Equation of Tractrix

Theorem

$x$-Axis Asymptote

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 $y$-axis of a cartesian plane with $T$ at the origin and $P$ therefore at the point $\tuple {0, a}$.

Let $T$ be dragged along the $x$-axis.

Cartesian Form

The equation of the tractrix along which $P$ travels is:

$x = a \map \ln {\dfrac {a \pm \sqrt {a^2 - y^2} } y} \mp \sqrt {a^2 - y^2}$

Parametric Form

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 $y$-axis of a cartesian plane with $T$ at the origin and $P$ therefore at the point $\tuple {0, a}$.

Let $T$ be dragged along the $x$-axis.

Formulation 1

The equation of the tractrix along which $P$ travels can be expressed in parametric form as:

\(\ds x\)

\(=\)

\(\ds a \paren {\ln \cot \dfrac \theta 2 - \cos \theta}\)

\(\ds y\)

\(=\)

\(\ds a \sin \theta\)

Formulation 2

The equation of the tractrix along which $P$ travels can be expressed in parametric form as:

\(\ds x\)

\(=\)

\(\ds a \paren {u - \tanh u}\)

\(\ds y\)

\(=\)

\(\ds a \sech u\)

$y$-Axis Asymptote

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.

Cartesian Form

The equation of the tractrix along which $P$ travels is:

$y = a \map \ln {\dfrac {a \pm \sqrt {a^2 - x^2} } x} \mp \sqrt {a^2 - x^2}$

Parametric Form

The equation of the tractrix along which $P$ travels can be expressed in parametric form as:

\(\ds x\)

\(=\)

\(\ds a \sin \theta\)

\(\ds y\)

\(=\)

\(\ds a \paren {\ln \cot \dfrac \theta 2 - \cos \theta}\)

Linguistic Note

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

The plural is tractrices.