Difference between revisions of "Elliptic hyperboloid of one sheet"
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Revision as of 12:42, 12 August 2011
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 semiaxis 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) 