Cayley's Ruled Cubic (1 Pinch Point at Inf.)

This classic from the 19th century one of the three essentially different types of ruled cubic surfaces: no, one, or two real pinch points. The example presented here has first been mentioned by A. Cayley in the 1850's and is the case of one pinch point; in the model shown, however, this pinch point is infinitly far away so that the hole surface is perfectly smooth away from the vertical double line.

Ruled cubic surfaces are some of the simplest mathematical surfaces which may be described by a single polynomial equation (of degree three for cubics). As many quadrics (degree 2) they are swept out by a family of straight lines (and thus contain infinitly many lines) in contrast to most other cubic surfaces which contain either 27 or even fewer lines.

The model also shows the circle which was used to construct this ruled surface so that it is easy to understand this construction when looking at the model.