Cayley's ruled cubic surface

In differential geometry, Cayley's ruled cubic surface is the ruled cubic surface

${\displaystyle z=xy-x^{3}/3\ .}$

In projective coordinates it is ${\displaystyle zw^{2}=xyw-x^{3}/3\ }$.

It contains a (double) line of self-intersection ${\displaystyle x=w=0\ }$and two pinch points.^[1]