The surfaces in the Olivier models are known as ruled surfaces
because they are formed by a collection of straight lines, or rulings.
One way to generate a ruled surface is to "sweep out" the surface by a
straight line in motion. Since a line is determined by a point and a
direction, we have the following characterization of ruled surfaces:
Given a curve ${\bf X}(t)$ and a collection of
direction vectors ${\bf v}(t)$, a ruled surface is the image
of the function
$$
f(s,t) = {\bf X}(t) + s{\bf v}(t).
$$
