Cartesian parametrization the coordinate lines of which are the curvature lines: . Another parametrization: . Transcendental surface. Total curvature: .

The Kuen surface is the surface with the above parametrization.
It comes from the pseudosphere by the "Bianchi transform", transformation that preserves the total curvature, hence the constant total curvature of this surface.

For surfaces with positive constant total curvature, see Sievert's surface (1886), and Rembs' surface (1929).

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© Robert FERRÉOL  2017