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Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein.

Author: L Rédei; Felix Klein
Publisher: Oxford : Pergamon Press, 1968.
Series: International series in pure and applied mathematics, v. 97.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein.
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Redei, L.
Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein.
Burlington : Elsevier Science, ©2014
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: L Rédei; Felix Klein
ISBN: 9781483282701 1483282708 9781483229904 1483229904
OCLC Number: 898422280
Notes: Translation of Begründung der euklidischen und nichteuklidischen Geometrien nach F. Klein.
65. Absolute involution of points on a proper line.
Description: 1 online resource (412 pages).
Contents: Front Cover; Foundation of Euclidean and Non-Euclidean Geometries According to F. Klein; Copyright Page; Table of Contents; PREFACE; CHAPTER 1. AXIOMS; 1. Axioms of incidence; 2. Axioms of betweenness; 3. Axiom of continuity; 4. Axioms of motion; CHAPTER 2. CONSEQUENCES OF THE SYSTEM OF AXIOMS I; 5. Simple properties of straight lines and planes; 6. Desargues configurations; 7. Linear subspaces; 8. The lattice of linear subspaces; 9. Basic projective configurations; 10. Projection and intersection; CHAPTER 3. SIMPLE CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II. 11. Segments. Triangles 12. Properties of segments; 13. Linear ordering; 14. Properties of triangles; 15. The tetrahedron; 16. Neighbourhoods; 17. Validity of the systems of Axioms I, II for the basic domain R'; 18. Generalization of the notion of space; 19. The extension and restriction of spaces; CHAPTER 4. PROJECTIVE CLOSURE; 20. Half-subspaces; 21. Half-pencils. Angles; 22. Some properties of pencils and bundles; 23. Coplanar Desargues configurations; 24. Improper pencils of lines; 25. Improper bundles of lines; 26. The projective closure R of R. 27. The projective axioms 28. The general case; CHAPTER 5. INVESTIGATION OF THE PROJECTIVE SPACE; 29. Preliminaries; 30. Theorem of duality in projective space; 31. Collineations; 32. The Erlangen programme; 33. Theorem of duality of the plane; 34. Perspectivities and projectivities; 35. Central collineations of the plane; 36. Separation; 37. Cyclic ordering; 38. Projective segments and angles; 39. Complete quadrangles. Harmonic points; 40. Preliminaries about coordinate systems; 41. Coordinates in projective scales; 42. Halving a projective scale. 43. Coordinates for dyadic sets of points on a lineCHAPTER 6. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III; 44. Preliminaries; 45. Theorem concerning the infinite point; 46. Coordinates in an affine line; 47. Coordinates on the basic projective configurations of the first degree; 48. Point-coordinates in an affine plane; 49. The fundamental theorem of projective geometry; 50. Point-coordinates in an affine space; 51. Vectors; 52. Homogeneous point- and plane-coordinates in space. Point- and line-coordinates in a plane. 53. Determination of all collineations of the space 54. Determination of the coordinate transformations of space; 55. Transformation of projective coordinates; 56. Cross ratio; 57. Imaginary points; 58. Fixed elements of projectivities; 59. Involutions; 60. Involutory collineations of a plane; CHAPTER 7. CONSEQUENCES OF THE SYSTEMS OF AXIOMS I, II, III, IV; 61. Extended motions; 62. The comparability of segments; 63. Reflections and rotations. Absolute polar plane; 64. Metric scales. Infinite and ultra-infinite points. Elliptic, parabolic and hyperbolic geometries.
Series Title: International series in pure and applied mathematics, v. 97.
Other Titles: Begründung der euklidischen und nichteuklidischen Geometrien nach F. Klein.

Abstract:

Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein.

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