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The Surface of Revolution produced by revolving the Tractrix

$\displaystyle x$ $\textstyle =$ $\displaystyle \mathop{\rm sech}\nolimits u$ (1)
$\displaystyle z$ $\textstyle =$ $\displaystyle u-\tanh u$ (2)

about the z-Axis is a tractroid given by
$\displaystyle x$ $\textstyle =$ $\displaystyle \mathop{\rm sech}\nolimits u\cos v$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle \mathop{\rm sech}\nolimits u\sin v$ (4)
$\displaystyle z$ $\textstyle =$ $\displaystyle u-\tanh u.$ (5)

See also Pseudosphere, Surface of Revolution, Tractrix

© 1996-9 Eric W. Weisstein