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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: |
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 = –2a), we get the ophiurides;
when it is the perpendicular bisector of the radial segment
passing by O(a = –2d, b = 0), we get the Maclaurin
trisectrix;
when the line (D) is tangent to the circle at
the point diametrically opposed to O (d = 2a), 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(2a,2b)), for which the
tangent at the vertex (here, x = 2a + 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 O outside of it. |
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