## 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 |

Riemannian geometry revisited | 198 |

Exercises | 205 |

4 | 211 |

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 |

Exercises | 166 |

DIFFERENTIAL GEOMETRY AND GAUSSIAN | 183 |

APPPENDIX | 277 |

BIBLIOGRAPHY | 293 |

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### Common terms and phrases

arbitrary arcs assume axis called Chapter circle clear common composition congruent Consequently consists constructed containing cosh course curvature curve defined definition denote describe discussion distance dy˛ equal equation Euclid's Euclidean geometry Example Exercise exist expression fact FIGURE Find fixed point flow diagram flow lines follows formula function geodesic geodesic segment given greater Hence hyperbolic geometry hyperbolic length hyperbolic plane hyperbolic rigid motion hyperbolic triangle integral interior intersect inversion joining known Lemma lies line segment maps measure Moebius obtained origin parallel perpendicular Poincaré positive Postulate proof Proposition Prove radius reader reflection respectively right angles rotation sides sinh sphere spherical straight line Suppose surface tangent Theorem transformation translation turn unit disk unit disk model upper half-plane vertices x-axis მყ

### References to this book

Mathematical Expeditions: Chronicles by the Explorers Reinhard Laubenbacher,David Pengelley Limited preview - 2000 |