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RATIONAL CIRCULAR CUBIC

Other names: elongated cissoid or hypercissoid (crunodal case), shrunk cissoid or hypocissoid (acnodal case). |

Polar equation: .
Cartesian equation: , i.e., in the case of a right cubic ( b = 0): .
Cartesian parametrization: . case depending on whether . |

The rational circular cubics are the circular
cubics that have a real singularity - here, *O*, which is necessarily
unique. The cubic 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.

The rational circular cubics have the property of being defined by four equivalent remarkable geometrical definitions.

1) They are exactly the cissoids
of a circle and a line with respect to a point *O* on the circle (here,
cissoid of the circle (*C*) passing through *O* with centre *A*(*a*,*b*)
and of the line (*D*):
with respect to *O*).

They are called *right* cubics if the diameter passing
by *O* is perpendicular to the line (here, if *b* = 0).

These cubics, for which (*D*) is an asymptote and
*O*
a singularity, are *crunodal*, *cuspidal*, or *acnodal*
depending on whether the line (*D'*), symmetrical image of (*D*)
about *O*, is *secant*, *tangent*, or *outside* the
circle (*C*).

Cissoidal construction, in the crunodal case: in dotted blue, the circle ( C) and the lines (D) and
(D'),
in blue, their homothetic image of ratio 2, for which the cissoid is the median curve. |
Cissoidal construction, in the acnodal case. |

When the line (*D'*) is tangent to the circle (),
we get the cissoids;

when it passes through the centre of the circle (*d
*=
–*a)*, we get the strophoids;

when it passes through the point diametrically opposed
to *O *(*d *= –2*a)*, we get the ophiurides;

when it is the perpendicular bisector of the radial segment
passing by *O*(*a = –*2*d, b = 0*), we get the Maclaurin
trisectrix;

when the line (*D*) is tangent to the circle at
the point diametrically opposed to *O *(*d = *2*a*), we
get the visiera;

In the right and acnodal case, we get the cubics
of Sluze.

2) They are the pedals
of parabolas. More precisely, the cissoid of a circle (*C*) and a
line (*D*) with pole *O* on the circle is the pedal with respect
to *O* of the parabola with focus *F, *the point diametrically
opposed to *O* (here, *F*(2*a,*2*b*)), for which the
tangent at the vertex (here, *x* = 2*a* + *d*) is the translated
image of (*D*) by the vector ;
here, this parabola's equation is .

These pedals are crunodal, cuspidal, or acnodal depending
on whether the point *O* is outside the parabola, on it, or inside
it.

Crunodal case: pedal of a parabola with a pole O extéoutside
of it. In dotted line, the initial curve, image of the previous parabola
by the homothety with centre O and ratio 1/2. |
Acnodal case: pedal of a parabola with a pole |

3) Therefore, they are the envelopes
of circles with a diameter whose extremities are a fixed point (here, *O*)
and a point describing a parabola (the previous parabola). In other words,
they are the cyclic curves with initial
curve a parabola, and a power of inversion equal to zero (figure above).

4) They are the inverses
of conics with respect to a point on the conic (here, inverse of the conic ,
where *r* is the inversion radius, with inversion centre *O*).

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

Crunodal case: inverse of a hyperbola |
Acnodal case: inverse of an ellipse |

See other definitions in the right
case.

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© Robert FERRÉOL 2017