Lines In Space - Part 6: Our Friend the Hyperbolic Paraboloid

Three mutually skew lines in space K, L, and M determine a unique hyperbolic paraboloid in which they are all embedded. The implicit equation for this is Q = KML-LMK = LKM - MKL = MLK - KLM. And a parametric equation for one of the two families (the one containing K, L, and M) of embedded lines sweeping out the surface is J = (cos(theta) + sin(theta) + 1)(LM) K +(-cos(theta) + sin(theta) + 1) (MK) L -sin(theta(KL)M. Looking at these two- and three- line constructions was so much fun that next time I will pursue the geometric patterns formed from four mutually skew lines in space.