# Ruled Cubics

__Models of Ruled Cubics__

Model VII 20 has a line of singularities ("double points") at which two 'sheets' of the surface intersect. The surface is generated with the director line (see the Ruled Surfaces page) as the almost vertical straight line passing into the "hole", and the director curve is the horizontal loop halfway up the model. This model illustrates some of the lines joining points on each of these, which generate the surface as a whole.

Model VII 21 also has a line of singularities (again double points), a segment of which (between two "cuspidal points") is the intersection of two sheets. Sadly the model is chipped, so the cuspidal points are difficult to see here. In the above construction, the director line is the near-vertical line running alongside the singular line, and the director curve is the horizontal circle in the middle. This model also illustrates lines joining points on each of these, which generate the surface as a whole.

__Cayley's Ruled Surface__

Model VII 22 is known as Cayley's ruled surface. This surface is especially interesting since the director line (the vertical line where the sheets intersect) intersects the director curve (the horizontal curve in the middle). In this case, the line of singularities (again double points) coincides with the director line, and there are two cuspidal points at infinity. This model again illustrates some lines joining points on each of these, which generate the surface as a whole.