Equation of Tractrix/Cartesian 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 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:
\(\ds \int \rd y\) | \(=\) | \(\ds \int \frac {\sqrt {a^2 - x^2} } x \rd x\) | Solution to Separable Differential Equation | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + \sqrt {a^2 - x^2} + C\) | Primitive of $\dfrac {\sqrt {a^2 - x^2} } x$ | ||||||||||
\(\ds \) | \(=\) | \(\ds a \map \ln {\frac {a - \sqrt {a^2 - x^2} } x} + \sqrt {a^2 - x^2} + C\) | after algebra |
Taking the negative square root:
\(\ds y\) | \(=\) | \(\ds -a \map \ln {\frac {a - \sqrt {a^2 - x^2} } x} - \sqrt {a^2 - x^2} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a \paren {\map \ln {a + \sqrt {a^2 - x^2} } - \ln x} - \sqrt {a^2 - x^2} + C\) | after algebra |
When $y = 0$ we have $x = a$.
Thus:
\(\ds 0\) | \(=\) | \(\ds a \map \ln {\frac {a + \sqrt {a^2 - a^2} } a} - \sqrt {a^2 - a^2} + C\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds -a \map \ln {\frac a a}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds -a \ln 1\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
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 $(2)$