Dini's Surface

A surface of constant Negative Curvature obtained by twisting a Pseudosphere and given by the parametric equations

$$x = a\cos u\sin v$$ (1)

$$y = a\sin u\sin v$$ (2)

$$z = a\{\cos v+\ln[\tan({\textstyle{1\over 2}}v)]\}+bu.$$ (3)

The above figure corresponds to $a=1$, $b=0.2$, $u\in [0, 4\pi]$, and $v\in (0, 2]$.

See also Pseudosphere

References

Geometry Center. ``Dini's Surface.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/dini/.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 494-495, 1993.

Nordstrand, T. ``Dini's Surface.'' http://www.uib.no/people/nfytn/dintxt.htm.