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DINI'S SURFACE

Surface studied by Dini in 1865.
Ulisse Dini (1845-1918): Italian mathematician. Other name: pseudospherical helicoid. |

Cartesian parametrization: (; cf. other parametrizations from that of the tractrix).
First fundamental quadratic form: . Total 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 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. |

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© Robert FERRÉOL
2017