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# Serie VII, Nr 21

## Surface of order three. Ruled surface, with a double line limited by two cuspidal
points at finite distance.

**Mathematical description**

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.

**Designer**

Dr. Carl Rodenberg

Darmstadt in 1881

**Company**

L. Brill in Darmstadt

**Original price**

Mark 12,-

**Dimension**

13-15 cm

**Material**

Plaster

**Literature**

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