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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.
a(x2z2-y2w2)-b(x2w^2-y2z2)+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.
Dr. Karl Rohn
Dresden in 1886
Martin Schilling in Halle a.S.
Schilling, M., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911