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The Advanced Geometry of Plane Curves and Their Applications
By C. Zwikker Dover Publications, Inc.
Copyright © 2014 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-15343-8
CHAPTER 1
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)). zi may stand for the point Pi (fig. 1) or for the vector from the origin to the point Pi. 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.
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) 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 analog on 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 the above notions we shall by way of introduction treat a number of curves and their analytical representation. It will strike us that simple functions always give rise to simple curves and vice versa.
a. z = 1 + ju is in fig. 2 represented by curve a; it is a straight line with a uniform u-scale. Its direction is the same as that of the imaginary axis (y-axis).
b. z = exp (ju) (Fig. 2, curve b)
Separating real and imaginary parts with the aid of Euler's rule),
exp (ju) = cos u + j sin u,
we see that the curve is a circle with radius 1 round the origin as centre. The u-scale is uniform, but we may change to non-uniform scales by a transformation of the parameter. Each function
z = exp (jf(u))
represents the same circle. Anticipating later results, we may well state that the simple analytical representation of the circle is the reason why complex calculus is so well-adapted to the treatment of technical problems.
c.
z = x + jy = √ 1 + ju.
In order to calculate real and imaginary parts we square the equation,
x2 + 2jxy - y2 1 + ju.
Separating now real and imaginary parts, we get
x2 - y2 = 1; 2 xy = u.
We recognize the first equation as that of the orthogonal hyperbola in Cartesian geometry. The curve is therefore a hyperbola (fig. 2, curve c).
The second equation fixes the u-scale along the curve, this scale being non-uniform in this case. Each curve
z = √ jf(u)
is identical with c.
d.
z = 1/1 - ju (Fig. 3)....
(1)
is a circle with radius 1/2 and with its centre on the real axis in the point z = 1/2. To show the circle character we split z into real and imaginary parts:
z = 1/1 - ju = 1 + ju/1 + u2
hence:
x = 1/1 + u2; y = u/1 + u2
and (x - 1/2)2 + y2 = 1/2, which is the Cartesian equation of the circle.
In connection with example b we might have represented the same circle by the equation
z = 1/2 (1 + exp (jv))....
(2)
now using the letter ν for the parameter to distinguish it from u.
The change from equation (1) to equation (2), however, is nothing but a transformation of the parameter. By putting
u = -tan ν/2
equation (1) is transformed into equation (2).
e. The evolvente of the circle z = exp (ju) is obtained by measuring on the tangent a distance u (BC in fig. 4). In complex notation the vector BC is
u (sin u - j cos u).
Adding this to the vector OB we find for the point C of the evolvente:
z = (cos u + j sin u) + u (sin u + - j cos u) = (1 - ju) exp (ju).
So that the equation of the circle evolvente is:
z = (1 - ju) exp (ju).
These five examples may suffice as a provisional introduction.
3. Intersection of two curves. Complex x- and y-values
The points of intersection of two curves
z1 = f1 (u); z2 = f2 (ν)
are found by solving the equation z1 = z2.
As this is a vector equation, it is equivalent to two normal equations, from which u and ν may be calculated. Introduction of these values in z1 or in z2 respectively gives us the points of intersection.
Example: Let us take for the first curve the circle z1 = r exp (ju) and for the second the straight line z2 = 1 + jν. From z1 = z2 it follows by taking real and imaginary parts apart:
r cos u = 1; r sin u = ν
from which u is easily eliminated by squaring and adding, giving:
ν = √ r2 - 1
so that the two points of intersection are: z = 1 ± j √ r2 - 1 which may be verified by applying elementary methods to fig. 5.
If r< 1 there are no points of intersection, ? is imaginary.
In this connection, however, one should carefully avoid a treacherous pitfall, for substituting in that case ν = ±j √ 1- r2 in z2 would make z = 1 ± √ 1 - r2 which are two points on the real axis and are certainly not the points of intersection.
This difficulty offers us an opportunity to enter somewhat deeper into the nature of the constant j. We may interpret the multiplication by j geometrically as a rotation in the x-y-plane counter-clockwise over an angle π/2. By repeating this operation the values obtained are the opposite of the original, thus j2z = -z. Therefore j can be treated in all calculations in the same way as √ -1.
(Continues...)
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