Focal Points, Ellipses and Ovals

Interactive three-ellipse and three-pins-and-a-string blob

• Move the pins.
• l-key => longer string
• s-key => shorter string

three-pins-and-a-string-blob ↑
3-ellipse ↓

Using a pin and a string one can draw a circle.

Using two pins and a string one can draw an ellipse.

Using three pins and a string one can draw a three-pins-and-a-string-blob. The blob is a curve made from six elliptical arcs. A three-pins-and-a-string-blob is shown in the first canvas.

Another way to go from two to three is to consider the focal points. An ellipse has two focal points. For any point on the ellipse the sum of the distances to the two focal points is constant.

A 3-ellipse has three focal points. For any point on the 3-ellipse the sum of the distances to the three focal points is constant. A 3-ellipse is shown in the second canvas.

A circle can be seen as a 1-ellipse. It is possible to make a n-ellipse for any positive integer n.

Interactive tree foci variant of Cassini oval

A Cassini oval is a curve defined by two focal points, just as an ellipse is. For all points on an ellipse, the sum of distances to the focal points is constant. For a Cassini oval, on the other hand, the product of distances to the focal points is constant. In the canvas above the curves are defined by three focal points.

Animated gifs

3-ellipse on tumblr.

Cassini swing on tumblr.