Dini's surface

DINI'S SURFACE

Surface studied by Dini in 1865.
Ulisse Dini (1845-1918): Italian mathematician.
Other names: Dini's helicoid, pseudospherical helicoid.

Cartesian parametrization:

(

; cf. other parametrizations from that of the tractrix).
First fundamental quadratic form:

.
Gaussian curvature:

Dini's surface is the surface generated by the application of a screw to a tractrix along its asymptote, in other words, the helicoid with generatrix a tractrix and axis the asymptote of this tractrix.

Its most important property is to have constant total curvature, like the pseudosphere (which is generated by the rotation of a tractrix around its asymptote).

Another of its properties is to be the only helicoid the curvature lines of which are meridians (Bianchi theorem, cf. [gray] p. 483).

Opposite, rotation of a half of Dini's surface around its axis.