Editing Hyperboloid (section)

== Properties ==

=== Hyperboloid of one sheet ===
==== 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]].

==== Plane sections ====
For simplicity the plane sections of the ''unit hyperboloid'' with equation <math> \ H_1: x^2+y^2-z^2=1</math> are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too.

*A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects <math>H_1</math> in an ''ellipse'',
*A plane with a slope equal to 1 containing the origin intersects <math>H_1</math> in a ''pair of parallel lines'',
*A plane with a slope equal 1 not containing the origin intersects <math>H_1</math> in a ''parabola'',
*A tangential plane intersects <math>H_1</math> in a ''pair of intersecting lines'',
*A non-tangential plane with a slope greater than 1 intersects <math>H_1</math> in a ''hyperbola''.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt)] (PDF; 3,4&nbsp;MB), S. 116</ref>
Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see [[circular section]]).

=== Hyperboloid of two sheets {{anchor|Two sheets}}===
[[File:Hyperboloid-2s.svg|thumb|hyperboloid of two sheets: generation by rotating a hyperbola]]
[[File:Hyperbo-2s-ca.svg|thumb|hyperboloid of two sheets: plane sections]]
The hyperboloid of two sheets does ''not'' contain lines. The discussion of plane sections can be performed for the ''unit hyperboloid of two sheets'' with equation
<math display="block">H_2: \ x^2+y^2-z^2 = -1.</math>
which can be generated by a rotating [[hyperbola]] around one of its axes (the one that cuts the hyperbola)

*A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects <math>H_2</math> either in an ''ellipse'' or in a ''point'' or not at all,
*A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does ''not intersect'' <math>H_2</math>,
*A plane with slope equal to 1 not containing the origin intersects <math>H_2</math> in a ''parabola'', 
*A plane with slope greater than 1 intersects <math>H_2</math> in a ''hyperbola''.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt)] (PDF; 3,4&nbsp;MB), S. 122</ref>
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see [[circular section]]).

''Remark:'' A hyperboloid of two sheets is ''projectively'' equivalent to a sphere.

===Other properties===

==== Symmetries ====
The hyperboloids with equations 
<math display="block">\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 , \quad \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1 </math>
are
*''pointsymmetric'' to the origin,
*''symmetric to the coordinate planes'' and
*''rotational symmetric'' to the z-axis and symmetric to any plane containing the z-axis, in case of <math>a=b</math> (hyperboloid of revolution).

==== Curvature ====
Whereas the [[Gaussian curvature]] of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a [[Hyperboloid model|model]] for hyperbolic geometry.