Geometry · A Comprehensive Course by Pedoe, Dan -- Read -- Imperial Library of Trantor

Index

Title Page Dedication Copyright Page Contents Preface Chapter 0: Preliminary Notions

0.1 The Euclidean plane 0.2 The affine plane 0.3 Notation 0.4 Complex numbers 0.5 The Argand diagram 0.6 The conjugate of a complex number 0.7 The triangle inequality Exercises 0.8 De Moivre's theorem Exercises 0.9 Real and imaginary in geometry 0.10 Equivalence relations Exercises 0.11 Mappings, or transformations 0.12 Products of mappings Exercises 0.13 Linear algebra

Chapter I: Vectors

1.1 Vectors in the Euclidean (and affine) plane 1.2 Addition of bound vectors 1.3 Vector spaces 1.4 Linear dependence Exercises 2.1 Notation 3.1 Points on a line, theorems I, II, III, IV 3.2 First applications 3.3 The centroid of a triangle Exercises 4.1 Theorem of Menelaus 4.2 Barycentric coordinates 4.3 Theorem of Ceva 4.4 Example Exercises 5.1 The parallel case of the Desargues theorem 5.2 The Desargues theorem Exercises 6.1 Reciprocal figures Exercises 7.1 Inner products in Euclidean space of two dimensions 7.2 Example 7.3 The nine-point circle Exercises 8.1 The supplement of vectors in a plane 8.2 Example 8.3 Orthologic triangles Exercises 9.1 The exterior product of two vectors 9.2 Geometrical interpretation of exterior product 9.3 A criterion for collinearity Exercises 9.4 Application of the criterion 9.5 The parallel case of Pappus' theorem Exercise 10.1 A necessary and sufficient condition for the concurrency of three lines 10.2 The theorem itself 10.3 Application of the concurrency conditions Exercises 11.1 The exterior product of three vectors 11.2 Applications 11.3 Example Exercises

Chapter II: Circles

12.1 The nine-point circle Exercises 13.1 The centers of similitude of two circles Exercises 14.1 Reflexions in a line 14.2 Product of reflexions in intersecting lines 14.3 Product of reflexions in parallel lines Exercises 14.4 Example Exercise 15.1 Applications of reflexions to minimum problems 16.1 Product of rotations Exercises 17.1 Another application of reflexions 18.1 The power of a point with respect to a circle 18.2 Geometrical interpretation of the power of a point 18.3 The Apollonius circle of two points Exercises 19.1 Harmonic division Exercises 20.1 Inversion 20.2 A property of inverse points Exercises 20.3 Inversion in a line 20.4 A construction for inverse points 20.5 Extending a segment with a pair of compasses Exercises 21.1 Inverses of lines and circles Exercises 22.1 The angle of intersection of two circles 22.2 Inversion preserves angles 22.3 Example Exercises 22.4 Another look at the angle of intersection 23.1 The effect of inversion on length 23.2 The cross-ratio of points on a line 23.3 The inverse of the center of a circle 23.4 Two celebrated theorems 24.1 An extension of Ptolemy's theorem 24.2 Fermat's principle and Snell's law Exercises 24.3 The Fermat problem Exercises 25.1 A fixed point theorem 25.2 The inverse of a circle and a pair of inverse points Exercises 25.3 Circles into concentric circles 25.4 The Steiner porism 25.5 The Steiner formula Exercises 26.1 The problem of Apollonius Exercises

Chapter III: Coaxal Systems of Circles

27.1 Pencils of circles 27.2 Pencil of circles as a locus Exercises 27.3 The radical axis of two circles Exercises 27.4 Justification of the term: coaxal system Exercise 27.5 Determination of a coaxal system Exercises 28.1 A theorem on radical axes 28.2 The circle orthogonal to three given circles Exercises 28.3 Another proof of the nine-point circle theorem 28.4 Clerk Maxwell's derivation of reciprocal figures 29.1 Canonical form for the equation of a coaxal system 29.2 The intersecting type of coaxal system 29.3 The tangent type of coaxal system 29.4 The non-intersecting type of coaxal system 29.5 Polar systems of coaxal circles Exercises 30.1 The inverse of a coaxal system Exercises 31.1 Geometry of the compasses 31.2 Determination of the center of the circle through three points Exercise 32.1 Constructions with a disk Exercises

Chapter IV: The Representation of Circles by Points in Space ofThree Dimensions

33.1 Vectors in Euclidean and affine space of three dimensions 33.2 The sum of two vectors 33.3 Linear dependence 33.4 Points on a line 33.5 Points in a plane Exercises 34.1 Inner products in Euclidean space of three dimensions 35.1 The polar plane of a point Exercises 36.1 The representation of circles by points of E3 36.2 The representation of a coaxal system Exercises 36.3 Deductions from the representation Exercises 37.1 Orthogonal circles and conjugate points 37.2 Polar lines and polar coaxal systems 37.3 Bundles of circles Exercises 38.1 Circles which cut three given circles at equal angles 39.1 The representation of inversion Exercises 40.1 An algebra of circles 40.2 An inner product for two circles 40.3 A theorem of Descartes Exercises

