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 | ![]() |
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This version need not be centered at the origin but is oriented parallel to the axes. | |
Up to rigid motions (rotations, translations, reflections) | ![]() |
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Up to similarity transformations | ![]() |
We ca normalize ![]() |
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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.