Elementary Geometry in Hyperbolic Space

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"Although the mathematical prerequisites are few, it is not, I think, a text that one can easily slip into halfway through: on the other hand, it does contain a lot of useful material and will be worth the effort necessary to master it." Mathematical Reviews I. Preliminaries 1. Quaternions 2. The hyperbolic functions 3. Trace relations 4. The fractional linear group and the cross ratio Notes to Chapter I II. The Möbius Group 1. Similarity transformations 2. The extended space. Orientation. Angular measure 3. Inversion 4. Circle- and sphere-preserving transformations 5. The Möbius group of the upper half-space Notes to Chapter II III. The Basic Notions of Hyperbolic Geometry 1. Lines and planes. Convexity 2. Orthogonality 3. The invariant Riemannian metric 4. The hyperbolic metric 5. Transformation to the unit ball Notes to Chapter III IV. The Isometry Group of Hyperbolic Space 1. Characterization of the isometry group 2. Classification of the motions 3. Reversals 4. The isometry group of a plane 5. The spherical and cylindric surfaces Notes to Chapter IV V. Lines 1. Line matrices 2. Oriented lines 3. Double crosses 4. Transversals 5. Pencils and bundles of lines Notes to Chapter V VI. Right-Angled Hexagons 1. Right-angled hexagons and pentagons 2. Trigonometric relations for right-angled hexagons 3. Trigonometric relations for polygons in a plane 4. Determination of a hexagon by three of its sides 5. The amplitudes of a right-angled hexagon 6. Transversals of a right-angled hexagon 7. The bisectors and radii of a right-angled hexagon 8. The medians of a right-angled hexagon 9. The altitudes of a right-angled hexagon Notes to Chapter VI VII. Points and Planes 1. Point and plane matrices 2. Incidence and orthogonality 3. Distances and angles 4. Pencils of points and planes 5. Bundles of points and planes 6. Tetrahedra Notes to Chapter VII VIII. Spherical Surfaces 1. Equations of spherical surfaces 2. An invariant of a pair of spherical surfaces 3. The power of a point with respect to a spherical surface 4. The radical plane of a pair of spherical surfaces 5. Linear families of spherical surfaces Notes to Chapter VIII IX. Area and Volume 1. Various coordinate systems 2. Area 3. Volume of some bodies of revolution 4. Volume of polyhedra Notes to Chapter IX References Index