Tractrix

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

formulas[edit]

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)

graphs y=y(x)[edit]

Tractrice.png
Tractrice1.png
Tractrice2.png
Tractrix1.png
TractrizFig1.png
TractrizFig2b.png
TractrizFig3.png


graphs x=x(y)[edit]

Tractrix.png


derivation[edit]

Tractrix-sketch-lettered-colour.png
 y' = {\mathrm d y \over \mathrm d x}  = \pm\ {y \over {A -x}}  = \pm\ {y \over {\sqrt {d^2-y^2} }}


General Tractrix[edit]

  • \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