It can also be represented by the parametric equations
x−x 0 =acos vsinh u, y −y 0 =bsin vsinh u, z −z 0 =ccosh u (7 .37)
where 0 ≤v≤ 2 πand 0 ≤u≤h, with both c > 0 and c < 0 producing a surface with
two parts. Here again the selection of hshould be of the same magnitude or less
than vor else the final image gets distorted. The hyperboloid of one sheet and some
hyperboloids of two sheets are illustrated in the figure 7-8. Note in this figure that
the axes x,yand zhave undergone various permutations. These permutations show
that the axis of symmetry for the hyperboloid of two sheets is always associated with
the term which has the positive sign. In a similar fashion one can do a permutation
of the symbols x,yand zin the equation describing the hyperboloid of one sheet to
obtain different axes of symmetry.
Figure 7-8. Hyperboloid of one sheet and several hyperboloids of two sheets.
In a similar fashion one can perform a permutation of the symbols x, y and zto
give alternative representations of any of the surfaces previously defined.
One can use the parametric equations to define a position vector r =r (u, v )from
which the coordinate curves r (u 0 , v) and r (u, v 0 ) can be constructed. The partial
derivatives of r (u, v )with respect to uand vproduce tangent vectors to the coordinate
curves and these tangent vectors can be used to construct a normal vector to each
point on the surface.
