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# Serie XIII, Nr 1

## Ruled surface with two real double lines and four Pinchpoints on each of the 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.

The surface consists of two congruent components and it is given by the following equation:
a(x^{2}z^{2}+y^{2}w^{2}) +
b(x^{2}w^2+y^{2}z^{2}) + 2cxyzw = 0.
It has two real double lines, both belonging to the two components.
The double lines are marked with a blue string.
In total there are 8 real pinch points, four on each double line and also four on each component.
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.
There are four of them lying on each double line.
**Title model and translation**

Regelfläche mit zwei reellen Doppelgeraden und vier Pinchpoints auf jeder derselben.

Ruled surface with two real double lines and four Pinchpoints on each of them.

**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 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