Pseudosphere

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

$$x = \mathop{\rm sech}\nolimits u\cos v$$ (1)

$$y = \mathop{\rm sech}\nolimits u\sin v$$ (2)

$$z = u-\tanh u$$ (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

$$\cosh^2 u+(v+c)^2=k^2.$$ (4)

See also Funnel, Gabriel's Horn, Tractrix

References

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

Geometry Center. ``The Pseudosphere.'' http://www.geom.umn.edu/zoo/diffgeom/pseudosphere/.

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