Solid Analytic Geometry

SOLID ANALYTIC GEOMETRY SOLID ANALYTIC GEOMETRY By ADRIAN ALBERT Professor of Mathematics The University of Chicago FIRST EDITION McGRAW-HILL BOOK COMPANY, INC. NEW YORK TORONTO LONDON 1949 SOLID ANALYTIC GEOMETRY Copyright, 1949, by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. THE MAPLE PRESS COMPANY, YORK, PA. PREFACE In a recent text on college algebra the author gave a brief pre senlation of what seems to him to be the best basis for a modern course on plane analytic geometry, that is, the algebraic vector approach. The present text contains an extension of this approach, yielding an exposition in full for the three-dimensional case, and thereby ties up the study of space analytic geometry with the theory of vectors and matrices. Chapters 1 and 2 contain a treatment of the equations of lines and planes. After a preliminary study of the linear operations on n-dimensional vectors, rectangular coordinates are introduced, three-dimensional vectors and inner products are interpreted geometrically, and, from a consideration of scalar products and axis translations, the parametric equations of a line are obtained. The vector approach then yields a very simple derivation of the normal form of an equation of a plane, and the standard forms of plane and line equations are rather immediate consequences. Chapter 3 contains an exposition of classical elementary sur face and curve theory. Chapter 4 contains the usual treatment of spheres, and Chapter 5 gives the classical descriptions of quadric surfaces in standard position. The latter chapterends with a rather novel classification of quadrics according to certain invariants. Chapter 6 is an exposition of that part of the theory of matrices which is needed for a complete development of the so-called principal axis transformation. A full account of the orthogonal reduction of a real quadratic form in n variables is given, and the theory is applied in Chapter 7 to the three-dimensional case of quadric surfaces. This latter chapter ends with a discussion of the symmetries of quadric surfaces. The first seven chapters of this text provide an exposition of the basic topics of solid analytic geometry, the material being adequate for a one-quarter course on the subject. The remain ing chapters contain additional material for longer courses or out side reading. Chapter 8, on spherical coordinates, contains a dis vi PREFACE cussion of some practical aspects of the theory of rotations and translations of axes in space. It is quite clear that rectangular coordinates are not as practical for actual measurements as are the coordinates of range, azimuth, and elevation, and in this chapter methods are developed for actual computation of the changes in these measurable coordinates after translations or rotations of axes. The chapter ends with a discussion of gno monic charts. Our final chapter contains a brief presentation of the elements of protective geometry. The effect on the theory of linear trans formations of the use of homogeneous coordinates is given, and the chapter contains a rigorous matrix proof of the invariance of the cross ratio under projective transformations. It is hoped that the use of modern algebraic techniques in this and in the earlier chapters of the present textwill serve to make the subject of solid analytic geometry fit better in the teaching of modern mathematics than it has in the past. ADRIAN ALBERT CHICAGO, ILL. March, 1949 CONTENTS PREFACE v 1 COORDINATES AND LINES I 1. Vectors 1 2. Scalar multiplication 2 3. Inner products 3 4. The angle between two vectors 4 5. Directed lines 5 6. Orthogonal projections 6 7. Rectangular coordinates in ordinary space 7 8. The length of a vector in space 10 9. Lines through the origin 10 10. The angle between two vectors in space 12 11. Translation of axes 13 12...