726 Polygonal triples
28.1 Double ruling ofS ....................
The surfaceS, being the surface of revolution of a rectangular hyperbola
about its conjugate axis, is a rectangular hyperboloid of one sheet. It
has a double ruling,i.e., through each point on the surface, there are two
straight lines lying entirely on the surface.
Figure 28.1:
LetP(x 0 ,y 0 ,z 0 )be a point on the surfaceS. A linethroughPwith
direction numbersp:q:rhas parametrization
: x=x 0 +pt, y=y 0 +qt, z=z 0 +rt.
Substitution of these expressions into (28.4) shows that the lineis en-
tirely contained in the surfaceSif and only if
px 0 +qy 0 = rz 0 , (28.7)
p^2 +q^2 = r^2. (28.8)
It follows that
r^2 = r^2 (x^20 +y 02 −z^20 )
= r^2 (x^20 +y 02 )−(px 0 +qy 0 )^2
=(p^2 +q^2 )(x^20 +y^20 )−(px 0 +qy 0 )^2
=(qx 0 −py 0 )^2.
This means
qx 0 −py 0 =r, =± 1. (28.9)
Solving equations (28.7) and (28.9), we determine the direction numbers
of the line. We summarize this in the following proposition.
