questions in geometry, such as the proofs of certain theorems, the solution of problems of construction, and the examination of geometric figures, high school courses in geometry use certain types of transformation: reflections, rotations, translations, similarities, and inversions. We shall give a general definition of the concept of a trans- formation and shall then examine those transformations that are most important for elementary geometry: orthogonal, affine, and projective transformations, and inversions. The examination of these types of transformation is import- ant for two reasons : 1. These transformations are the "simplest," in the sense that they preserve the fundamental material of geometry : line segments and angles are preserved by orthogonal transforma- tions; straight lines by orthogonal, affine, and projective transformations; and lines and circles taken together by inversions. 2. The division of geometry into elementary, affine, pro- jective, and other "geometries" is determined by those geo- metric properties of the figures which are required to remain fixed. Thus each "geometry" is associated with a group of transformations : precisely those that leave the required prop- erties invariant. For example, elementary geometry is concerned I 2 Introduction with such properties of geometric figures as angles between lines, lengths, parallelism, and, in fact, all those properties of figures which are preserved under translation (and similarity). Affine geometry studies precisely those properties of geometric figures that are preserved under affine transformations: the straightness of lines, parallelism, the ratio between the lengths of two segments on a line, and so on; however, affine trans- formations do not, in general, preserve lengths or angles, and affine geometry does not therefore concern itself with such metrical properties of figures. Projective geometry studies those properties that are preserved by projective transformations. Such properties are, for example, the straightness of a line, the cross ratio of four points on a line (in particular, harmon- icity), and others. But projective transformations may change lengths and angles, and even take parallel lines into non- parallel ones. Thus in projective geometry there is no concept of parallelism, just as in affine geometry (and also projective geometry) there is no concept of length or angle. We note finally that it is sometimes possible to replace the study of some more general transformation by the study of an affine transformation which is a sufficiently good approxima- tion to it. This is done in hydromechanics when investigating a complicated flow of liquid, in the theory of elasticity when study- ing the deformation of solid bodies, and in other fields. On the whole, we shall confine our attention to transforma- tions of the plane. The investigation of transformations of 3- space is analogous, and we shall only consider it separately when it presents special features of its own. CHAPTER I General Definitions I. Sets and Functions There are two concepts in mathematics in terms of which the basic definitions and theorems may be formulated in the most exact way possible. These are the concepts of a set and a func- tion. These concepts themselves are regarded in mathematics as fundamental; that is, they are not defined in terms of anything else. Their meaning is commonly made clear by examples, and this is how we too shall start. As one example of a set, consider a circle. This is the set of all those points of the plane that lie at a certain given distance from a given point (the center). As another example, consider the set of all points M lying within a triangle ABC. This is a set characterized by the following property: for each point M in it, M and A lie on the same side of the line BC: M and B lie on the same side of the line CA, and M and C lie on the same side of the line AB. Lines and planes can be defined in mathematics as sets of points in space satisfying quite definite conditions. In what follows we shall identify lines and planes with the sets of points of which they consist, although in elementary geometry all these concepts (point, line, plane) are regarded as fundamental. 3 4 /. General Definitions We meet with the concept of a function very early in our study of mathematics : the polynomial y = α χη + tf!*""1 + ··· + α 0 η gives us a very simple example of an algebraic function. Examples of more complicated functions are y = ax, y = log x, y = sin x fl y = tan x. In examining a function we must always pay attention to two questions : 1. What is the set of values for which the function is defined (the domain of definition of the function)? 2. What is the set of values assumed by the function? For example, if we restrict our attention to real numbers, the domain of definition of the function y = x2 is the set of all real numbers, while its range of values is the set of all nonnegative numbers. For the function y = log x the domain of definition is the set of all positive numbers, and the range of values is the set of all real numbers. For the function y = sin x the domain is the set of all real numbers, while the range of values is the set of all those y whose absolute value does not exceed 1. For the function y = log sin x the domain is the set of all those numbers which lie in an interval of the form 2kn <x<(2k+ 1)π where k is any integer. The range is the set of all nonpositive numbers. If we go one step further to the function y = \Jlog sin x we find that the domain is the set of points of the form χ = (π/2) + 2kn