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CYCLIC CURVE

From the Greek Kuklos: circle, wheel. |

Polar equation:
where
is the pedal of the initial curve, O the centre of inversion and
p
the power.
Condition for the points of the cyclic curve to be real: . |

The *cyclic* curve associated to a curve
(the *initial curve*), a *centre* *O* and a *power p*
is the envelope of the circles (*C*) the centres of which describe
and such that *O* has a constant power *p* with respect to these
circles.

The circle with centre *O* and radius
is then referred to as the *directrix* circle (or circle of *inversion*)
(*C*_{0}); the condition
of having a constant power means:

- in the case where *p* is positive, that
(*C*) remains orthogonal to the directrix circle.

- in the case where *p* = 0, that (*C*)
passes by *O*.

- in the case where *p* is negative, that
(*C*) remains "pseudo-orthogonal" to the directrix circle, in other
words that it cuts it in two diametrically opposed points.

In the case where *p* is positive, the circle (*C*)
is real only for the points on the initial curve located outside the directrix
circle; for the interior points, the portion of cyclic curve obtained only
has imaginary points.

Consider the projection *N* of *O* on the tangent
(*T*_{0}) to
at (*M*_{0}) (*N*
therefore describes the pedal of
with respect to *O*).

The two characteristic points *M* and *M*'
of the circle (*C*) centred on (*M*_{0})
are the intersection points between (*C*) and (*ON*). Therefore,
they are defined by

When *p *is different from 0, the cyclic curve is
anallagmatic,
with inversion centre *O* and power *p*. Conversely, any anallagmatic
curve has as many cyclic generations as it has inversion centres.

When *p* = 0, there is only one characteristic point
*M*
such that *N* is the middle of [*OM*] and the cyclic curve is
none other than the orthotomic
of with
respect to *O*.

The cyclic curves with initial curve a parabola are the
circular
cubics (rational if *p* = 0) and the cyclic curves with initial
curve a centred conic are the bicircular
quartics (rational if *p* = 0).

The analogue of the notion of cyclic curve for surfaces
is that of cyclid.

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© Robert FERRÉOL 2017