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# Serie XIII, Nr 5

## Ruled surface with self intersection forming a line and with four Pinchpoints on the
line.

**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 two components and is given by the following equation:
a(*x*^{4}+*y*^{4}) +
2b*x*^{2}*y*^{2} +
c(*xw*-*yz*)^{2} = 0,
where b/a<-1.
It has one real double line, where the two sheets intersect themself.
The double line is marked with a blue string and on it are four 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 Selbstberührungsgeraden und vier Pinchpoints auf derselben.

Ruled surface with self intersection forming a line and with four Pinchpoints on the line.

**Text on Sticker and translation**

Regelfl. mit selbstberührungsgerad.

Ruled surface with curve with self intersection.

**Designer**

Dr. Karl Rohn

Dresden in 1886

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 42

**Dimension**

18-18-18 cm

**Material**

String

**Literature**

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