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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
In the frame ( |

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 (???).

- 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

- when the inversion circle is a singleton or is tangent to the initial circle (, or

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

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© Robert FERRÉOL 2017