# Elliptic hyperboloid of one sheet

From Diffgeom

## 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) |