51 Ruled hyperboloid
52 Quartic ruled surface
53 Intersection
54 Sextic ruled surface
55 Twisted cubic
56 Quartic scroll
57 Osculating hyperboloid
58 Ruled elliptic hyperboloid
59 Intersecting cones
60 Helicoid
61 Conoid
62 Double helicoid
63 Intersecting cylinders
65 Ruled cone
66 Ruled surface
67 Ruled hyperbolic paraboloid
68 Doubly ruled hyperboloid
A ruled surface (or scroll) is a surface swept out by a straight line as it moves through space. For example, a cylinder is formed by moving a straight line around a curve in a plane, keeping it perpendicular to the plane at all times; a cone is formed by moving a line so that it stays fixed at one point, but changes direction (65); and a helicoid is formed by moving a straight line along another straight line, keeping it perpendicular but rotating it as it moves (60).
These models show ruled surfaces using stretched string to represent the position of the straight line at various times. The straight lines in the ruling are called generators of the surface. Some quadratic surfaces are ruled: hyperboloids of one sheet, hyperbolic paraboloids, and quadratic cones and cylinders. The first two are doubly ruled, that is, they have two distinct ways of being generated by a moving straight line (67 and 68). A general way to form a ruled surface is to take three curves in space, and move a straight line so that it intersects all three curves at all times. This procedure, applied to the three lines in one of the rulings of a hyperboloid of one sheet or a hyperbolic paraboloid, will give the other ruling. This procedure is also illustrated in model 56 (Baker No. 84), with two straight lines and an ellipse, and in model 52, with two circles and a straight line. Models 53, 55, 59, and 63 show space curves obtained by taking the intersection of two ruled surfaces.