Previous
Serie XIII
Next

# 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 **P**^{3}.

The surface consist of two sheets and 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.

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