Projective Geometry, Volume 2

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CHAPTER I	1

Assumption	3

Double points of projectivities	4

Complex geometry 6 Imaginary elements adjoined to a real space	7

Harmonic sequence	9

Order in a net of rationality	13

10 Cuts in a net of rationality	14

11 Assumption of continuity	16

Foci of the ellipse and hyperbola	189

Focus and axis of a parabola	193

Eccentricity of a conic	196

Synoptic remarks on conic sections	199

Focal properties of collineations	201

Homogeneous quadratic equations in three variables	202

Nonhomogeneous quadratic equations in two variables	208

Euclidean classification of point conics	210

12 Chains in general	21

13 Consistency categoricalness and independence of the assumptions	23

14 Foundations of the complex geometry	29

16 Modula spaces	32

Recapitulation	36

CHAPTER II	37

The two senseclasses on a line	40

Sense in any onedimensional form	43

Separation of point pairs	44

Segments and intervals	45

Linear regions	47

Algebraic criteria of sense	49

Pairs of lines and of planes	50

The triangle and the tetrahedron 27 Algebraic criteria of separation Cross ratios of points in space	55

Euclidean spaces 29 Assumptions for a Euclidean space	59

Sense in a Euclidean plane	61

31 Sense in Euclidean spaces	63

32 Sense in a projective space	64

Intuitional description of the projective plane	67

THE AFFINE GROUP IN THE PLANE	70

64	106

53 Algebraic formulas for line reflections	115

Displacements and symmetries Congruence	123

The group of displacements	131

Introduction of order relations	138

The Euclidean geometry A set of assumptions	144

Area	151

Pencils of circles	157

Measure of line pairs	163

CHAPTER V	170

Interior and exterior of a conic	174

Doub 76 Double points of projectivities	179

Rulerandcompass constructions	180

Conjugate imaginary elements	182

79 Projective affine and Euclidean classification of conics	186

Classification of line conics	212

89 Polar systems	215

CHAPTER VI	219

Correspondence between the complex line and the real Euclidean plane	222

The inversion group in the real Euclidean plane	225

Correspondence between the real Euclidean plane and a complex pencil of lines	238

The real inversion plane	241

Order relations in the real inversion plane	244

Types of circular transformations	246

Chains and antiprojectivities	250

Tetracyclic co�rdinates	253

Involutoric collineations	257

The projective group of a quadric	259

Real quadrics	262

The complex inversion plane	264

Function plane inversion plane and projective plane	268

Projectivities of onedimensional forms in general	271

107 Projectivities of a quadric	273

108 Products of pairs of involutoric projectivities	277

Conjugate imaginary lines of the second kind	281

The principle of transference	284

117 Generalization to n dimensions	305

129 Representation of rotations and onedimensional projectivities	342

CHAPTER VIII	350

Interpretation of hyperbolic geometry in the inversion plane	357

Angular measure	363

154	385

168 Sense in overlapping convex regions	424

170 Pencils of rays	431

173 One and twosided regions	437

179 Elementary transformations of triads of lines	446

Polygons in space	480

Odd and even polygons in the projective plane	489

Orientation of space 430	496

Copyright

Title	Projective Geometry, Volume 2 Projective Geometry, John Wesley Young
Authors	Oswald Veblen, John Wesley Young
Publisher	Ginn, 1918
Original from	the University of California
Digitized	Nov 21, 2008

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