Lines on a Cubic Surface
The level sets of the cubic polynomial . When , the surface contains six affine lines. When , the surface contains 24 affine lines. For each line, one coordinate is constant, so the lines can be grouped into three sets of eight (shown in different colors) depending upon which coordinate plane the lines are parallel to.
Snapshot 1: , surface contains no affine lines
Snapshot 2: , surface contains six affine lines
Snapshot 3: , surface contains 24 affine lines
Snapshot 4: , the surface contains 24 affine lines. There are four Eckardt points where three of these lines simultaneously intersect.