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PSEUDOSPIRAL OF PIRONDINI

Curve studied by Puiseux in 1844 and by Pirondini in
1892 and 1905, after whom it is named.
Geminiano Pirondini 1857 - 1914: Italian mathematician. |

Intrinsic equation 1:
with n real number different from -1.
Intrinsic equation 2: . Cartesian parametrization: . Curvilinear abscissa: s = at.
Radius of curvature: . |

The pseudospiral (of Pirondini) of index *n* is the
curve the curvature of which is proportional to the *n*-th power of
the curvilinear abscissa. It is a generalisation of the
clothoid
(case *n* = 1), that also includes the cases of the circle (*n*
= 0), of the logarithmic spiral
(*n* = -1), of the involute
of a circle (*n* = -1/2), and of a limit case of alysoid
(*n* = -2).

It assumes, for *t* > 0, the following shapes:

n > 0 |
-1 < n < 0 |
n < -1 |

An important property, that explains the case of the involute
of a circle, is that the evolute
of the pseudospiral of index *n* is a pseudospiral of index
(intrinsic equation ).

The radial of
the pseudo-spiral of index *n* is the Archimedean
spiral of index , .

Its Mannheim curve
is the curve with Cartesian equation: .

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© Robert FERRÉOL
2017