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# Serie XIII, Nr 4

## Ruled surface with two conjugated imaginary double 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 model represents a ruled surface of degree four embedded in **P**^{3}.

The two lines are complex conjugated so the real picture of this surface does not show the
singularities. The equation of the surface, in four homogeneous variables, reads:

(x^{2}+y^{2})^{2} +
(z^{2}+w^{2})^{2} +
2b(xz-yw)^{2}+2c(xw+yz)^{2} = 0
with either *b*<-1 or *c*<-1.
On the surface are two double lines, they are complex conjugated so the real picture of this surface
does not show the singularities.
**Title model and translation**

Regelfläche mit zwei conjugiert imaginärien Doppelgeraden.

Ruled surface with two conjugated imaginary double lines.

**Text on Sticker and translation**

Regelfl. mit 2 conjug. Imag. Doppelger.

Ruled surface with two conjugated imaginary double lines.

**Designer**

Dr. Karl Rohn

Dresden in 1886

**Company**

Martin Schilling in Halle a.S.

**Original price**

Mark 42

**Dimension**

18-18-18 cm

**Material**

String

**Literature**

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