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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 onetoone 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  
Nonparametric equation  
Centroid of a triangle  
Algebraic value of the area of a triangle  
Exercises 32 through 37  
The Circle  p. 63 
Nonparametric 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 BellermannMorley 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  
Nonsimilitude 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  
NonAntisimilitude Antigraphy  p. 179 
Circular transformation  
Limit points  
Inversion  
Noninvolutoric antigraphies  
Elliptic antigraphy  
Hyperbolic antigraphy  
Symmetric points  
Determination of the affix of the center of a circle by the method of H. PfliegerHaertel  
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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