The applet

The Geogebra applet shown below draws the image curve of a circumference $B_r$ centered in the origin with radius r in the complex plane under a transformation defined by a complex polynomial

$p(z) = a_{n}z^{n}+ \cdots + a_{3}z^{3} + a_{2}z^{2}+a_{1}z + a_{0}=0.$

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Created with GeoGebra

The applet opens a window that shows a left panel called Algebra panel and a right panel called Graphics panel. A graphics panel appears in a new window called Graphics 2.

The coefficients and the degree of the polynomial can be input in either panel of the first window. In the Algebra panel, you can edit the values using the keyboard. The coefficients must have real and imaginary parts, even if they are 0. The value of n is an integer between 4 and 12, that defines the polynomial degree if this is at least 4. Polynomials of lower degree are defined setting to 0 the apropriate coefficients.

In the first Graphics panel, the coefficients are represented by points of the complex plane that can be dragged to the desired value. A slider in the same panel can be used to establish the degree.

A point z can be defined introducing its modulus and argument in the Algebra panel or by dragging its affix in the Graphics panel.

The polynomial value at this point is automatically computed and shown both in the Algebra panel and in the second window.

Finally, a circle $C_r$ centered in the origin whose radius is the modulus of z is drawn in the Graphics panel of the first window. The values of the polynomial p on the points of $C_r$ form a curve that is represented in the other graphical window.

We will call original window to the graphical panel of the first window and image window to the second graphical window.

There is a tool bar with two buttons. The single arrow button allows to select and dragg objects. The arrow cross icon enables the mouse to move the graphics views or to zoom them by using the mouse wheel.