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RATIONAL CUBIC


Cartesian parametrization:  where P, Q and R are three polynomials with real coefficients
 of degree less than or equal to 3.
Replacing t by  gives the parametrization .

A cubic is rational if and only if it has a singularity (which is necessarily real, but possibly at infinity).

The cubic is called crunodal, acnodal or cuspidal depending on whether the singularity is a double point, an isolated point, or a cuspidal point.

When the singularity is not at infinity and there is at least one asymptote at finite distance, the curve can be constructed as a cissoid of Zahradnik.
Examples of rational cubics that are not cissoids of Zahradnik:
 - the anguinea, the witch of Agnesi , the curve  and Newton's trident: the singularity is at infinity.
 - the semicubical parabola, the Tschirnhausen cubic, the duplicatrix cubic, and more generally any rational divergent parabola, as well as the parabolic folia: there are no asymptotes.
 
 
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© Robert FERRÉOL  2017