Cartesian Ovals

A module of Egg Math for Chickscope.

René Descartes, a famous French mathematician and philosopher (perhaps best known for the deduction "I think, therefore I am") invented the Cartesian coordinates that bear his name, and also discovered an interesting way to modify the pins-and-string construction for ellipses to produce more egg-shaped curves. In an ordinary ellipse, there are two fixed points (called the foci) in the midddle of the figure. The sum of the distances from a point on the ellipse to each of the two foci is the same for all points on the ellipse. In a Cartesian oval, there are still two foci. However now the distance from a point to the one focus plus TWICE its distance to the other focus is what remains the same for all points on the curve:

A Cartesian Oval is the figure consisting of all those points for which the sum of the distance to one focus plus twice the distance to a second focus is a constant.
You can draw such curves using pins and string if you modify the usual method for ellipses as follows: instead of joining the ends of the string to make a loop, tie the one end of the string to one of the pins, and attach the other end to the point of your pen or pencil. Loop the string once around the other pin and use the pencil point to pull the string tight. Now drag the pencil around. This is demonstrated by the computer simulation. Actually, the virtual pin-and-string allows variations on Descarte's idea that would be impossible to draw using real pins and string. As far as the computer is concerned, there's nothing special about the number two; it can equally well take the sum of the distance to one focus plus 3 times (or 2.45 times, or..) the distance to the other focus. You can experiment with the various possibilities and decide for yourself which is best for eggs.

In the drawing below, you can

When you drag in the area around the oval, the lengths of the two segments are shown at upper left, and their weighted sum appears below them, colored in red if you're on (or outside) the oval.

If you see this message, it means that your browser doesn't support Java, or that Java is disabled. (In Netscape, see Options -> Network Prefs -> Languages ). If it worked, you'd see something like this:
[Image of Cartesian Oval applet]

Related outside links:

Return to The Shape of an Egg, or the main EggMath page.