Tractrix

A plane transcendental curve whose equation in rectangular Cartesian coordinates has the form

$$ x = \pm a \mathop{\rm ln} \ \frac{a + \sqrt {a ^ {2} - y ^ {2} } }{y } \mp \sqrt {a ^ {2} - y ^ {2} } . $$

The tractrix is symmetric about the origin (see Fig.), the $ x $- axis being an asymptote. The point $ ( 0, a) $ is a cusp with vertical tangent. The length of the arc measured from the point $ x = 0 $ is:

$$ l = a \mathop{\rm ln} { \frac{a}{y} } . $$

The radius of curvature is:

$$ r = a \mathop{\rm cot} { \frac{x}{y} } . $$

The area bounded by the tractrix and its asymptote is:

$$ S = { \frac{\pi a ^ {2} }{2} } . $$

Figure: t093570a

The rotation of the tractrix around the $ x $- axis generates a pseudo-sphere. The length of the tangent, that is, of the segment between the point of tangency $ M $ and the $ x $- axis, is constant. This property enables one to regard the tractrix as the trajectory of the end of a line segment of length $ a $, when the other end moves along the $ x $- axis. The notion of a tractrix generalizes to the case when the end of the segment does not move along a straight line, but along some given curve; the curve obtained in this way is called the trajectory of the given curve.

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