# Elliptic hyperboloid of one sheet

## 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.