Tractrix

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

Contents

[edit] 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)

[edit] graphs y=y(x)

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


[edit] graphs x=x(y)

Tractrix.png


[edit] derivation

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} }}


[edit] 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
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