INTRODUCTION TO THE GEOMETRY OF COMPLEX NUMBERS ROLAND DEAUX Translated by HOWARD EVES DOVER PUBLICATIONS, INC. Mineola, New York Bihliographical Note This Dover edition, first published in 2008, is an unabridged republication of the revised French edition of the work, translated by Howard Eves, and origi­ nally published In 1956 by Frederick Ungar Publishing Company, New York. Lihraryoj Congress Cataloging-in-Puhlication Data Deaux, Roland, 1893- [Introduction ilIa geometrie des nombres complexes. English] Introduction to the geometry of complex numbers / Roland Deaux ; trans­ lated by Howard Eves. -_. Dover ed. p. cm. Includes index. Prevo ed. of. New York: F. Ungar Pub. Co., 1956. ISBN-13: 978-0-486-46629-3 ISBN-IO: 0-486-46629-9 I. N umbers, Complex. 2. Geometry, ProJecllve. I. Title. QA471.D3732008 512.7'88- -dcn 2007042751 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 PREFACE TO THE AMERICAN EDITION The presentation of this book to English-speaking students permits us to return to its reason for existence. It is addressed to beginners, if we understand by this term all those who, whatever might be their mathematical knowledge beyond the usual studies, have not considered the use of complex numbers for establishing geometric properties of plane figures. We declare, in fact, that a student accustomed to the classical methods of analytic geometry or of infinitesimal geometry is not, ipso facto, prepared to solve problems, even be they elementary, by appealing to complex numbers. He is even very often much embarrassed. Now such obligations arise in the applied sciences, particularly in electrical studies, where the graphical interpretation of the calcu­ lation is a convenient picture of the circumstances of the phenomenon studied. Finally, to assure the reader an easy and complete understanding, we have not hesitated to develop things from the beginning, by placing emphasis on the constructions connected with the algebraic operations. In deference to the practical end endorsed, we have only considered constructions related to the concepts and methods of elementary geometry. One will find in other works (see, for example, the first cited book by Coolidge) constructions which, following the ideas of Klein, possess the character of invariance under the group of circular transformations. The French edition has undergone a complete revision in the theory of unicursal bicircular quartics, in that of antigraphies, and in the proof of the theorem of Schick (art. 128). In addition, the statements of 136 exercises have been inserted after the various sections. They will allow the reader to test his new knowledge and to interest himself at the same time with some aspects of the geometry of the triangle; a theory is well understood and will acquire power only after its successful application to the 5 6 PREFACE solution of proposed problems. At the start, some indications of the path to follow, but not always suppressing personal effort, are offered to assist the solver. One can find in the Belgian journal Mathesis, when such a reference is given, a complete solution by means of complex numbers and often accompanied by developments which enlarge the horizon of the problem. The reader desirous of increasing his geometric knowledge in the domain to which the present work serves as an introduction will consult with profit, in addition to the books already cited, the re­ markable work Inversive Geometry (G. Bell and Sons, Ltd., London, 1933) by Frank Morley and F. V. Morley. He will also find in Advanced Plane Geometry (North-Holland Publishing Company, Amsterdam, 1950), by C. Zwikker, a study of numerous curves by means of complex coordinates. It is pleasant to express our gratitude to Professor Howard Eves of the University of Maine, who was so willing to take on the burdensome task of the translation and whose judicious remarks have improved the texture of several proofs, and our thanks to the Frederick Ungar Publishing Company of New York for the attentive care that it has brought to the presentation of this book. Virelles-lez-Chimay (Belgium) ROLAND DEAUX August, 1956 FOREWORD The present work briefly develops the lectures which we have given since 1930 to the engineering candidates who chose the section of electromechanics at the Faculte polytechnique de Mons. A memoir by Steinmetz 1 emphasized the simplifying role that can be played by the geometric interpretation of complex numbers in the study of the characteristic diagrams of electrical machines. This idea has proved fruitful in its developments, and many articles published in such journals as the Archiv fur Elektrotechnik can be comprehended only by the engineer acquainted with the meaning and the results of a plane analytic geometry adapted to representations in the Gauss plane. One there encounters, for example, unicursal bicircular quartics discussed by commencing with their parametric equation in complex coordinates; the