The Advanced Geometry of Plane Curves and Their Applications

C. Zwikker

• Publisher: Dover Publications
• Year: 2011
• Print ISBN:
  9780486442761
• eBook ISBN:
  9780486153438
• Topic: Mathématiques
• Subtopic: Géométrie

"Of chief interest to mathematicians, but physicists and others will be fascinated ... and intrigued by the fruitful use of non-Cartesian methods. Students ... should find the book stimulating." — British Journal of Applied Physics
This study of many important curves, their geometrical properties, and their applications features material not customarily treated in texts on synthetic or analytic Euclidean geometry. Its wide coverage, which includes both algebraic and transcendental curves, extends to unusual properties of familiar curves along with the nature of lesser known curves.
Informative discussions of the line, circle, parabola, ellipse, and hyperbola presuppose only the most elementary facts. The less common curves — cissoid, strophoid, spirals, the leminscate, cycloid, epicycloid, cardioid, and many others — receive introductions that explain both their basic and advanced properties. Derived curves-the involute, evolute, pedal curve, envelope, and orthogonal trajectories-are also examined, with definitions of their important applications. These range through the fields of optics, electric circuit design, hydraulics, hydrodynamics, classical mechanics, electromagnetism, crystallography, gear design, road engineering, orbits of subatomic particles, and similar areas in physics and engineering. The author represents the points of the curves by complex numbers, rather than the real Cartesian coordinates, an approach that permits simple, direct, and elegant proofs.

CHAPTER I

THE COMPLEX PLANE

1. Introduction

The representation of a complex number

z = x + jy (j² = – 1)

as a point with coordinates x and y in the mathematical plane dates from CASPAR WESSEL (1799) and from GAUSS (1831)¹). z_i may stand for the point P_i (fig. 1) or for the vector from the origin to the point P_i. Addition and subtraction of two complex numbers may be accomplished geometrically by vectorial addition, respectively subtraction of the vectors denoted by z₁ and z₂. In Ch. II we shall extend this remark in such a way that we shall find for each analytical operation its geometrical equivalent.

Fig. 1

This geometrical representation of the operations performed on complex numbers has been an important aid in the study of complex functions and in the treatment of complex impedances and admittances in electric engineering. The reverse procedure, using analytical calculations with complex numbers as a means of detecting or proving geometrical properties of plane figures has so far received but little attention.

It will appear throughout this book that the application of complex numbers in analytical geometry often has marked advantages over the conventional Cartesian geometry, especially as regards physical and technical problems.

Now suppose x and y to be functions of a real parameter u and suppose we plot the corresponding z values in the plane for all values of u, we find a continuous locus of z values, a curve and a scale of u-values attached to it. The equation

z =f (u),

where f (u) stands for a complex function of a real parameter u, is the equation of a curve and is equivalent to the Cartesian formula:

f (x, y) = 0

or rather we should say that it gives more than the Cartesian formula, as it also fixes a scale along the curve.

The method of representing a curve as the locus of the extremities of vectors, the components of which change in a continuous way with a parameter has been used in three-dimensional geometry ²) and it might look as if we were taking a two-dimensional cross-section of this more general treatment. This is not the case. The important property of the two-dimensional vector of being represented by a complex number has no simple analogon in three dimensions and the possibility of applying the often surprisingly simple calculation with complex numbers gives a special charm to two-dimensional vectorial geometry, or, expressed in a less exact way, to the geometry of the complex plane.

In the actually occurring cases various quantities may be introduced as the parameter. In kinematics, for example, it will usually be the time, while in purely geometrical problems the length of the curve, measured from a fixed point will be preferably used, especially in considerations of a more general nature. Further, the abscissa x or the ordinate y and even the radius of curvature may occasionally be the most suitable one. Again, in electrotechnical problems the usual one is the angular frequency along the contours representing impedances as functions of the frequency and we shall, besides, meet many cases, in which still other quantities will play the part of parameter.

By changing from one parameter u to another v, being a real function of u, we do not change the character of the curve, the only thing that changes geometrically is the scale along the curve. It is therefore possible to represent one curve by different equations and we shall in each case choose the one that is most adapted to the problem in hand.

2. First examples

In order to familiarize the reader with th...

Table of contents

• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• CHAPTER I - THE COMPLEX PLANE
• CHAPTER II - THE GEOMETRICAL INTERPRETATION OF ANALYTIC OPERATIONS APPLIED TO COMPLEX NUMBERS
• CHAPTER III - THE STRAIGHT LINE
• CHAPTER IV - THE TRIANGLE
• CHAPTER V - THE CIRCLE
• CHAPTER VI - ALGEBRAIC CURVES
• CHAPTER VII - THE ELLIPSE
• CHAPTER VIII - HYPERBOLA
• CHAPTER IX - THE PARABOLA
• CHAPTER X - INVOLUTES, EVOLUTES, ANTICAUSTICS
• CHAPTER XI - PEDALS AND OTHER DERIVED CURVES
• CHAPTER XII - AREAS AND OTHER INTEGRALS
• CHAPTER XIII - ENVELOPES
• CHAPTER XIV - ORTHOGONAL TRAJECTORIES
• CHAPTER XV - KINKED CURVES
• CHAPTER XVI - SPIRALS
• CHAPTER XVII - LEMNISCATE
• CHAPTER XVIII - CYCLOID
• CHAPTER XIX - EPI- AND HYPOCYCLOIDS
• CHAPTER XX - CARDIOID AND LIMAÇON
• CHAPTER XXI - GEAR WHEEL TOOTH PROFILES
• APPENDIX I - ANALYTICAL FORMULAS
• APPENDIX II - TABLE OF PLANE CURVES
• APPENDIX III - TABLE OF OPERATIONS
• APPENDIX IV - TABLE OF DERIVED CURVES
• APPENDIX V - HISTORICAL NOTES
• INDEX
• DOVER PHOENIX EDITIONS