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
- <math>x={\rm sech}^{-1}(y)-\sqrt{1-y^2}<math>
The evolute of a tractrix is a catenary.
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: <math>(x,\, y) = (t-\cos\,(\phi),\, \sin\,(\phi)) \ <math>
and because the follower faces the leader
- 2: <math>{dy \over dx}=-\tan\,(\phi) \ <math>
But, taking the differential of 1,
- <math>(dx,\, dy)=(dt+\sin(\phi)\,d\phi,\ \cos(\phi)\, d\phi) \ <math>
so by 2
- <math>{\cos(\phi)\, d\phi \over dt+\sin(\phi)\, d\phi}=-\tan(\phi) \ <math>
- <math> dt + \sin \phi \, d\phi = -{\cos \phi \, d\phi \over \tan \phi} <math>
- <math> dt = -\left( \sin \phi + {\cos^2 \phi \over \sin \phi} \right) d\phi <math>
- <math> {dt \over d\phi} = -{\sin^2 \phi + \cos^2 \phi \over \sin \phi} <math>
- <math>{dt\over d\phi}=-\csc(\phi)<math>
- <math> t = \int -\csc \phi \, d\phi = -\ln | \csc \phi - \cot \phi | \ <math>
- <math> e^{-t} = \csc \phi - \cot \phi = \tan\left( {\phi \over 2} \right) <math> (due to a half-angle formula)
- <math>\phi=2\arctan(e^{-t}) = -{\rm gd}(t) + {\pi \over 2} \ <math>
where gd is the Gudermannian function.
Having found φ, now find cos φ and sin φ:
- <math> \cos \phi = \cos \left( -{\rm gd}(t) + {\pi \over 2} \right) <math>
- <math> = \cos(-{\rm gd}(t)) \cos {\pi \over 2} - \sin (-{\rm gd}(t)) \sin {\pi \over 2} = - \sin (-{\rm gd}(t)) <math>
- <math> = \tanh (t) \ <math>
- <math> \sin \phi = \sin \left( -{\rm gd}(t) + {\pi \over 2} \right) <math>
- <math> = \sin(-{\rm gd}(t)) \cos {\pi \over 2} + \cos (-{\rm gd}(t)) \sin {\pi \over 2} = \cos({\rm gd}(t)) <math>
- <math> = {\rm sech} (t) \ <math>
Putting these results back into 1 gives a parametric form
- <math>(x,\,y)=(t-\tanh(t),\,{\rm sech}(t))\ <math>
which immediately gives the form at top.
The arc length
- <math>s=\int\sqrt{(dx)^2+(dy)^2}=\int|\tanh(t)|\,dt=\sgn(t)\ln(\cosh(t))<math>
gives a natural parametrization
- <math>(x,y)=({\sgn}(s)({\rm arccosh}(e^{|s|})-\sqrt{1-e^{-2|s|}}),e^{-|s|})<math>
See also
- Hyperbolic functions for tanh, sech, csch, arccosh
- Trigonometric function for sin, cos, tan, arccot, csc
- Signum function for sgn
- Natural logarithm for ln
- Calculus
- Cartesian coordinate system
- Pseudosphere (related surface)
External links
Template:Math-stub An illustration would be really nice.zh:曳物线