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.