## The Real Projective Plane, Volume 1Along with many small improvements, this revised edition contains van Yzeren's new proof of Pascal's theorem (§1.7) and, in Chapter 2, an improved treatment of order and sense. The Sylvester-Gallai theorem, instead of being introduced as a curiosity, is now used as an essential step in the theory of harmonic separation (§3.34). This makes the logi cal development self-contained: the footnotes involving the References (pp. 214-216) are for comparison with earlier treatments, and to give credit where it is due, not to fill gaps in the argument. H.S.M.C. November 1992 v Preface to the Second Edition Why should one study the real plane? To this question, put by those who advocate the complex plane, or geometry over a general field, I would reply that the real plane is an easy first step. Most of the prop erties are closely analogous, and the real field has the advantage of intuitive accessibility. Moreover, real geometry is exactly what is needed for the projective approach to non· Euclidean geometry. Instead of introducing the affine and Euclidean metrics as in Chapters 8 and 9, we could just as well take the locus of 'points at infinity' to be a conic, or replace the absolute involution by an absolute polarity. |

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### Contents

A Comparison of Various Kinds of Geometry | 1 |

13 Central projection | 2 |

14 The line at infinity | 4 |

15 Perspective triangles | 6 |

16 The directed angle or cross | 8 |

17 Hexagramma mysticum | 9 |

18 An outline of subsequent work | 10 |

Incidence | 12 |

73 Construction for a projectivity on a conic | 96 |

74 Construction for the invariant points of a given hyperbolic projectivity | 98 |

75 Involution on a conic | 99 |

76 A generalization of Steiners construction | 102 |

77 Trilinear polarity | 103 |

Affine Geometry | 105 |

82 Intermediacy | 106 |

83 Congruence | 107 |

22 The axioms of incidence | 14 |

23 The principle of duality | 15 |

24 Quadrangle and quadrilateral | 17 |

25 Harmonic conjugacy | 18 |

26 Ranges and pencils | 21 |

28 The invariance and symmetry of the harmonic relation | 23 |

Order and Continuity | 25 |

32 Sense | 27 |

33 The SylvesterGallai theorem | 29 |

34 Ordered correspondence | 30 |

35 Continuity | 34 |

37 Order in a pencil | 37 |

OneDimensional Projectivities | 39 |

42 The fundamental theorem of projective geometry | 41 |

43 Pappuss theorem | 43 |

44 Classification of projectivities | 45 |

45 Periodic projectivities | 48 |

47 Quadrangular set of six points | 52 |

48 Projective pencils | 54 |

TwoDimensional Projectivities | 55 |

52 Perspective collineation | 57 |

53 Involutory collineation | 59 |

54 Correlation | 61 |

55 Polarity | 62 |

56 Polar and selfpolar triangles | 66 |

57 The selfpolarity of the Desargues configuration | 68 |

58 Pencil and range of polarities | 70 |

59 Degenerate polarities | 71 |

Conics | 73 |

62 Elliptic and hyperbolic polarities | 74 |

63 How a hyperbolic polarity determines a conic | 76 |

64 Conjugate points and conjugate lines | 78 |

65 Two possible definitions for a conic | 80 |

66 Construction for the conic through five given points | 83 |

67 Two triangles inscribed in a conic | 85 |

68 Pencils of conics | 87 |

Projectivities on a Conic | 92 |

72 Pascal and Brianchon | 94 |

84 Distance | 109 |

85 Translation and dilatation | 113 |

86 Area | 114 |

87 Classification of conies | 117 |

88 Conjugate diameters | 119 |

89 Asymptotes | 121 |

Euclidean Geometry | 126 |

92 Circles | 128 |

93 Axes of a conic | 131 |

94 Congruent segments | 133 |

95 Congruent angles | 134 |

96 Congruent transformations | 138 |

Continuity | 147 |

102 Proving Archimedess axiom | 148 |

103 Proving the line to be perfect | 149 |

104 The fundamental theorem of projective geometry | 152 |

105 Proving Dedekinds axiom | 153 |

The Introduction of Coordinates | 155 |

111 Addition of points | 156 |

112 Multiplication of points | 158 |

113 Rational points Defining | 161 |

115 The onedimensional continuum | 163 |

116 Homogeneous coordinates | 165 |

The Use of Coordinates | 169 |

122 Analytic geometry | 171 |

123 Verifying the axioms of incidence | 173 |

124 Verifying the axioms of order and continuity | 174 |

125 The general collineation | 178 |

126 The general polarity | 181 |

127 Conies | 185 |

affine and areal coordinates | 191 |

Certesian and trilinear coordinates | 193 |

The Complex Projective Plane | 200 |

How to Use Mathematica | 202 |

Bibliography | 214 |

217 | |

### Common terms and phrases

AB/C affine geometry angle asymptotes axes axioms axis bisectors centre Chapter circle coincide collinear points collineation concurrent lines congruent conic conjugate diameters conjugate lines conjugate points conjugates with respect construction coordinates corresponding points deduce Desargues Desargues configuration Desargues's diagonal triangle distinct elliptic equation Euclidean geometry EXERCISES fixed line fixed point four points given line given point H(AB harmonic conjugate harmonic homology Hence hexagon Hint incident invariant point involution of conjugate isogonal conjugate line at infinity line joining locus meet midpoint parabola parallelogram Pascal Pascal's theorem perpendicular perspectivity point at infinity points and lines pole projective geometry Proof Prove quadrangular quadrilateral range real numbers reflexion relation secant segment self-conjugate points self-dual system self-polar triangle Show Staudt tangents three points tion transformation triangle ABC triangle of reference triangle PQR trilinear polar Veblen and Young vertex vertices