# Serie XIII, Nr 3

## Ruled surface with two real double lines and four Pinchpoints on both lines.

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 P3.
The surface consist of two sheets and is given by the following equation:
```
a(x2z2-y2w2)-b(x2w^2-y2z2)+2cxyzw.
```
It has two real double lines, where the two sheets intersect each other. The double lines are marked with a blue string and cross each other perpendicular. There are four real pinch points on one of the double lines. One the other are four imaginary 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 zwei reellen Doppelgeraden und vier Pinchpoints auf einer derselben.
Ruled surface with two real double lines and four Pinchpoints on both lines.

Text on Sticker and translation
Regelfl. mit 2 reellen doppelgeraden.
Ruled surface with two real double lines.

Designer
Dr. Karl Rohn
Dresden in 1886

Company
Martin Schilling in Halle a.S.

Original price
Mark 45

Dimension
18-18-18 cm

Material
String

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