Chapter V: Mappings of the Euclidean Plane

41.1 Translations 41.2 Rotations 41.3 Reflexions 41.4 Inversion 41.5 Central dilatations Exercises 42.1 Isometries 42.2 Coordinate transformations Exercises 43.1 The main theorem on isometries of the Gauss plane 43.2 An auxiliary theorem 43.3 Isometries are collineations 43.4 Isometries and parallel lines 43.5 Determination of an isometry 43.6 An auxiliary theorem Exercises 44.1 Subgroups 44.2 Conditions for a subgroup 44.3 Cosets 44.4 The identity of cosets 44.5 Right and left cosets Exercises 45.1 Conjugate and normal subgroups 45.2 An equivalent definition of normal subgroup 45.3 Subgroups of index two 45.4 Isomorphisms 45.5 Automorphisms Exercises 46.1 Translations 46.2 Dilative rotations 46.3 Central dilatations 46.4 Group property of isometries Exercises 47.1 Similarity transformations 47.2 Main theorem for similitudes 47.3 An auxiliary theorem 47.4 Similitudes are collineations 47.5 Similitudes and parallel lines 47.6 Determination of a similitude 47.7 An auxiliary theorem 47.8 Group property of similitudes 47.9 Condition for direct similarity 47.10 Orientation Exercises 48.1 The group of translations 48.2 The fixed points of a mapping 48.3 The fixed point of a direct isometry 48.4 Direct isometries and rotations 48.5 Rotation groups Exercises 49.1 The fixed points of indirect isometries 49.2 Indirect isometries as reflexions 49.3 Canonical form for reflexions Exercises 50.1 Involutory isometries 50.2 The composition of two reflexions 50.3 Opposite isometries without fixed points 50.4 Hjelmslev's theorem 50.5 Direct and opposite similitudes Exercises 51.1 Line reflexions 51.2 Condition for two lines to be perpendicular 51.3 The incidence of a point and a line 51.4 Isometries as reflexions Exercises

Chapter VI: Mappings of the Inversive Plane

52.1 Moebius transformations 52.2 The determinant of a Moebius mapping 52.3 The map of a circle or line under a Moebius transformation 52.4 Dissection of a Moebius transformation 52.5 The group of Moebius transformations Exercises 53.1 Fixed points of a Moebius transformation 53.2 The determination of a Moebius transformation 53.3 The cross-ratio of four complex numbers 53.4 An invariant under Moebius transformations 53.5 The reality of the cross-ratio 53.6 Lines in the inversive plane and the point ∞ Exercises 54.1 The map of the inside of a circle 54.2 Map of a circle into itself 54.3 Conformal property of Moebius transformations Exercises 55.1 Special Moebius transformations 55.2 Theorems on M-transformations 55.3 Interchanging two points Exercises 56.1 The Poincaré model of a hyperbolic non-Euclidean geometry 56.2 The congruence of p-segments Exercises 57.1 The SAS theorem 57.2 The parallel axiom Exercises 58.1 The angle-sum of a p-triangle 58.2 The exterior angle is greater than either of the interior and opposite angles 58.3 Non-Euclidean distance Exercises 59.1 Similar p-triangles are p-congruent 59.2 p-circles in the p-plane Exercises

Chapter VII: The Projective Plane and Projective Space

60.1 The complex projective plane 60.2 The principle of duality in the projective plane Exercises 61.1 A model for the projective plane 62.1 The normalization theorem for points on a line 62.2 The Desargues theorem Exercises 63.1 The normalization theorem for points in a projective plane Exercises 64.1 Change of triangle of reference 64.2 Harmonic ranges 64.3 Abstract projective planes p. 262. Exercises 65.1 Projective space of three dimensions 65.2 Moebius tetrahedra Exercises

Chapter VIII: The Projective Geometry of n Dimensions

66.1 Projective space of n dimensions 66.2 The principle of duality in Sn 66.3 The intersection and join of linear subspaces of Sn Exercises 67.1 The propositions of incidence in Sn. p. 278. 67.2 Some examples 67.3 Projection 67.4 General subspaces in Sn Exercises 68.1 Dedekind's law for linear subspaces in Sn 68.2 Dedekind's law and projections Exercises 69.1 Cross-ratio Exercises 70.1 Projective transformations 70.2 Change of coordinate system Exercises 71.1 The fundamental theorem for projective transformations 71.2 Projections as projective transformations Exercises 72.1 Derivation of the tetrahedral complex 73.1 Singular projectivities Exercise 74.1 Fixed points of projectivities Exercises 75.1 The Jordan normal form for a collineation Exercises 76.1 Special collineations in Sn 76.2 Cyclic collineations in Sn Exercises

Chapter IX: The Projective Generation of Conics and Quadrics

77.1 The conic 77.2 Quadratic forms Exercises 78.1 Pointconics and line-conics 78.2 Projective correspondences on a conic 78.3 The polar of a point Exercises 79.1 Conics through four points Exercises 80.1 Conics in the real Euclidean plane 80.2 Conics in the real affine plane 80.3 The Euclidean group Exercises 81.1 The quadric 81.2 Intersection of a plane and a quadric surface Exercises 82.1 Quadric cones 83.1 Transversals of three skew lines Exercises 84.1 The quadratic form in four variables 85.1 The polar of a point 85.2 Polar lines with regard to a quadric 85.3 Conjugate lines 85.4 Correlations 85.5 Polarities and null-polarities 85.6 The dual of a quadric Exercises 86.1 Another look at Pascal's theorem 86.2 Steiner's theorem on Pascal lines Exercises 87.1 Stereographic projection Exercises 88.1 The sphere as a quadric 88.2 Tetracyclic coordinates 88.3 The Euclidean group in three dimensions 88.4 The affine group in three dimensions Exercises

Chapter X: Prelude to Algebraic Geometry

89.1 Curves on a quadric 89.2 Generation of C3 89.3 The space cubic as a residual curve Exercises 90.1 Normal equations for a space cubic Exercises 91.1 Rational surfaces Exercises 92.1 The cubic surface Exercises 93.1 The intersection of two quadrics 93.2 The intersection of three quadrics 93.3 Sets of eight associated points 93.4 A theorem for plane cubic curves Exercises 94.1 Applications of the associated points theorem 94.2 Oriented circles Exercises

Appendix I Appendix II Appendix III Bibliography and References Index

← Prev
Back
Next →

← Prev
Back
Next →