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Elliptic hyperboloid of one sheet

703 bytes added, 12:55, 12 August 2011
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| Up to similarity transformations || <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} - z^2 = 1</math> || We ca normalize <math>c</math> to 1 using a similarity transformation. || <math>x = a\cos u \cosh v, y = b \sin u \cosh v, z = \sinh v</math> || ||
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| Up to all affine transformations (''not permissible if we want to study geometric structure'') || <math>x^2 + y^2 - z^2 = 1</math> || || || ||
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==Basic topology==
 
Topologically, the elliptic hyperboloid of one sheet is homeomorphic to the infinite [[right circular cylider]]. It is a non-compact regular surface. it divides its complement in <math>\R^3</math> into two pieces, one of which is homeomorphic to <math>\R^3</math> and the other is homeomorphic to the complement of a line in <math>\R^3</math>.
 
==Ruling==
 
The elliptic hyperboloid of one sheet is a [[ruled surface]], i.e., every point on the surface is contained in a line that also lies on the surface.
 
Below is an explicit parametrization using a ruling:
 
{{fillin}}
 
 
==Particular cases==
 
In the case <math>a = b</math>, we get a [[circular hyperboloid of one sheet]].
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