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ANALLAGMATIC CURVE

Notion studied by Moutard in 1864.
Anallagmatic literally means: without change
(from the Greek
allagma "change"). |

General polar equation: with . |

A curve is said to be *anallagmatic* if it is globally
invariant by inversion.

Any anallagmatic curve can be generated as a cyclic curve, i.e. as the envelope of circles, orthogonal to the circle of inversion (in the case of a positive power), the centres of which describe one of the deferential curves of the curve (in the plural form, because we can indeed have several generations of this type there):

In the algebraic case, it is always circular.

Examples of anallagmatic curves:

- the straight line (with respect
to one of its points)

- the circle

- the strophoids
(the circle of inversion is centered at the focus (on the loop) and pass
by the double point, the deferential curve is a parabola passing by the
double point).

- More generally all circular
cubics (the pole of inversion is the point of the cubic where the tangent
is parallel to the real asymptote).

- the Pascal's
snails (with respect to the point of abscissa
on *Ox*).

- the Casinian
and Cartesian ovals.

- more generally the bicircular
quartics and circular ones with double point.

- anallagmatic
spirals.

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© Robert FERRÉOL
2016