Editing Hyperboloid (section)

==== Lines on the surface ====
*A hyperboloid of one sheet contains two pencils of lines. It is a [[doubly ruled surface]]. 
If the hyperboloid has the equation
<math> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}= 1</math>  then the lines

<math display="block">g^{\pm}_{\alpha}:
 \mathbf{x}(t) = \begin{pmatrix} a\cos\alpha \\ b\sin\alpha \\ 0\end{pmatrix}
  + t\cdot \begin{pmatrix} -a\sin\alpha\\ b\cos\alpha\\ \pm c\end{pmatrix}\ ,\quad t\in \R,\ 0\le \alpha\le 2\pi\  </math>
are contained in the surface.

In case <math>a = b</math> the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines <math>g^{+}_{0}</math> or <math>g^{-}_{0}</math>, which are skew to the rotation axis (see picture). This property is called ''[[Christopher Wren|Wren]]'s theorem''.<ref>K. Strubecker: ''Vorlesungen der Darstellenden Geometrie.'' Vandenhoeck & Ruprecht, Göttingen 1967, p. 218</ref> The more common generation of a one-sheet hyperboloid of revolution is rotating a [[hyperbola]] around its [[Semi-major and semi-minor axes#Hyperbola|semi-minor axis]] (see picture; rotating the hyperbola around its other axis gives a two-sheet hyperbola of revolution).

A hyperboloid of one sheet is ''[[projective geometry|projectively]]'' equivalent to a [[hyperbolic paraboloid]].