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Half the Surface of Revolution generated by a Tractrix about its Asymptote to form a Tractroid. The Cartesian parametric equations are

$\displaystyle x$ $\textstyle =$ $\displaystyle \mathop{\rm sech}\nolimits u\cos v$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle \mathop{\rm sech}\nolimits u\sin v$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle u-\tanh u$ (3)

for $u\geq 0$.

It has constant Negative Curvature, and so is called a pseudosphere by analogy with the Sphere, which has constant Positive curvature. An equation for the Geodesics is

\cosh^2 u+(v+c)^2=k^2.
\end{displaymath} (4)

See also Funnel, Gabriel's Horn, Tractrix


Fischer, G. (Ed.). Plate 82 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 77, 1986.

Geometry Center. ``The Pseudosphere.''

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 383-384, 1993.

© 1996-9 Eric W. Weisstein