
Introduction to the Geometry of Complex Numbers
by Deaux, Roland; Eves, Howard-
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Summary
Table of Contents
Geometric Representation of Complex Numbers | |
Fundamental Operations | p. 15 |
Complex coordinate | |
Conjugate coordinates | |
Exponential form | |
Case where r is positive | |
Vector and complex number | |
Addition | |
Subtraction | |
Multiplication | |
Division | |
Scalar product of two vectors | |
Vector product of two vectors | |
Object of the course | |
Exercises 1 through 11 | |
Fundamental Transformations | p. 26 |
Transformation | |
Translation | |
Rotation | |
Homothety | |
Relation among three points | |
Symmetry with respect to a line | |
Inversion | |
Point at infinity of the Gauss plane | |
Product of one-to-one transformations | |
Permutable transformations | |
Involutoric transformations | |
Changing coordinate axes | |
Exercises 12 through 16 | |
Anharmonic Ratio | p. 35 |
Definition and interpretation | |
Properties | |
Case where one point is at infinity | |
Real anharmonic ratio | |
Construction | |
Harmonic quadrangle | |
Construction problems | |
Equianharmonic quadrangle | |
Exercises 17 through 31 | |
Elements of Analytic Geometry in Complex Numbers | |
Generalities | p. 55 |
Passage to complex coordinates | |
Parametric equation of a curve | |
Straight Line | p. 56 |
Point range formula | |
Parametric equation | |
Non-parametric equation | |
Centroid of a triangle | |
Algebraic value of the area of a triangle | |
Exercises 32 through 37 | |
The Circle | p. 63 |
Non-parametric equation | |
Parametric equation | |
Construction and calibration | |
Particular cases | |
Case ad - bc = 0 | |
Example | |
Exercises 38 through 45 | |
The Ellipse | p. 75 |
Generation with the aid of two rotating vectors | |
Construction of the elements of the ellipse | |
Theorem | |
Ellipse, hypocycloidal curve | |
Cycloidal Curves | p. 83 |
The Bellermann-Morley generation with the aid of two rotating vectors | |
Theorems | |
Epicycloids, hypocycloids | |
Unicursal Curves | p. 88 |
Definition | |
Order of the curve | |
Point construction of the curve | |
Circular unicursal curves | |
Foci | |
Conics | p. 95 |
General equation | |
Species | |
Foci, center | |
Center and radius of a circle | |
Parabola | |
Hyperbola | |
Ellipse | |
Exercises 46 through 59 | |
Unicursal Bicircular Quartics and Unicursal Circular Cubics | p. 106 |
General equation | |
Double point | |
Point construction of the cubic | |
Inverse of a conic | |
Limacon of Pascal, cardioid | |
Class of cubics and quartics considered | |
Foci | |
Construction of the quartic | |
Exercises 60 through 71 | |
Circular Transformations | |
General Properties of the Homography | p. 126 |
Definition | |
Determination of the homography | |
Invariance of anharmonic ratio | |
Circular transformation | |
Conservation of angles | |
Product of two homographies | |
Circular group of the plane | |
Definitions | |
Exercises 72 through 74 | |
The Similitude Group | p. 134 |
Definition | |
Properties | |
Center of similitude | |
Determination of a similitude | |
Group of translations | |
Group of displacements | |
Group of translations and homotheties | |
Permutable similitudes | |
Involutoric similitude | |
Application | |
Exercises 75 through 83 | |
Non-similitude Homography | p. 145 |
Limit points | |
Double points | |
Decomposition of a homography | |
Definitions | |
Parabolic homography | |
Hyperbolic homography | |
Elliptic homography | |
Siebeck's theorem | |
Exercises 84 through 91 | |
Mobius Involution | p. 158 |
Equation | |
Sufficient condition | |
Properties | |
Determination of an involution | |
Theorem | |
Construction of the involution defined by two pairs of points AA', BB' | |
Exercises 92 through 96 | |
Permutable Homographies | p. 166 |
Sufficient condition | |
Theorems | |
Harmonic involutions | |
Theorems | |
Simultaneous invariant of two homographies | |
Transform of a homography | |
Exercises 97 through 102 | |
Antigraphy | p. 174 |
Definition | |
Properties | |
Antisimilitude | p. 175 |
Equation | |
Properties | |
Symmetry | |
Double points | |
Construction of the double point E | |
Exercises 103 and 104 | |
Non-Antisimilitude Antigraphy | p. 179 |
Circular transformation | |
Limit points | |
Inversion | |
Non-involutoric antigraphies | |
Elliptic antigraphy | |
Hyperbolic antigraphy | |
Symmetric points | |
Determination of the affix of the center of a circle by the method of H. Pflieger-Haertel | |
Schick's theorem | |
Exercises 105 through 112 | |
Product of Symmetries | p. 193 |
Symmetries with respect to two lines | |
Symmetry and inversion | |
Product of two inversions | |
Homography obtained as product of inversions | |
Antigraphy obtained as product of three symmetries | |
Assorted exercises 113 through 136 | |
Index | p. 206 |
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