Cassini Ovals and Other Curves

Fix two points and in the plane and consider the locus of a point so that the sum of the distances from to and equals some constant. For different arithmetic operations (sum, difference, quotient, or product), this set takes on different shapes.

If the sum of the distances is constant, the shape is an ellipse with foci at the two points.

If the difference of the distances is constant, the shape is a hyperbola with foci at the two points.

If the product of the distances is constant, the resulting family of curves are called Cassini ovals.

If the quotient of the distances is constant, the resulting curves are circles.

The slider controls a value that is proportional to the square of the constant used to determine the black curve.