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For one obtains a hyperboloid of one sheet,
* For a hyperboloid of two sheets, and
* For a double cone.

One can obtain a parametric representation of a hyperboloid with a different coordinate axis as the axis of symmetry by shuffling the position of the term to the appropriate component in the equation above.

Generalised equations

More generally, an arbitrarily oriented hyperboloid, centered at , is defined by the equation

where is a matrix and

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