RATIONAL QUARTIC

The rational quartics are the quartics with genus zero, in other words, with one, two, or three singularities, that decrease the genus by 3 (in the complex projective plane).

 Cartesian parametrization: where P, Q and R are three polynomials with real coefficients, the maximum of the degrees of which is 4. Replacing t by , we get a trigonometric parametrization: .

Examples of rational quartics, with their linearised trigonometric parametrization:
- the epitrochoids with parameter q = 1, or limaçons of Pascal (a = 2: cardioid).
- the hypotrochoids with parameter q = 3: (a = 1: regular trifolium , a = 2: deltoid ).
- the fish curves and the ovoid quartics .
- the piriform quartic .
- the folia (including the bifolia and the trifolia)

- the Lissajous curve , the basin .

- the besaces (b = 0: lemniscate of Gerono, and its biaxial inverse)
- the cross curve and the double u - the bullet nose curve and the Külp quartic (and, more generally, the Granville egg)
- the kappa .
- the Delanges trisectrix .
- the lemniscate of Bernoulli - the bicorn - the parabolic trifolium: - the Kampyle of Eudoxus - the Jerabek curve and, more generally, the focal conchoids of conics.
- the conchoid of Nicomedes - the scyphoid - the curve of the tightrope walker .
- the Alain curves.
- the Rosillo curves.
- the kieroids.
- the ramphoid.