Equation of Tractrix

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Theorem

Let $S$ be a cord of length $a$ situated as a (straight) line segment whose endpoints are $P$ and $T$.

Let $S$ be aligned along the $x$-axis of a cartesian plane with $T$ at the origin and $P$ therefore at the point $\tuple {a, 0}$.

Let $T$ be dragged along the $y$-axis.


Tractrix.png


Cartesian Form

The equation of the tractrix along which $P$ travels is:

$y = a \, \map \ln {\dfrac {a \pm \sqrt {a^2 - x^2} } x} \mp \sqrt {a^2 - x^2}$


Parametric Form

The equation of the tractrix along which $P$ travels can be expressed in parametric form as:

$x = a \sin \theta$
$y = a \paren {\ln \cot \dfrac \theta 2 - \cos \theta}$