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# Serie XIII, Nr 7

## Ruled surface with a triple line and two constant tangential planes along 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:
u_{3}(*x*,*y*) + *z*v_{3}(*x,y*) + u_{4}(*x,y*) = 0,
where u_{3} and v_{3} are homogeneus functions of degree three and u_{4} is a homogeneus function of
degree four.
The functions u_{3} and v_{3} have two of the same factors.
The surface contains a triple line, and has two tangent planes that are constant along the line.
The double line is marked with a blue string and on it are two 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 dreifachen Geraden und zwei constanten Tangentialebenen längs
derselben.

Ruled surface with a triple line and two constant tangential planes along the line.

**Text on Sticker and translation**

Missing sticker.

**Designer**

Dr. Karl Rohn

Dresden in 1886

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 45

**Dimension**

18-18 cm

**Material**

String

**Literature**

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