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Serie IX, Nr 9

Surface of order four with a double line.


Mathematical description
This model represents a cross-cap, the self-intersection of a one sided surface. It is a Dupin cyclide with a double curve, the self-intersection of the surface with two different tangential planes. It is given by the following polynomial on the real projective plane:

f(x,y,z)=(xz,yz,0.5(z^2-x^2)).
In three dimensional space the surface is given with its spherical coördinates:
                    x(u,v)=cos(u)sin(2v)
                    y(u,v)=sin(u)sin(2v)
                    z(u,v)=cos2(v)-cos2(u)sin2(v),
for u∈[0,2π) and v∈[0,π/2].

Title model and translation
Fläche vierter Ordnung mit einer Doppelgeraden.
Surface of order four with a double line.

Text on Sticker and translation
Fl. 4. Ord. mit 1 doppelgeraden.
Surface of order four with one double line.

Designer
Dr. Kummer
Berlin in 1883

Company
L. Brill in Darmstadt

Original price
Mark 7

Dimension
9-3 cm

Material
Plaster

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