Cassini oval

The Cassini oval (named after Giovanni Domenico Cassini ) the locus of all points in the plane, for which the product of its distances from two given points and is the same. From Giovanni Domenico Cassini, these curves have also been proposed after the discovery of Kepler 's laws as planetary orbits. A special case of the Cassini curve is the lemniscate.

Equations

The curve can be divided into Cartesian coordinates by the equation

Describe where and was set. In polar coordinates, the equation

Deriving from the definition

The problem was addressed in a rectangular Cartesian coordinate system of the plane, and so that is valid. Then, for a point on the curve according to the definition:

For the transition to polar coordinate transformation is necessary. It arises with the trigonometric Pythagoras:

This is a quartic equation, it is in particular here to the biquadratic special case which can be solved as a quadratic equation in:

Shape of the curve

The shape of the Cassini curve can be divided into five different cases: