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ANALLAGMATIC SPIRAL

Curve studied and named by C. Masurel in 2013. |

Polar equation:
( n > 0) i.e. .
Polar parametrization, for : . Polar tangential angle: . Curvilinear abscissa: for , for n = 1 (coming from ). |
n = 2. |

The anallagmatic spirals are the curves with the above polar equation.

As indicated by their name, and as proven by their equation, they are invariable under inversion (with pole *O* and square of the radius of inversion equal to *a*).

The branch outside the reference circle has an asymptote: the Archimedean spiral of index 1/n: |
and the inside branch has an asymptote: the Archimedean spiral of index -1/n: |

The anallagmatic spirals are the "wheel" associated to the linear pursuit curves (see wheel-road couple).
More precisely, if an anallagmatic spiral with parameter n rolls, like in the opposite figure, on the pursuit curve with parameter n (= speed of the dog / speed of the master), then the pole of the spiral describes the asymptote (case n1) or the tangent at the vertex (case n < 1) of the pursuit curve. |

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© Robert FERRÉOL
2017