next curve | previous curve | 2D curves | 3D curves | surfaces | fractals | polyhedra |

CURVE OF THE TIGHTROPE WALKER

Curve studied and named by Bérard in 1810. |

The *curve of the tightrope walker* is the locus
of the feet of a tightrope walker walking on a rope attached to a fixed
point at one of its ends, passing by a pulley located at the same height
as the attached end, and tightened by a counterweight attached to the other
end.

If *k* is the ratio of the mass of the counterweight
over the mass of the man, the end is attached at *O* and the pulley
at *A*(*a*, 0), and the axis *Oy* points towards the bottom,
the laws of statics give:

Cartesian equation: ,
with .
Cartesian parametrization: . Portion of a rational quartic with equation . Special cases: a = b: right
strophoid; a = 0: Kappa. |

If k < 1 (the weight of the walker is greater
than that of the counterweight), the curve has a vertical asymptote x
= b = ka: therefore, it is remarkable that the walker does not fall
until they reach a limit abscissa. Otherwise, the curve is an arc joining
O to A. |

View of the complete curves; in bold, the case k
= 1, which is a right
strophoid.
When |

Without a pulley, the curve described by the feet of a tightrope walker is an ellipse. |

Compare with the curve
of the bucket of water.

next curve | previous curve | 2D curves | 3D curves | surfaces | fractals | polyhedra |

© Robert FERRÉOL 2017