Projective Geometry

by
Format: Paperback
Pub. Date: 2006-07-21
Publisher(s): Dover Publications
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Summary

Highlighted by numerous examples, this book explores methods of the projective geometry of the plane. It derives the projective properties of the conic and discusses representation by the general equation of the 2nd degree, concluding with a study of the relationship between Euclidean and projective geometry. 1960 edition.

Table of Contents

CHAPTER I INTRODUCTION: THE PROPOSITIONS OF INCIDENCE
1. Historical note
1(2)
2. The projective method
3(2)
3. Desargues' theorem
5(2)
4. The analytical method
7(5)
5. Analytical proof of Desargues' theorem
12(1)
6. Pappus' theorem
13(2)
7. The fourth harmonic point
15(4)
8. The complete quadrangle
19(2)
CHAPTER II RELATED RANGES AND PENCILS: INVOLUTIONS
9. Related ranges
21(1)
10. The cross ratio
22(3)
11. Cross ratio property of a (1-1) correspondence
25(3)
12. Ranges in perspective
28(3)
13. Related ranges on the same base; double points
31(3)
14. Related pencils
34(1)
15. Involution on a line
35(2)
16. Cross ratio property of an involution
37(2)
17. Involution property of the complete quadrangle
39(1)
18. An algebraic representation of an involution
40(3)
19. Pencils in involution
43(1)
CHAPTER III THE CONIC
20. Introduction
44(1)
21. Projective definition of the conic
44(2)
22. Related ranges on a conic
46(1)
23. Involution on a conic
47(1)
24. The conic as an envelope
48(2)
25. Desargues' theorem
50(1)
26. Pascal's theorem
51(3)
27. Pole and Polar
54(5)
28. Properties of two conies
59(5)
29. Pencils of conics
64(6)
CHAPTER IV ABSOLUTE ELEMENTS: THE CIRCLE: FOCI OF CONICS
30. Introduction
70(1)
31. Absolute elements
71(1)
32. The circle
72(6)
33. The conic and the absolute points
78(1)
34. Central properties of conics; conjugate diameters
79(1)
35. Foci and axes of a conic
80(3)
36. The director circle
83(1)
37. Confocal conics
84(2)
38. The auxiliary circle
86(2)
39. Some properties of the parabola
88(1)
40. Some properties of the rectangular hyperbola
89(3)
41. The hyperbola of Apollonius
92(2)
42. The Fr├ęgier point
94(1)
CHAPTER V THE EQUATION OF A LINE AND OF A CONIC: ALGEBRAIC CORRESPONDENCE ON A CONIC: THE HARMONIC LOCUS AND ENVELOPE
43. The equation of a line
95(2)
44. The equation of a conic
97(3)
45. Tangent, pole and polar
100(1)
46. The line-equation of a conic
101(1)
47. Special forms for the equation of a conic
102(2)
48. Correspondence between points of a conic
104(2)
49. The symmetrical (22) correspondence of points on a conic
106(1)
50. The harmonic envelope
107(3)
51. A conic associated with three conics of a pencil
110(3)
CHAPTER VI METRICAL GEOMETRY
52. Introduction
113(1)
53. Projective definition of distance and angle
114(1)
54. The absolute conic
114(2)
55. Algebraic expressions for distance and angle
116(1)
56. Real and complex points and lines
117(1)
57. Real and complex conics
118(1)
58. Metrical geometry
119(1)
59. Distance and angle in Euclidean geometry
120(4)
60. The Euclidean equivalents of simple projective elements
124(3)
INDEX 127

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