## Non-Euclidean GeometryThe 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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### Contents

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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### Common terms and phrases

absolute polar line angle of parallelism axioms axis bundle centre circle Clifford parallels Clifford surface Clifford translation collinear points collineation common perpendicular congruent transformation conic conjugate points coordinates coplanar cosh Coxeter cross ratio deduce define definition diameter double points elliptic geometry equation equidistant curve Euclid Euclidean geometry Euclidean plane flat pencil follows formulae given line harmonic conjugate harmonic homology Hence horocycle horosphere hyperbolic geometry hyperbolic plane ideal point inversive distance involution Klein length line at infinity lines and planes Lobatschewsky meet metrical mid-lines mid-points non-Euclidean non-Euclidean geometry one-dimensional opposite ordinary line ordinary point orthogonal pairs parallel lines permutable point-pairs points and lines points at infinity pole product of reflections projective geometry Proof quadric quaternion real numbers real projective geometry relation represented respect rotation segment self-polar sides sinh space sphere spherical tangent tetrahedron theorem triangle ABC ultra-infinite ultra-parallel vertices