Elementary Geometry in Hyperbolic Space

✍ Scribed by Werner Fenchel

• Publisher: De Gruyter
• Year: 1989
• Tongue: English
• Leaves: 240
• Series: De Gruyter Studies in Mathematics; 11
• Edition: Reprint 2011
• Category: Library

✦ Synopsis

"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

✦ Table of Contents

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