procedure consisting of returning to classical geometry by the separation of the real and the imaginary will more often than not be devoid of interest, for it hides the clarity of certain interpretations which employ the relation between a com­ plex number and the equivalent vectors capable of representing this number. The study of such questions explains the appearance, particularly in Switzerland and Germany, of works in which geometric expositions accompany considerations on alternating currents.2 There does not exist a treatise of this sort in Belgium, and this lack will perhaps justify the book which we are presenting to the public. Since, on the other hand, the geometrical results are independent of the technical reasons which called them forth, the service rendered to engineers will be little diminished if we stay in the purely mathematical domain. We most certainly do not wish to affirm by this that some technical illustrations of the theories which we 1 Die Anwendung komplexer GrofJen in der Elektrotechnik, ETZ, 1893, pp. 597, 631, 641, 653. 2 O. BLOCH, Die Ortskurven der graphischen Wechselstromtechnik, ZUrich, 1917. G. l!AUFFE, Ortskurven der Starkstromtechnik, Berlin, 1932. G. OBERDORFER, Die Ortskurven der Wechselstromtechnik, Munich, 1934. 7 8 FOREWORD develop, presented by an informed specialist, would be lacking in interest or timeliness; on the contrary, all our best wishes attend the efforts of the engineer who would contribute to the papers of Belgian scientific activity a work which certain foreign countries have judged useful to undertake. An engineer looking for a geometric solution to a problem born in an electrotechnical laboratory will be able, in general, to limit himself to the first two chapters. It is interesting, however, to note that H. PBieger-Haertell arrived at the determination of the affix of the center of a circle, not by the method that we give for finding the foci of a conic (article 60), but by utilizing a property of Mobius transformations (article 127); and this shows that the engineer is able to take advantage of these theories too. The table of contents states the subject matter of each of the articles, and therefore it appears unnecessary here to take up again the detailed sequence of material treated. To the friends of geometry, we hope they find in the reading of these pages the pleasure that we have experienced in writing them. No serious difficulty needs to be surmounted. Variety is assured by the appeals that we make to algebra, to the classical notions of analytic geometry, to modern plane geometry, and to some results furnished by kinematics, and the third chapter revives in a slightly modified form the essentials of the projective geometry of real binary forms, thus putting in relief all the pertinency of the title given by Mobius to the first of his papers on the subject: 2 Ueber eine Methode, um von Relationen, welche der Longimetrie angehOren, zu entsprechenden Siitzen der Planimetrie zugelangen (1852). We would like to emphasize, moreover, the importance of those circular transformations of which the particular case of inversion is so often considered, but of which the direct involutoric case does not seem to have received the same favor, which it so well deserves. The obligation of limiting ourselves has not permitted us to treat the antigraphy with the same amplification as the homography in the complex plane. Concerning these matters the reader will consult with profit the Vorlesungen iiber projektive Geometrie (Berlin, 1934) of C. Juel, the two books of J. L. Coolidge, A treatise on the circle 1 Zur Theone tier Kreisdiagramme: Archiv fur Elektrotechnik, vol. XII, 1923, pp. 486-493. • Werke, 2, pp. 191-204. FOREWORD 9 and the sphere (Oxford, 1916), The geometry of the complex domain (Oxford, 1924), as well as the Lefons de geometrie projective complexe (paris, 1931) of E. Cartan. At a time when difficulties of all sorts create much anxiety for the most cautious editors, the firm of A. De Boeck has not, however, been afraid to undertake the publication of this work. For its noble courage, and for the attentive care that it has brought to the composi­ tion as well as to the presentation of this book, we express our thanks and our gratitude. R. DEAUX Mons, October 23, 1945 TABLE OF CONTENTS CHAPTER ONE GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS I. Fundamental Operations 15 1. Complex coordinate. 2. Conjugate coordinates. 3. Exponential form. 4. Case where r is positive. 5. Vector and complex number. 6. Addition. 7. Sub­ traction. 8. Multiplication. 9. Division. 10. Scalar product of two vectors. 11. Vector product of two vectors. 12. Object of the course. Exercises 1 through 11 II. Fundamental Transformations ........... .. 26 13. Transformation. 14. Translation. 15. Rotation. 16. Homothety. 17. Relation among three points. 18. Symmetry with respect to a line. 19. Inversion. 20. Point at infinity of the Gauss plane.