## Geometry IVolume I of this 2-volume textbook provides a lively and readable presentation of large parts of classical geometry. For each topic the author presents an esthetically pleasing and easily stated theorem - although the proof may be difficult and concealed. The mathematical text is illustrated with figures, open problems and references to modern literature, providing a unified reference to geometry in the full breadth of its subfields and ramifications. |

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### Contents

examples and applications | 4 |

Affine spaces | 32 |

Barycenters the universal space | 67 |

Projective spaces | 85 |

applications | 111 |

Projective quadrics | 116 |

Projective lines crossratios homographies | 122 |

Complexifications | 142 |

Euclidean conies | 218 |

The sphere for its own sake | 255 |

Triangles spheres and circles | 279 |

Elliptic and hyperbolic geometry | 318 |

Convex sets | 331 |

The space of spheres | 349 |

Bibliography | 363 |

Index of notations | 379 |

Affine quadrics | 146 |

Euclidean vector spaces | 151 |

Projective conics | 170 |

Euclidean affine spaces | 200 |

385 | |

401 | |

409 | |

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### Common terms and phrases

affine space algebraic angles apply arbitrary associated assume bases basis bounded called canonical chapter circle closed compact complex condition consider construction containing continuous converse convex set coordinates COROLLARY cross-ratio curve defined definition denoted determined differential dimension direction discuss distance distinct elements equal equivalent Euclidean exactly example exists fact field Figure finite fixed point follows formula four function geometry given gives holds hyperplane implies intersection introduce invariant inverse measure metric natural necessary NOTE O+(E Observe obtained oriented orthogonal particular plane points polygon positive problem projective projective space Proof properties PROPOSITION prove reader reflection relation remark resp respect restriction result rotation satisfying segment Show sides similarity sphere structure subset subspace supporting tangent theorem triangle unique vector space volume write