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.
Blinn
James F
JF
Microsoft Research, USA. blinn@microsoft.com
D000465:Algorithms; D003196:Computer Graphics; D009716:Numerical Analysis, Computer-Assisted