# Equation of Tractrix/Cartesian Form

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## Contents

## 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**.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.12$: The Hanging Chain. Pursuit Curves: Example $(2)$