# Complex Cubics

The following four GeoGebra sketches illustrate the effects of complex cubics on simple geometric objects. The complex cubics are of the form f(z)=(z-A)(z-B)(z-C) where A, B, and C are the vertices of a triangle.

As you drag the various shapes around, you will notice the changes in the image curve. Where "kinks" in the curve occur means the pre-image curve is passing through a critical point of the mapping. For a complex cubic, these critical points are the foci of the Stiener Inellipse.

#### Complex Cubic Tool

In all three of these files, there is a complex cubic tool that takes as input A, B, C and z, and as output gives f(z). This makes it EASY to find the images of your own objects under the cubic.

#### Image of a Circle

The first file shows the image of a circle under a complex cubic.

#### Image of the Axis

The second file shows the image of two perpendicular lines under a complex cubic.

#### Image of a Line

The third file shows the image of a line under a complex cubic.

#### Image of the Critical Points

There is alot of stuff in this Critical Points of a Complex Cubic file. In this sketch, the Stiener Inellipse (the inconic tangent to the sides of the triangle at the midpoints) is shown, as are the critical points in purple, and their images in bright green. There is also a blue circle and its image in red. As you drag the blue circle close to the critical points, notice how the image circle is deformed. You should use the complex cubic tool to explore the images of lines and how they react around the critical points. You can also confirm by measurement that the critical points are the foci of the ellipse.