Editing Hyperboloid (section)

{{short description|Unbounded quadric surface}}
{{distinguish|text=[[hyperbolic paraboloid]], a saddle-like surface}}
{| class=wikitable align=right
|- align=center
|[[File:Hyperboloid1.png|150px]]<br />Hyperboloid of one sheet
|[[File:DoubleCone.png|160px]]<br />[[conical surface]] in between
|[[File:Hyperboloid2.png|150px]]<br />Hyperboloid of two sheets
|}
In [[geometry]], a '''hyperboloid of revolution''', sometimes called a '''circular hyperboloid''', is the [[surface (mathematics)|surface]] generated by rotating a [[hyperbola]] around one of its [[Hyperbola#Equation|principal axes]]. A '''hyperboloid''' is the surface obtained from a hyperboloid of revolution by deforming it by means of directional [[scaling (geometry)|scaling]]s, or more generally, of an [[affine transformation]].

A hyperboloid is a [[quadric surface]], that is, a [[surface (mathematics)|surface]] defined as the [[zero set]] of a [[polynomial]] of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a [[conical surface|cone]] or a [[cylinder]], having a [[central symmetry|center of symmetry]], and intersecting many [[plane (geometry)|planes]] into hyperbolas. A hyperboloid has three pairwise [[perpendicular]] [[rotational symmetry|axes of symmetry]], and three pairwise perpendicular [[reflection symmetry|planes of symmetry]].

Given a hyperboloid, one can choose a [[Cartesian coordinate system]] such that the hyperboloid is defined by one of the following equations:
<math display="block"> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} =  1,</math> 
or
<math display="block"> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = -1.</math>
The coordinate axes are axes of symmetry of the hyperboloid and the origin is the center of symmetry of the hyperboloid. In any case, the hyperboloid is [[asymptotic]] to the cone of the equations:
<math display="block"> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 .</math>

One has a hyperboloid of revolution if and only if <math>a^2=b^2.</math> Otherwise, the axes are uniquely defined ([[up to]] the exchange of the ''x''-axis and the ''y''-axis).

There are two kinds of hyperboloids. In the first case ({{math|+1}} in the right-hand side of the equation): a '''one-sheet hyperboloid''', also called a '''hyperbolic hyperboloid'''. It is a [[connected set|connected surface]], which has a negative [[Gaussian curvature]] at every point. This implies near every point the intersection of the hyperboloid and its [[tangent plane]] at the point consists of two branches of curve that have distinct tangents at the point. In the case of the one-sheet hyperboloid, these branches of curves are [[line (geometry)|lines]] and thus the one-sheet hyperboloid is a [[doubly ruled]] surface.

In the second case ({{math|−1}} in the right-hand side of the equation): a '''two-sheet hyperboloid''', also called an '''elliptic hyperboloid'''. The surface has two [[connected component (topology)|connected component]]s and a positive Gaussian curvature at every point. The surface is ''convex'' in the sense that the tangent plane at every point intersects the surface only in this point.