# Tractrix

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

## formulas

Cartesian Coordinates (en)

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

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

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

Parametric equations (en)

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

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

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

(all formulas from w:de:Traktrix)

## derivation

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

## General Tractrix

• ${\displaystyle \mathrm {curve} \,k:A_{0}\in k}$
• ${\displaystyle \mathrm {point} \,P_{0}}$
• ${\displaystyle \mathrm {parameter} \,t}$
• ${\displaystyle d={\overline {A_{0}P_{0}}}}$
• ${\displaystyle {\mathbf {A}}(t):A(0)=A_{0}}$
• ${\displaystyle {{\mathbf {P}}(t)}:P(0)=P_{0}}$
${\displaystyle {\mathbf {A}}(t)={{\mathbf {P}}(t)}+d\cdot {\frac {{\mathbf {\dot {P}}}(t)}{|{\mathbf {\dot {P}}}(t)|}}}$
${\displaystyle t:{{\mathbf {\dot {P}}}(t)}\neq 0}$