Non-Euclidean Geometry

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Cambridge University Press, 17 de set de 1998 - 336 páginas
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The MAA is delighted to be the publisher of the sixth edition of this book, updated with a new section 15.9 on the author's useful concept of inversive distance. Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called colineations. They lead in a natural way to elliptic isometries or "congruent transformations". Following a recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing an elliptic or hyperbolic polarity which transforms points into lines (in two dimensions) or planes (in three dimensions) and vice versa. This treatment can be enjoyed by anyone who is familiar with algebra up to the elements of group theory. - Publisher.
 

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Conteúdo

THE HISTORICAL DEVELOPMENT
1
FOUNDATIONS
16
POLARITH5S
48
SECTION PAGE
55
HOMOGENEOUS COORDINATES
71
ELLIPTIC GEOMETRY IN ONE DIMENSION
95
ELLIPTIC GEOMETRY IN TWO DIMENSIONS
109
section page
115
SECTION PAGE
186
HYPERBOLIC GEOMETRY IN TWO DIMENSIONS
199
CIRCLES AND TRIANGLES
213
THE USE OF A GENERAL TRIANGLE
224
SECTION PAGE
235
AREA
241
EUCLIDEAN MODELS
252
CONCLUDING REMARKS
267

ELLIPTIC GEOMETRY IN THREE DIMENSIONS
128
DESCRIPTIVE GEOMETRY
157
EX EUCLIDEAN AND HYPERBOLIC GEOMETRY
179
SECTION PAGE
289
Bibliography
317
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