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