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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

Bicircular quartic.

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.
 
 
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© Robert FERRÉOL  2017