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TRACTORY

Animation by Alain Esculier

Other names: tractrix (but we chose to preserve this
name for the case where the base is linear), (recalcitrant) donkey curve.
Website: The traces of a bike Thérèse Eveilleau's bike |

Differential equation:
where f(x,_{0}y)
= 0 is the equation of the base.
_{0}Cartesian parametrization: where satisfies . |

A curve (C) travelled along by a point *M* is a *tractory*
of a curve (C_{0}) - the *base* - travelled
along by a point *M*_{0} if the length
of the segment line [*M*_{0}*M*]
- the *leash* - remains constant and the line (*M*_{0}*M*)
remains tangent to (C). For example, the back wheels of a car (or better,
of a bike) describe a tractory of the curve described by the front wheels
(forward or reverse gear).

Conversely, the curve (C_{0})
can be derived from the curve (C) by copying a constant length on the tangent
to (C); (C_{0}) is then called *equitangential
curve* of (C).

As opposed to the notion of pursuit
curve, this notion is not kinematic (the tractory does not depend on
the speed of the point *M*); however, in a steady state, the pursuit
curve of a dog running after a hare at the same speed is a tractory of
the hare's trajectory.

Property: the normal at *M*_{0}
to (C_{0}) and the normal at *M* to the
tractory (C) intersect at the centre of curvature of the tractory (hence
the construction of the evolute of the tractory).

Remark: the equitangential curve can be seen as the glissette of a point on a line sliding on the curve.

Examples:

- the tractory of a line, which is the tractrix

- the tractory of a circle

- the tractory of the curve of constant gyration (defined by the fact that the leash forms with the tangent an angle proportional to the curvilinear abscissa).

- See an animation of a tractory of
a sinusoid at http://did.mat.uni-bayreuth.de/geonet/beispiele/verfolgung/Sintraktrix.html

Below, an animated view of an equitangential line (in blue) of a nephroid
(in red); the nephroid is therefore a tractory of this curve.

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© Robert FERRÉOL 2017