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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