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# Serie VII, Nr 23

## Surface of order three. Cayley's ruled surface with a cuspidal point (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 the Cayley's ruled surface. It is a cubic ruled surface in which the director line
meets the director curve. The Cayley's surface is the unique cubic surface having four ordinary double
points, the maximum possible for a cubic surface.
**Title model and translation**

Fläche dritter Ordnung. Cayley'sche Regelfläche mit (unendlich fernen im Endlichen gelegenen)
Cuspidalpunkte.

Surface of order three. Cayley's ruled surface with a cuspidal point (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 16,50

**Dimension**

13-15 cm

**Material**

Plaster

**Literature**

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