Projective Geometry

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.

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 Frégier 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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