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