Ruled Surfaces

 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 $$f(s,t) = {\bf X}(t) + s{\bf v}(t).$$