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Schaum's Outline Series Theory and Problems of Projective Geometry

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The purpose of this book is to provide a first course in Projective Geometry for undergraduate majors in mathematics and for prospective teachers of high school geometry. For the former it will furnish an introduction to the important concept of projective spaces; for the latter it will introduce a more general geometry from which, by proper specialization, the familiar metric geometry is obtained. Since only the real geometry of one and two dimensions is considered here, every theorem may be illustrated by a diagram in the construction of which nothing more than a straight edge is required.

Chapter 1 begins with a brief survey of the geometry of Euclid and his associates. That part of this geometry which is concerned solely with the incidence of points and lines is called projective. The projective plane is then obtained by properly modifying the fundamental plane of Euclidean geometry. In Chapters 1-6 and 8-12, the reader will find the basic propositions of plane projective geometry developed entirely by synthetic methods.

Chapter 7 is concerned with an axiomatic approach. In the course of providing models for the axioms, certain finite projective geometries are introduced. Since this leads eventually to the geometry of points defined by number triples over the field of real numbers of Chapter 15, the reading of the chapter may be postponed until that time.

In Chapter 13 the procedure is reversed. Taking the projective plane as funda- mental, modifications are made to obtain the affine plane in which parallel lines reappear. Additional modifications are made in Chapter 14 in order to define per- pendicular lines and thus permit a return to the metric plane. Of interest here is the fact, observed perhaps for the first time by the reader, that so much of the metric geometry with which he is familiar depends on parallelism rather than on perpendicu- larity. Also to be noted is the great variety of metric theorems which often follow from a single projective theorem.

In Chapters 15-17 the reader is introduced to analytic methods in projective geom- etry. In these chapters an acquaintance with matrix algebra is assumed. For those who would wish a brief review, the Appendix will be found helpful.

The final chapter parallels Chapters 13-14. Beginning with the set of all projective transformations of the plane onto itself, the reader is led by successive steps to the familiar rigid motions of Plane Analytic Geometry.

The author wishes to take this opportunity to express his gratitude to the staff of the Schaum Publishing Company for their splendid cooperation.
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