Difference between revisions of "Hyperbolic paraboloid"
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Revision as of 13:13, 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. If we're considering the surface up to similarity transformations, the parameters are unique up to projective equivalence.
The surface, however, is unique up to affine transformations, which include transformations that do not preserve the affine structure.
Implicit and parametric descriptions
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  
Up to rigid motions (rotations, translations, reflections)  
Up to similarity transformations  Here and we've done a similarity transformation.  
Up to all affine transformations (not permissible if we want to study geometric structure) 