Tractroid

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)