Research Catalog

Title

Non-Euclidean geometry.

Author

Coxeter, H. S. M. (Harold Scott Macdonald), 1907-2003.

Publication

Toronto, University of Toronto Press, 1957.

Items in the Library & Off-site

Details

Description

309 pages illustrations; 23 cm.

Summary

A text which surveys real projective geometry, the elliptic metric, and supplies applicable definitions and theorems.

Series Statement

Mathematical expositions ; no. 2

Uniform Title

Mathematical expositions ; no. 2.

Subject

• Geometry, Non-Euclidean
• Niet-Euclidische meetkunde
• Géométrie non-euclidienne

Bibliography (note)

• Includes bibliographical references (pages 293-300).

Contents

1. The Historical Development of Non-Euclidean Geometry -- 1.1 Euclid -- 1.2 Saccheri -- 1.3 Gauss, Wachter, Schweikart, Taurinus -- 1.4 Lobatschewsky -- 1.5 Bolyai -- 1.6 Riemann -- 1.7 Klein --2. Real Projective Geometry: Foundations -- 2.1 Definitions and Axioms -- 2.2 Models -- 2.3 The principle of duality -- 2.4 Harmonic Sets -- 2.5 Sense -- 2.6 Triangular -- 2.7 Ordered Correspondences -- 2.8 One-dimensional Projectivities -- 2.9 Involutions -- 3. Real Projective Geometry: Polarities, Conics and Quadrics -- 3.1 Two-Dimensional Projectivites -- 3.2 Polarities in the plane -- 3.3 Conics -- 3.4 Projectivities on a conic -- 3.5 The fixed points of a collineation -- 3.6 Cones and reguli -- 3.7 Three-dimensional projectivities -- 3.8 Polarities in Space -- 4. Homogeneous Coordinates -- 4.1 The von Staudt-Hessenberg calculus of points -- 4.2 One-dimensional projectivities -- 4.3 Coordinates in one and two dimensions -- 4.4 Collineations and coordinate transformations -- 4.5 Polarities -- 4.6 Coordinates in three dimensions -- 4.7 Three-dimensional projectivities -- 4.8 Line Coordinates for the generators of a quadratic -- 4.9 Complex projective geometry -- 5. Elliptic Geometry in One Dimension -- 5.1 Elliptic geometry -- 5.2 Models -- 5.3 Reflections and translations -- 5.4 Congruence -- 5.5 Continuous translation -- 5.6 The length of a Segment -- 5.7 Distance in terms of cross ratio -- 5.8 Alternative treatment using the complex line -- 5.8 Alternative treatment using the complex line -- 6. Elliptic Geometry in Two dimensions -- 6.1 Spherical and Elliptic Geometry -- 6.2 Reflection -- 6.3 Rotations and angles -- 6.4 Congruence -- 6.5 Circles -- 6.6 Composition of Rotations -- 6.7 Formulae for Distance and angle -- 6.8 Rotations and Quaternions -- 6.9 Alternative treatment using the complex plane -- 7. Elliptic Geometry in Three Dimensions -- 7.1 Congruent Transformations -- 7.2 Clifford Parallels -- 7.3 The Stephanos-Cartan representation of Rotations by points -- 7.4 Right Translations and Left Translations -- 7.5 Right parallels and left parallels -- 7.6 Study's coordinates for a line -- 7.9 Complex Space -- 8. Descriptive Geometry -- 8.1 Klein's Projective model for hyperbolic geometry -- 8.2 Geometry in a convex region -- 8.3 Veblen's axioms of order -- 8.4 Order in a pencil -- 8.5 The geometry of lines and planes through a fixed point -- 8.6 Generalized bundles and pencils -- 8.7 Ideal points and lines -- 8.8 Verifying the projective axioms -- 8.9 Parallelism -- 9. Euclidean and Hyperbolic Geometry -- 9.1 the introduction of congruence -- 9.2 Perpendicular lines and places -- 9.3 Improper bundles and pencils -- 9.4 the absolute polarity -- 9.5 The Euclidean case -- 9.6 the hyperbolic case -- 9.7 The Absolute -- 9.8 the geometry of a bundle -- 10. Hyperbolic Geometry in Two Dimensions -- 10.1 Ideal Elements -- 10.2 Angle-bisectors -- 10.3 Congruent Transformations -- 10.4 some famous constructions -- 10.5 an alternative expression for distance -- 10.6 the angle of parallelism --10.7 Distance and angle in terms of poles and polar -- 10.8 Euclidean Geometry as a limiting case -- 11. Circles and Triangles -- 11.1 various definitions for a circle -- 11.2 the circle as a special conic -- 11.3 Spheres -- 11.4 the in- and ex-circles of a triangle -- 11.5 the circum-circles and centroids -- 11.6 The Polar triangle and the orthocenter -- 12. The Use of a General Triangle of Reference -- 12.1 Formulae for Distance and Angle -- 12.2 The General Circle -- 12.3 Tangential Equations -- 12.4 Circum-Circles and Centroids -- 12.5 In- and ex-circles -- 12.6 The Orthocenter -- 12.7 Elliptic trigonometry -- 12.8 the radii -- 12.9 Hyperbolic Trigonometry -- 13. Area -- 13.1 Equivalent regions -- 13.2 The choice of a unit -- 13.3 The area of a triangle in elliptic geometry -- 13.4 Area in hyperbolic geometry -- 13.5 The extension to three dimensions -- 13.6 The differential of distance -- 13.7 Arcs and areas of circles -- 13.8 two surfaces which can be developed on the Euclidean place -- 14 Euclidean Models -- 14.1 The meaning of "elliptic: and "hyperbolic" 14.2 Beltrami's model -- 14.3 The differential of distance -- 14.4 Gnomonic Projection -- 14.5 Development on Surfaces of constraint curvature -- 14.6 Klein's conformal model of the elliptic place -- 14.7 Klein's conformal model of the hyperbolic place -- 14.9 Conformal models of non-Euclidean Space -- 15. Concluding Remarks -- 15.1 Hjelmslev's mid-line -- 15.2 The Napier chain -- 15.3 The Engel Chain -- 15.4 Normalized canonical -- 15.5 Curvature -- 15.6 Quadratic forms -- 15.7 the volume of a tetrahedron -- 15.8 A brief historical survey of construction problems.

LCCN

58000715

OCLC

• ocm00529659
• 529659
• SCSB-279788

Owning Institutions

Princeton University Library