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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 P3.
The surface consist of two components and is given by the following equation:
u3(x,y) + zv3(x,y) + u4(x,y) = 0,
where u3 and v3 are homogeneus functions of degree three and u4 is a homogeneus function of degree four. The functions u3 and v3 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