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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(x0,y0) = 0 is the equation of the base.
Cartesian parametrization:  where  satisfies .

A curve (C) travelled along by a point M is a tractory of a curve (C0) - the base - travelled along by a point M0 if the length of the segment line [M0M] - the leash - remains constant and the line (M0M) 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 (C0) can be derived from the curve (C) by copying a constant length on the tangent to (C); (C0) 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 M0 to (C0) 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.

when the tractory (C) slides on a line at a fixed point, the glissette of the end of the leash is the base (C0).

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