## The Poincaré Half-plane: A Gateway to Modern GeometryThe Poincare Half-Planeprovides an elementary and constructive development of this geometry that brings the undergraduate major closer to current geometric research. At the same time, repeated use is made of high school geometry, algebra, trigonometry, and calculus, thus reinforcing the students' understanding of these disciplines as well as enhancing their perception of mathematics as a unified endeavor. |

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### Contents

EUCLIDEAN RIGID MOTIONS | 35 |

INVERSIONS | 51 |

THE HYPERBOLIC PLANE | 63 |

EUCLIDEAN VERSUS HYPERBOLIC GEOMETRY | 79 |

THE ANGLES OF THE HYPERBOLIC TRIANGLE | 93 |

HYPERBOLIC AREA | 109 |

THE TRIGONOMETRY OF THE HYPERBOLIC | 119 |

The general hyperbolic triangle | 125 |

A review of lengths and areas on surfaces | 190 |

Gausss formula for the curvature at a point | 196 |

Exercises | 205 |

The unit disk model and its flow diagrams | 213 |

Explicit rigid motions of the unit disk model | 221 |

Regular tesselations of the unit disk model | 228 |

A BRIEF HISTORY OF NONEUCLIDEAN | 247 |

Exercises | 254 |

COMPLEX NUMBERS AND RIGID MOTIONS | 131 |

ABSOLUTE GEOMETRY AND THE ANGLES | 161 |

DIFFERENTIAL GEOMETRY AND GAUSSIAN | 183 |

APPPENDIX | 277 |

293 | |

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### Common terms and phrases

absolute geometry arbitrary axioms axis bowed geodesic Chapter chord Common Notions complex numbers composition congruent constructed containing coordinates cosh cross ratio curve defined definition denote dxdy elliptic elliptic geometry equation Euclidean center Euclidean circle Euclidean geometry Euclidean line segment Euclidean plane Euclidean radius Euclidean rigid motions Euclidean straight lines Example Exercise fact FIGURE fixed point flow diagram flow lines follows function Gauss Gaussian curvature glide-reflections Hence hyperbolic area hyperbolic distance hyperbolic geodesic hyperbolic geometry hyperbolic length hyperbolic plane hyperbolic reflection hyperbolic rigid motion hyperbolic rotation hyperbolic triangle interior intersect inversion Lemma line segment maps mathematicians Moebius rigid motion Moebius transformation non-Euclidean geometry orthogonal parallel Parallel Postulate Poincare metric positive proof of Proposition Prove q.e.d. Proposition reader Riemann metric right angles sides sin2 sinh sphere straight geodesic tangent Theorem translation triangle ABC trigonometry unit circle unit disk model upper half-plane vertices xy plane

### References to this book

Mathematical Expeditions: Chronicles by the Explorers Reinhard Laubenbacher,David Pengelley Limited preview - 1999 |