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: