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CIRCULAR CUBIC

Reduced Cartesian equation: (with (0,1,0) as a point at infinity, i.e. a vertical asymptotic direction) |

The circular
cubics are the cubics passing by cyclic
points. It can be proved that they are the cyclic
curves for which the initial curve is a parabola. In other words, they
are the envelopes of circles the centres of which describe a parabola,
and such that a fixed point (the pole) has a constant power with respect
to these circles (i.e. these circles have an orthogonal or pseudo-orthogonal
intersection with a fixed circle, the directrix circle).

They are rational iff this power is zero.

A circular cubic can be obtained by four definitions of
this type in the general case. It loses one if it has a symmetry axis.
Independently, it loses two or three of these definitions when it is rational
(depending on the kind of singularixty).

The poles are the points of the curve where the tangent
is parallel to the asymptote.

Examples:

- the circular hyperbolic
cubic is
the envelope of the circles
where .
The initial curve is
and the directrix circle .

- the circular rational cubics.

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© Robert FERRÉOL 2017