Equation of Tractrix/Parametric 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 can be expressed in parametric form as:
 * $x = a \sin \theta$
 * $y = a \paren {\ln \cot \dfrac \theta 2 - \cos \theta}$

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


 * [[File:Tractrix.png]]

Consider the upper part of the tractrix.

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

Let the angle formed by cord $S$ and the $y$-axis be $\theta$.

Then $x = a \sin \theta$.

Substituting this into the Cartesian Form of the equation of the tractix: