Equation of Tractrix/Cartesian Form

Definition
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.

The equation of the tractrix along which $P$ travels is:
 * $y = a \, \map \ln {\dfrac {a \pm \sqrt {a^2 - x^2} } x} \mp \sqrt {a^2 - x^2}$

Proof
Consider $P$ when it is at the point $\tuple {x, y}$.


 * [[File:Tractrix.png]]

The cord $S$ is tangent to the locus of $P$.

Thus from Pythagoras's Theorem:
 * $\dfrac {\d y} {\d x} = - \dfrac {\sqrt {a^2 - x^2} } x$

Hence:

Taking the negative square root:

When $y = 0$ we have $x = a$.

Thus:

Hence the result.