# 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 $\tuple {x, y}$.

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:

 $\displaystyle \int \rd y$ $=$ $\displaystyle \int \frac {\sqrt {a^2 - x^2} } x \rd x$ Separation of Variables $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle -a \, \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + \sqrt {a^2 - x^2} + C$ Primitive of $\dfrac {\sqrt {a^2 - x^2} } x$ $\displaystyle$ $=$ $\displaystyle a \, \map \ln {\frac {a - \sqrt {a^2 - x^2} } x} + \sqrt {a^2 - x^2} + C$ after algebra

Taking the negative square root:

 $\displaystyle y$ $=$ $\displaystyle -a \, \map \ln {\frac {a - \sqrt {a^2 - x^2} } x} - \sqrt {a^2 - x^2} + C$ $\displaystyle$ $=$ $\displaystyle a \paren {\map \ln {a + \sqrt {a^2 - x^2} } - \ln x} - \sqrt {a^2 - x^2} + C$ after algebra

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

Thus:

 $\displaystyle 0$ $=$ $\displaystyle a \, \map \ln {\frac {a + \sqrt {a^2 - a^2} } a} - \sqrt {a^2 - a^2} + C$ $\displaystyle \leadsto \ \$ $\displaystyle C$ $=$ $\displaystyle -a \map \ln {\frac a a}$ $\displaystyle \leadsto \ \$ $\displaystyle C$ $=$ $\displaystyle -a \ln 1$ $\displaystyle$ $=$ $\displaystyle 0$

Hence the result.

$\blacksquare$

## Linguistic Note

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

The plural is tractrices.