Editing Hyperboloid (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.