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RATIONAL SEXTIC
The rational
sextics are the sextics of genus
zero, therefore with between one or ten singularities making the genus
decrease by 10 (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 6.
Replacing t by , we get a trigonometric parametrization: . 
Most of the remarkable rational sextics are part of the family of the rational curves of degree 6 that are bounded, with Oy as a symmetry axis, and parametrization , as it will be noted thereafter.
Examples of rational sextics:
 the quadrifolium
and its conchoids:
(a = 0: quadrifolium, a = 1 : Ceva
trisectrix , a = 2: double
egg: a > 2: peanut
curve)
 the Dürer
folium and its conchoids:
(a = 0: Dürer folium, a = 1: Freeth's
nephroid )
 the epitrochoids
with parameter q = 2:
(a = 1 : Dürer
folium, a = 3: nephroid).
 the hypotrochoids
with parameter q = 4:
(a = 1: quadrifolium, a
= 3: astroid).
 the epitrochoids
with parameter q = 1/2: .
 the hypotrochoids
with parameter q = 5/2 (5branch star):
.
 the cornoid:
 the Maltese
cross:
 the double
drop of water:
 the Cayley
sextic:
 the Lissajous
curve:
 the windmill:
 the bow
tie curve:
 the beetle
curves: (including
the quadrifolium: )
 the Talbot
curves: .
In order of apparition: a = 0: quadrifolium a = pi/4: Maltese cross a= pi/2: double drop of water a = 3pi/4: double egg a = piarctan(1/2): Dürer folium a= piarctan(1/3): curve similar to the cornoid This curves are the projections on the planes containing Oy of the 3D curve: 
Other examples:
A beautiful sextic with a triple point where the tangents coincide, which belongs to the family of the basins:  
Another one, which cannot be decomposed, though it seems to be composed of a circle and an eight: . 

This third one also seems to be composed of a circle
and a cardioid:
. This curve is associated to the triangles an altitude, a median, and a bisector of which are concurrent (triangles considered by E. LEMOINE in 1885 in mathesis): if the vertex of the altitude is fixed on (0,1) and that of the median on (0,0), then this curve is the locus of the third vertex of such a triangle. The locus of the intersection point is a septic. 
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© Robert FERRÉOL 2017