Parametric Cartesian equation:
x = 1/cosh(t), y = t - tanh(t)
Click below to see one of the Associated curves.
|Definitions of the Associated curves||Evolute|
|Involute 1||Involute 2|
|Inverse curve wrt origin||Inverse wrt another circle|
|Pedal curve wrt origin||Pedal wrt another point|
|Negative pedal curve wrt origin||Negative pedal wrt another point|
|Caustic wrt horizontal rays||Caustic curve wrt another point|
The study of the tractrix started with the following problem being posed to Leibniz:
What is the path of an object dragged along a horizontal plane by a string of constant length when the end of the string not joined to the object moves along a straight line in the plane?
He solved this using the fact that the axis is an asymptote to the tractrix.
The evolute of a tractrix is a catenary. Among the properties of the tractrix are the fact that the length of a tangent from its point of contact to an asymptote is constant. The area between the tractrix and its asymptote is finite.
When a tractrix is rotated around its asymptote then a pseudosphere results. This is a surface of constant negative curvature and was used by Beltrami in 1868 in his concrete realisation of non-euclidean geometry.
You can see this
surface of revolution.
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