Tractrix (from the Latin verb trahere ‘pull, drag’) is the curve along which a small object (tractens) moves when pulled on a horizontal plane with a piece of thread by a puller (tractendus) which moves rectilinearly.
Let the object initially be in the -plane on the -axis in the point and the puller in the origin; is the of the pulling thread. Then the puller begins to move along the -axis in the positive direction. The object follows drawing the path curve so that the line determined by the thread is at every the tangent of the curve. This condition gives in the point the differential equation
with the initial condition . The solution is
Here the minus alternative is for the case that the puller moves in the negative direction from the origin. In fact, both branches, corresponding to both signs, belong to the tractrix. The branches meet in the cusp point .
The substitution gives for the tractrix the parametric
Another one is
where and are the hyperbolic functions cosinus hyperbolicus and tangens hyperbolica.
It is obvious that the line, on which the puller goes, is the asymptote of the tractrix. The curve thus has the property that its tangent, between the asymptote and the point of tangency, has the ().
The differential equation of the orthogonal curves of the tractrix is
whence they are the circles .
The area between the tractrix and its asymptote is . This may be calculated ordinarily as
integrating by parts and using the area of a quarter-circle yield
(see this entry (http://planetmath.org/GrowthOfExponentialFunction) for ). Another way to determine is differential-geometric: as the object draws the tractrix from above to down, the thread turns and thus sweeps an area equal to a half-circle.
The envelope of the normal lines of the tractrix, i.e. the evolute of the tractrix is the catenary (or “chain curve”) .
|Date of creation||2013-03-22 15:18:32|
|Last modified on||2013-03-22 15:18:32|
|Last modified by||pahio (2872)|