Non-Euclidean GeometryCambridge University Press, 17 de set. de 1998 - 336 páginas 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. |
Conteúdo
THE HISTORICAL DEVELOPMENT | 1 |
SECTION | 9 |
1 | 11 |
SECTION | 14 |
4 | 28 |
POLARITIES | 48 |
3 | 55 |
HOMOGENEOUS COORDINATES | 71 |
EUCLIDEAN AND HYPERBOLIC GEOMETRY | 179 |
HYPERBOLIC GEOMETRY IN TWO DIMENSIONS | 199 |
1 | 213 |
THE USE OF A GENERAL TRIANGLE | 224 |
SECTION PAGE | 235 |
AREA | 241 |
EUCLIDEAN MODELS | 252 |
CONCLUDING REMARKS | 267 |
6 | 81 |
ELLIPTIC GEOMETRY IN ONE DIMENSION | 95 |
5 | 101 |
ELLIPTIC GEOMETRY IN TWO DIMENSIONS | 109 |
ELLIPTIC GEOMETRY IN THREE DIMENSIONS | 128 |
DESCRIPTIVE GEOMETRY | 157 |
SECTION PAGE | 289 |
BIBLIOGRAPHY | 317 |
48 | 323 |
328 | |
331 | |
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Termos e frases comuns
absolute polar according angle angle of parallelism applying axioms axis belong bundle called centre circle Clifford collinear collineation common congruent conic consider consists construction contains coordinates coplanar correlation corresponding cosh deduce define definition determined dimensions direct distance distinct double points elliptic geometry equal equation expressed fact fixed follows formulae four given harmonic Hence horocycle hyperbolic ideal point intersection invariant inverse involution joining length lies lines means meet namely numbers obtain opposite ordinary point pairs parallel particular pencil perpendicular plane points at infinity polar line pole positive preserves projectivity PROOF proved quadric reflection region relation represented respect result rotation segment self-conjugate sense separate shows sides Similarly space sphere surface tangent theorem translation triangle unit vertices