f(z) = b0 + b1z + b2z2 + ... + bnzn
In each example given here (n = 2, 3, and 4) the polynomial is monic, that is, bn = 1. The values of r, and bi (for i = 0, .. n-1) may be controlled by the user.
The idea is to understand f(Cr), the locus of f(w) for all w on Cr. The Cabri figure shows this locus and as well, a particular point "z" which may be Animated to move around Cr. As it does so, the point "f(z)" traverses the curve f(Cr).
It is interesting to see the global changes in f(Cr) as r and bi (for i = 0, .. n-1) are changed. For example, for relatively small values of r, changes in the number of times f(z) goes around b b0 depending on the magnitudes of the various coefficients of f. For example, when the degree of f is 3, if b1 and b2 are small enough, f(z) winds three times around b0, but if b1 is moved far enough outside the circle of radius 1, f(z) winds only two times around b0. As the values of the coefficients of f and the radius r are modified, one might see loops getting larger or smaller. When loops shrink, a cusp might appear and then cease to exist as the curve becomes smooth again, leaving little or no evidence of its existence except perhaps for a hint of a dimple.
When z is animated on Cr it is also interesting to watch the micro behaviour of f(z) as it moves along the locus, in some parts of the curve moving faster and other parts moving slower, and even sometimes coming to a momentary stop and reversing its direction at a cusp.
Explorers might wish to consider these questions:
These are the Cabri source files:
One of the files above gives a figure for the product of two complex numbers, a and b. It may be worth mentioning that the construction of the product numbers depends on knowing both 0 and 1. The number 0 is the dividing point between the positive and negative numbers and the direction from 0 to 1 determines which numbers are positive on the x-axis. The use of "ray(0,1)", the ray starting at 0 and pasing through 1, is useful for this purpose. The construction used requires the product of two positive numbers, |a| and |b|, the moduli of a and b, respectively. Also used is the sum of arg(a) and arg(b), the arguments (angles) of a and b. The reader may appreciate the "plain text" version of the product construction. The vocabulary used this plain text version is an attempt to formalize an interlingua between human language and computing languages such as Cabri or Maple source files. The author solicits your comments on this form of exposition.