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When a cubic surface has more than 4 double points must be singular along and entire curve. In this case, the surface is a ruled surface, i.e., it is swept out by a line moving in space. It has a parametrization of the form:
x(u,v) = b(u) + vδ(u),where b is called the director line (or directrix) and δ is the director curve. The model illustrates a ruled surface with a double line which is the complete intersection of the two sheets of the surface. Its cuspidal points are imaginary conjugated.
Title model and translation
Fläche dritter Ordnung. Regelfläche, deren Doppelgerade völlig von reellen Flächenteilen umgeben ist.
A ruled surface of order three, whose double straight line is completely surrounded by a real sheet of the surface.
Text on Sticker and translation
Dr. Carl Rodenberg
Darmstadt in 1881
L. Brill in Darmstadt
Schilling, M., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911