# Equation of Tractrix/Parametric 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 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 $\tuple {x, y}$.

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

## Linguistic Note

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

The plural is **tractrices**.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Tractrix: $11.21$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.12$: The Hanging Chain. Pursuit Curves: Example $(2)$