Equation of Tractrix/Parametric Form

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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 $\tuple {x, y}$.


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:

\(\ds y\) \(=\) \(\ds a \, \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} - \sqrt {a^2 - x^2}\)
\(\ds \) \(=\) \(\ds a \, \map \ln {\frac {a + \sqrt {a^2 - \paren {a \sin \theta}^2} } {a \sin \theta} } - \sqrt {a^2 - \paren {a \sin \theta}^2}\)
\(\ds \) \(=\) \(\ds a \, \map \ln {\frac {a + a \cos \theta} {a \sin \theta} } - a \cos \theta\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds a \, \map \ln {\frac 1 {\tan \theta/2} } - a \cos \theta\) Half Angle Formula for Tangent: Corollary $1$
\(\ds \) \(=\) \(\ds a \paren {\ln \cot \frac \theta 2 - \cos \theta}\) Definition of Cotangent

$\blacksquare$


Linguistic Note

The word tractrix derives from the Latin traho (trahere, traxi, tractum) meaning to pull or to drag.

The plural is tractrices.


Sources

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