# Changes

## Elliptic hyperboloid of one sheet

, 12:55, 12 August 2011
no edit summary
| Up to similarity transformations || $\frac{x^2}{a^2} + \frac{y^2}{b^2} - z^2 = 1$ || We ca normalize $c$ to 1 using a similarity transformation. || $x = a\cos u \cosh v, y = b \sin u \cosh v, z = \sinh v$ || ||
|-
| Up to all affine transformations (''not permissible if we want to study geometric structure'') || $x^2 + y^2 - z^2 = 1$ || || || ||
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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 $\R^3$ into two pieces, one of which is homeomorphic to $\R^3$ and the other is homeomorphic to the complement of a line in $\R^3$.

==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 $a = b$, we get a [[circular hyperboloid of one sheet]].