## Cassini Ovals

#### A module of Egg Math for Chickscope.

Giovanni Cassini, an astronomer who discovered the moons of Saturn, was interested in oval curves as descriptions of orbits. Like the Cartesian Ovals, the ovals of Cassini are based on a modification of the pins-and-string construction for ellipses and produce more egg-shaped curves. The Cassini ovals are defined by the condition that for all points on the curve, the product of the distances to two fixed points (foci) is a constant:

 An Oval of Cassini is the figure consisting of all those points for which the product of their distances to two fixed points (called the foci) is a constant.

Unfortunately, we don't know of any way to modify a real life pin-and- string device to draw these ovals of Cassini. But what can't be done in the real world can easily be done in the virtual world! Our computer simulation of the pin-and-string device has no trouble with Cassini's definition.

In this drawing, 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 Cassini's ovals by adjusting the slider at the bottom.
• The product of the distance from the left pin times the distance to the right pin is given by the slider. This can yield one convex curve, or a curve with a pinch in the middle, or two curves, depending on the distance-product and the separation between the pins.
When you drag in the area around the oval, the lengths of the two segments are shown at upper left, and their product 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: