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 k goes to infinity, the curve approaches the Kappa.

 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.

© Robert FERRÉOL  2017