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Tractrix

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A tractrix is a curve of pursuit in which the leader travels in a constant direction and the follower maintains a constant distance.

With one unit between leader and follower, the leader on the x-axis, and the follower starting at (0,1), the tractrix is

$x={\rm sech}^{-1}(y)-\sqrt{1-y^2}$

The evolute of a tractrix is a catenary.

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Derivation

Suppose the leader to be at (t,0) and φ to be the angle of the leash to the horizontal. Then the follower is at

1: $(x,\, y) = (t-\cos\,(\phi),\, \sin\,(\phi)) \$

and because the follower faces the leader

2: ${dy \over dx}=-\tan\,(\phi) \$

But, taking the differential of 1,

$(dx,\, dy)=(dt+\sin(\phi)\,d\phi,\ \cos(\phi)\, d\phi) \$

so by 2

${\cos(\phi)\, d\phi \over dt+\sin(\phi)\, d\phi}=-\tan(\phi) \$

$dt + \sin \phi \, d\phi = -{\cos \phi \, d\phi \over \tan \phi}$

$dt = -\left( \sin \phi + {\cos^2 \phi \over \sin \phi} \right) d\phi$

${dt \over d\phi} = -{\sin^2 \phi + \cos^2 \phi \over \sin \phi}$

${dt\over d\phi}=-\csc(\phi)$

$t = \int -\csc \phi \, d\phi = -\ln | \csc \phi - \cot \phi | \$

$e^{-t} = \csc \phi - \cot \phi = \tan\left( {\phi \over 2} \right)$ (due to a half-angle formula)

$\phi=2\arctan(e^{-t}) = -{\rm gd}(t) + {\pi \over 2} \$

where gd is the Gudermannian function.

Having found φ, now find cos φ and sin φ:

$\cos \phi = \cos \left( -{\rm gd}(t) + {\pi \over 2} \right)$

$= \cos(-{\rm gd}(t)) \cos {\pi \over 2} - \sin (-{\rm gd}(t)) \sin {\pi \over 2} = - \sin (-{\rm gd}(t))$

$= \tanh (t) \$

$\sin \phi = \sin \left( -{\rm gd}(t) + {\pi \over 2} \right)$

$= \sin(-{\rm gd}(t)) \cos {\pi \over 2} + \cos (-{\rm gd}(t)) \sin {\pi \over 2} = \cos({\rm gd}(t))$

$= {\rm sech} (t) \$

Putting these results back into 1 gives a parametric form

$(x,\,y)=(t-\tanh(t),\,{\rm sech}(t))\$

which immediately gives the form at top.

The arc length

$s=\int\sqrt{(dx)^2+(dy)^2}=\int|\tanh(t)|\,dt=\sgn(t)\ln(\cosh(t))$

gives a natural parametrization

$(x,y)=({\sgn}(s)({\rm arccosh}(e^{|s|})-\sqrt{1-e^{-2|s|}}),e^{-|s|})$

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External links

Tractrix (http://mathworld.wolfram.com/Tractrix.html) on MathWorld

This mathematics-related article is a stub. You can help Biocrawler by expanding it (http://www.biocrawler.com/w/index.php?title=Tractrix&action=edit).

An illustration would be really nice.zh:曳物线

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