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# Serie XIII, Nr 2

## Ruled surface with two real double lines 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 consists of two sheets 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,
where *a*>0 and *b*<0.
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 perpendicularly.
There are no real pinch points on this surface.
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 ohne Pinchpoints.

Ruled surface with two real double lines and without Pinchpoints.

**Text on Sticker and translation**

missing sticker.

**Designer**

Dr. Karl Rohn

Dresden in 1886

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 49

**Dimension**

18-18-18 cm

**Material**

String

**Literature**

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