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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 (G0) (the initial curve), a centre O and a power p is the envelope of the circles (C) the centres of which describe (G0) 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) (C0); 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 (T0) to (G0) at (M0) (N therefore describes the pedal of (G0) with respect to O).
The two characteristic points M and M' of the circle (C) centred on (M0) are the intersection points between (C) and (ON). Therefore, they are defined by

and are real iif ON2³ p. We deduce from this the polar equation given in the header section.

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 (G0) 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