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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 represents a ruled surface with a double line of which only a segment, bounded by 2 cuspidal real points, is the intersection of the two real sheets.
Title model and translation
Fläche dritter Ordnung. Regelfläche, deren Doppelgerade völlig von reellen Flächenteilen umgeben ist.
Surface of order three. Ruled surface, with a double line limited by two cuspidal points at finite distance.
Text on Sticker and translation
Regelfläche 3. Ordnung.
Ruled surface of order three.
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