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RATIONAL BICIRCULAR QUARTIC

Polar equation: .
Cartesian equation: . Case depending on whether . |

The rational bicircular quartics are the bicircular
quartics with a real singularity - here, *O*- that is necessarily
unique (the cyclic points are the other two singularities); the quartic
is called *crunodal*, *cuspidal*, or *acnodal*, depending
on whether this singularity is a double point with different tangents,
a cuspidal point, or an isolated point.

Like the rational circular cubics, they have the property of having 4 equivalent remarkable geometrical definitions.

1) They are the pedals
of centred conics (here pedal with respect to *O* of the conic ).

They are crunodal, cuspidal, or acnodal depending on
whether the point *O* is outside, on, or inside the conic.

crunodal case |
cuspidal case |
acnodal case |

Examples: when the conic is a circle, we get the limaçons
of Pascal (including the cardioid)
and when *O* is the centre of the conic, we get the Booth
curves (including the lemniscate
of Bernoulli).

This definition as a pedal implies a definition as the roulette of a conic rolling on an equal conic, and also as a curve of the three-bar linkage in the case of the antiparallelogram.

2) They are the envelopes of the circles with diameters
the ends of which are a fixed point (here *O*) and a point describing
a centred conic.

crunodal case |
cuspidal case |
acnodal case |

3) They are the inverses
of conics with respect to a point that is not on the conic (here, the conic
with equation:
where *p* is the square of the inversion radius).

The quartic is crunodal, cuspidal, or acnodal depending
on whether the conic is a hyperbola, a parabola or an ellipse.

crunodal case |
cuspidal case |
acnodal case |

4) They are the cissoids
of two circles with respect to one of the points of one of these circles,
the first one being the circle with centre
passing by *O* and the second one being the circle with centre
and radius *a*. The quartic is crunodal, cuspidal, or acnodal depending
on whether these circles intersect, are tangent or are disjoint.

crunodal case |
cuspidal case |
acnodal case |

See also, as a special case, the Dürer
conchoid.

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© Robert FERRÉOL 2017