Hyperboloid
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Hyperboloid of two sheets
In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation
- [ + - =1] (hyperboloid of one sheet),
or
- [- - + =1] (hyperboloid of two sheets)
If, and only if,
a =
b, it is a
hyperboloid of revolution. A hyperboloid of one sheet can be obtained by revolving a hyperbola around its transversal axis. Alternatively, a hyperboloid of two sheets of axis AB is obtained as the
set of points P such that AP−BP is a
constant, AP being the distance between A and P. Points A and B are then called the
foci of the hyperboloid. A hyperboloid of two sheets can be obtained by
revolving a
hyperbola around its focal axis.
A hyperboloid of one sheet is a doubly ruled surface; if it is a hyperboloid of revolution, it can also be obtained by revolving a line about a skew line.
An elliptic hyperboloid of one sheet. The wires are straight lines. For any point on the surface, there are two straight lines lying entirely on the surface which pass through the point. This illustrates the doubly ruled nature of this surface.
A degenerate hyperboloid is of the form
- [ + - =0;]
if
a =
b then this will give a
cone, if not then it gives an
elliptical cone.
See also
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