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Serie XVII, Nr 3a

Surfaces of constant positive curvature. The elliptic type.


Mathematical description
This model is an example of surface of positive constant curvature. In this case, the surface is a surface of Enneper of elliptic type. The Enneper surface is a minimal surface that intersects itself. A minimal surface is a surface with zero mean curvature. The mean curvature H of a surface is defined as:
H = k1+k2
2
where k1 and k2 are the principal curvatures of the surface (i.e., the maximum and minimum of the normal curvature K at a given point of a surface).
The surface is given by the following parametrization:

                    x = r cos(φ)
                    y = r sin(φ)
                    z = (ln(tan(0.5v)) + 4a cos(v)) / √(3),
where φ, a and r are given by:
                    φ = -0.5u + arctan(2tan(u))
                    a = 2 / (4 - 3sin2(v)cos2(u))
                    r = a √(4 + 12sin2(u)) sin(v) / √(3)
The parameter u in the interval |u|<π/2 and the parameter v in 0<v<π. This gives a quarter of the model.

Title model and translation
Flächen von constantem positiven Krümmungsmass. Der elliptische Typus.
Surfaces of constant positive curvature. The elliptic type.

Text on Sticker and translation
Fläche von const. Posit. Krümm.-Mass m. einem System ebener Krümm.-Lin. I. Elliptischer Typus.
Surface of constant positive curvature with a system. I. Elliptic type.

Designer
Dr. Sievert
Nürnberg in 1886

Company
Martin Schilling in Halle a.S.

Original price
Mark 21 (two pieces)

Dimension
16-16-8 cm

Material
Plaster

Literature
Schilling, M., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911