# Tractrix

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Tractrix is the curve along which a small object moves when pulled with a pole by a puller.

## formulas

Cartesian Coordinates (en)

$y(x) = \pm\ d \cdot \operatorname{arcosh} {d \over x} \mp \sqrt{d^2 -x^2}$

$y(x) = \pm\ d \cdot \ln \left| {d +\sqrt{d^2-x^2} \over x} \right| \mp \sqrt{d^2-x^2}$

$x(y) = \pm\ d \cdot \left( \operatorname{arcosh} {d \over y} -\sqrt{1-\left( {y \over d} \right)^2} \right)$

Parametric equations (en)

$x(t) = \pm\ d \cdot ( t - \tanh t );$   $\quad y(t) = d \cdot \operatorname{sech} t$
$t = \operatorname{arcosh} {d \over y}$

$x(\omega) = \pm\ d \cdot \left( \cos \omega + \ln \tan {\omega \over 2} \right);$   $y(\omega) = d \cdot \sin \omega$
$\omega,\ \sin\omega = {y \over d}\ ,\ 0 \le \omega \le {\pi \over 2}$

$x(\lambda) = \pm\ d \cdot \left( {{\lambda^2 -1} \over {\lambda^2 +1}} - \ln \lambda \right);$   $y(\lambda) = 2d \cdot {{\lambda} \over {\lambda^2 +1}}$
$\lambda = \tan {\omega \over 2}$

(all formulas from w:de:Traktrix)

## derivation

$y' = {\mathrm d y \over \mathrm d x}$$= \pm\ {y \over {A -x}}$$= \pm\ {y \over {\sqrt {d^2-y^2} }}$

## General Tractrix

• $\mathrm{curve}\, k: A_0 \in k$
• $\mathrm{point}\, P_0$
• $\mathrm{parameter}\, t$
• $d = \overline {A_0P_0}$
• $\bold A(t): A(0) = A_0$
• ${\bold P(t)}: P(0) = P_0$
$\bold A(t) = {\bold P(t)} +d \cdot \frac{ {\bold {\dot P}(t)} }{ {| {\bold {\dot P}}(t) |} }$
$t: {\bold{\dot P}(t)} \neq 0$