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Serie XIII, Nr 10

Ruled surface with one real double curve of third order 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 P3.
The surface consist of just one component and is given by the following equation:
a (x2 + z2) + b y2 + 2 (c - a)xz = 0,
where b < 0 and (a+b)2 < c2. The surface contains a double curve of order three and has no 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 Doppelcurve dritter Ordnung und vier Pinchpoints.
Ruled surface with one real double curve of third order and without Pinchpoints.

Text on Sticker and translation
Regelfl. m. einer Doppelcurve 3. Ordn.
Ruled surface of order three with one real double lines.

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