## Cartesian Ovals

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

- Click and drag the pins (green and yellow dots), or
- Click and drag around the oval, or
- Look at other members of the family of Descartes' ovals by
adjusting the slider at the bottom.
The distance from the left pin plus a multiple
**p** --
given by the slider -- of the distance to the right pin is a constant
(here, **1+p**);

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.

#### Related outside links:

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