Editing Hyperboloid (section)

==In more than three dimensions==
Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a [[pseudo-Euclidean space]] one has the use of a [[quadratic form]]:
<math display="block">q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_{k+1}^2+\cdots + x_n^2\right), \quad k < n .</math> 
When {{math|''c''}} is any [[constant (mathematics)|constant]], then the part of the space given by
<math display="block">\lbrace x \ :\ q(x) = c \rbrace </math>
is called a ''hyperboloid''. The degenerate case corresponds to {{math|1=''c'' = 0}}.

As an example, consider the following passage:<ref>Thomas Hawkins (2000) ''Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869—1926'', §9.3 "The Mathematization of Physics at Göttingen", see page 340, Springer {{ISBN|0-387-98963-3}}</ref>
<blockquote>... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates {{math|(''y''<sub>1</sub>, ..., ''y''<sub>4</sub>)}}, its equation is {{math|''y''{{su|b=1|p=2}} + ''y''{{su|b=2|p=2}} + ''y''{{su|b=3|p=2}} − ''y''{{su|b=4|p=2}} {{=}} −1}}, analogous to the hyperboloid {{math|''y''{{su|b=1|p=2}} + ''y''{{su|b=2|p=2}} − ''y''{{su|b=3|p=2}} {{=}} −1}} of three-dimensional space.{{refn|Minkowski used the term "four-dimensional hyperboloid" only once, in a posthumously-published typescript and this was non-standard usage, as Minkowski's hyperboloid is a three-dimensional submanifold of a four-dimensional Minkowski space <math>M^4.</math><ref>{{Citation|author=Walter, Scott A.| year=1999 | contribution=The non-Euclidean style of Minkowskian relativity|editor=J. Gray|title=The Symbolic Universe: Geometry and Physics 1890-1930|pages=91–127|publisher=Oxford University Press|contribution-url=http://scottwalter.free.fr/papers/1999-symbuniv-walter.html}}</ref>}}</blockquote>

However, the term '''quasi-sphere''' is also used in this context since the sphere and hyperboloid have some commonality (See {{section link||Relation to the sphere}} below).