Skew Lines - Skew Lines and Ruled Surfaces

Skew Lines and Ruled Surfaces

If one rotates a line L around another line L' skew but not perpendicular to it, the surface of revolution swept out by L is a hyperboloid of one sheet. For instance, the three hyperboloids visible in the illustration can be formed in this way by rotating a line L around the central white vertical line L'. The copies of L within this surface make it a ruled surface; it also contains a second family of lines that are also skew to L' at the same distance as L from it but with the opposite angle. An affine transformation of this ruled surface produces a surface which in general has an elliptical cross-section rather than the circular cross-section produced by rotating L around L'; such surfaces are also called hyperboloids of one sheet, and again are ruled by two families of mutually skew lines. A third type of ruled surface is the hyperbolic paraboloid. Like the hyperboloid of one sheet, the hyperbolic paraboloid has two families of skew lines; in each of the two families the lines are parallel to a common plane although not to each other. Any three skew lines in R3 lie on exactly one ruled surface of one of these types (Hilbert & Cohn-Vossen 1952).