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 
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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