Introduction to the Geometry of Complex Numbers
by Deaux, Roland; Eves, HowardFormat: Paperback
Pub. Date: 2008-03-05
Publisher(s): Dover Publications
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Summary
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 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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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