CARTESIAN CURVE

 René Descartes (1596-1650): French philosopher, mathematician and physicist.

 Reduced Cartesian equation: . If  then the equation can be written: and a cyclic generation stems from the circle with centre O and radius R as the initial curve, the circle with centre A(a,0) and radius  as inversion circle, and inversion power p.  In the frame (A,  , ): Cartesian equation:  . Polar equation: .

The Cartesian curves are the bicircular quartics with two cusps at infinity.
They are the curves that can be defined as cyclic curves with a circle as the initial curve (called initial circle). In other words, they are the envelopes of the circles whose centres describe the initial circle and such that a fixed point has a constant power with respect to these circles.
When p < 0, we get the genuine complete Cartesian oval (???).

Different cases when = radius of the inversion circle:
- when the initial circle is inside the inversion circle (), the curve does not have real points.
- when the inversion circle is a singleton and is inside or outside the initial circle ( or  and ) we get, twice, the genuine complete Cartesian ovals again.
- when the inversion and initial circles intersect at two distinct points B and C (), we get the special Cartesian curves, that have a tripolar equation:  with ??. These curves are inverses of Cartesian ovals.
- when the inversion circle is a singleton or is tangent to the initial circle ( or  r = 0), we get a limaçon of Pascal.

To sum up, the family of Cartesian curves is composed of complete Cartesian ovals and their inverses.