Introduction to the Geometry of Complex Numbers

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Format: Paperback
Pub. Date: 2008-03-05
Publisher(s): Dover Publications
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Geared toward readers unfamiliar with complex numbers, this text explains how to solve the kinds of problems that frequently arise in the applied sciences, especially electrical studies. To assure an easy and complete understanding, topics are developed from the beginning, with emphasis on constructions related to algebraic operations. 1956 edition.

Table of Contents

Geometric Representation of Complex Numbers
Fundamental Operationsp. 15
Complex coordinate
Conjugate coordinates
Exponential form
Case where r is positive
Vector and complex number
Scalar product of two vectors
Vector product of two vectors
Object of the course
Exercises 1 through 11
Fundamental Transformationsp. 26
Relation among three points
Symmetry with respect to a line
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 Ratiop. 35
Definition and interpretation
Case where one point is at infinity
Real anharmonic ratio
Harmonic quadrangle
Construction problems
Equianharmonic quadrangle
Exercises 17 through 31
Elements of Analytic Geometry in Complex Numbers
Generalitiesp. 55
Passage to complex coordinates
Parametric equation of a curve
Straight Linep. 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 Circlep. 63
Non-parametric equation
Parametric equation
Construction and calibration
Particular cases
Case ad - bc = 0
Exercises 38 through 45
The Ellipsep. 75
Generation with the aid of two rotating vectors
Construction of the elements of the ellipse
Ellipse, hypocycloidal curve
Cycloidal Curvesp. 83
The Bellermann-Morley generation with the aid of two rotating vectors
Epicycloids, hypocycloids
Unicursal Curvesp. 88
Order of the curve
Point construction of the curve
Circular unicursal curves
Conicsp. 95
General equation
Foci, center
Center and radius of a circle
Exercises 46 through 59
Unicursal Bicircular Quartics and Unicursal Circular Cubicsp. 106
General equation
Double point
Point construction of the cubic
Inverse of a conic
Limacon of Pascal, cardioid
Class of cubics and quartics considered
Construction of the quartic
Exercises 60 through 71
Circular Transformations
General Properties of the Homographyp. 126
Determination of the homography
Invariance of anharmonic ratio
Circular transformation
Conservation of angles
Product of two homographies
Circular group of the plane
Exercises 72 through 74
The Similitude Groupp. 134
Center of similitude
Determination of a similitude
Group of translations
Group of displacements
Group of translations and homotheties
Permutable similitudes
Involutoric similitude
Exercises 75 through 83
Non-similitude Homographyp. 145
Limit points
Double points
Decomposition of a homography
Parabolic homography
Hyperbolic homography
Elliptic homography
Siebeck's theorem
Exercises 84 through 91
Mobius Involutionp. 158
Sufficient condition
Determination of an involution
Construction of the involution defined by two pairs of points AA', BB'
Exercises 92 through 96
Permutable Homographiesp. 166
Sufficient condition
Harmonic involutions
Simultaneous invariant of two homographies
Transform of a homography
Exercises 97 through 102
Antigraphyp. 174
Antisimilitudep. 175
Double points
Construction of the double point E
Exercises 103 and 104
Non-Antisimilitude Antigraphyp. 179
Circular transformation
Limit points
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 Symmetriesp. 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
Indexp. 206
Table of Contents provided by Ingram. All Rights Reserved.

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