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# Serie XIII, Nr 10

## Ruled surface with one real double curve of third order and without
Pinchpoints.

**Mathematical description**

A ruled surface is a surface that can be swept out by a moving line 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.
In this case, the surface has order four and it is embedded in **P**^{3}.

The surface consist of just one component and is given by the following equation:
a (*x*^{2} + *z*^{2}) + b *y*^{2} + 2 (c -
a)*xz* = 0,
where b < 0 and (a+b)^{2} < c^{2}.
The surface contains a double curve of order three and has no pinch points.
A pinch point is a singular point such that every neighborhood of the point intersects itself.
Pinch points are also called Whitney singularities or branch points.
**Title model and translation**

Regelfläche mit einer Doppelcurve dritter Ordnung und vier Pinchpoints.

Ruled surface with one real double curve of third order and without Pinchpoints.

**Text on Sticker and translation**

Regelfl. m. einer Doppelcurve 3. Ordn.

Ruled surface of order three with one real double lines.

**Designer**

Dr. Karl Rohn

Dresden in 1886

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 45

**Dimension**

18-18 cm

**Material**

String

**Literature**

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