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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 curveand, 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.
See also the rational bicircular quartics and the superposition curves.
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© Robert FERRÉOL  2017