# Difference between revisions of "Elliptic hyperboloid of one sheet"

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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> || || | | 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> || || | + | | 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>. | ||

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+ | ==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. | ||

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+ | 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]]. |

## Latest revision as of 12:55, 12 August 2011

## Contents

## Definition

The surface type is *not* unique up to isometry or even up to similarity transformations, but rather, depends on multiple nonzero parameters . If we're considering the surface up to rigid isometries, the parameters are unique up to transposition of and , which we can avoid by stipulating that .

The surface, however, *is* unique up to affine transformations, which include transformations that do not preserve the affine structure.

### Implicit and parametric descriptions

Degree of generality | Implicit description | What the parameters mean | Parametric description | What the additional parameters mean | Comment |
---|---|---|---|---|---|

Arbitrary | Fill this in later |
This version need not be centered at the origin and need not be oriented parallel to the axes. | |||

Up to rotations | are positive numbers representing the semi-axis lengths. | This version need not be centered at the origin but is oriented parallel to the axes. | |||

Up to rigid motions (rotations, translations, reflections) | |||||

Up to similarity transformations | We ca normalize to 1 using a similarity transformation. | ||||

Up to all affine transformations (not permissible if we want to study geometric structure) |

## 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 into two pieces, one of which is homeomorphic to and the other is homeomorphic to the complement of a line in .

## 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:

*Fill this in later*

## Particular cases

In the case , we get a circular hyperboloid of one sheet.