EggMath: The Shape of an Egg
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 pinsandstring
construction for ellipses to produce more
eggshaped 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 pinandstring
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.
FOR FURTHER
READING...
 Page 6 of 15 
