Effects with Cassini Ellipses

Standard Ellipse

By: Arsham Skrenes

There are a number of ways of describing an ellipse mathematically:

Cartesian equation:

Parametric form:

Gauss-mapped form:

This is probably due to the fact that ellipses have a number of interesting properties. One such example is the fact that they can have two foci. I use this feature to demonstrate an interesting way of constructing an ellipse. To begin, you simply hammer in two nails into a single board. You then attach the ends of a single length of string to each nail. Finally, pushing outwards against the string with a pen you will draw a perfect ellipse with the foci at the nails. To illustrate this, click on the following graphic:

In this demonstration you will notice that the string stayed at a fixed length (you can visually see this by looking at the colors of the string or reading the numerical length above the diagram). This is because a standard ellipse has the following property:

All points on the ellipse represent the SUM of the distances from the two fixed points (the foci).