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Serie XVII, Nr 3b

Surfaces of constant positive curvature. The cyclic 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 cyclic type. The model is constructed by Kuen. 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).
A minimal surface is a surface with zero mean curvature. The surface is given by the following parametrization:

                    ρ = 17.33 / √(3) * √(1+3sin2(w) / (4-3sin2(w)) * sin(φ)
                    z = 4.33/√(3) log(tan(0.5 φ)) + 34.66/&radic(3) * cos(φ) / (4 - 3sin2(φ) cos2(w))
                    tan(ψ) = tan(w) / (1 + 2tan2(w)),
with the parameters w and φ.

Title model and translation
Flächen von constantem positiven Krummungsmass. Der cyclische Typus.
Surfaces of constant positive curvature. The cyclic type.

Text on Sticker and translation
Fläche von const. posit. Krüm. mit einem System ebener Krüm. II. Cyclischer Typus.
Surface of constant positive curvature with a system. II. Cyclic type.

Designer
Dr. Sievert
Nürnberg in 1886

Company
Martin Schilling in Halle a.S.

Original price
Mark 21 (two pieces)

Dimension
16-15-14 cm

Material
Plaster

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