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Elliptic hyperboloid of one sheet

1,873 bytes added, 12:42, 12 August 2011
Created page with "==Definition== The surface type is ''not'' unique up to isometry or even up to similarity transformations, but rather, depends on multiple nonzero parameters <math>a,b,c</math>...."
==Definition==

The surface type is ''not'' unique up to isometry or even up to similarity transformations, but rather, depends on multiple nonzero parameters <math>a,b,c</math>. If we're considering the surface up to rigid isometries, the parameters are unique up to transposition of <math>a</math> and <math>b</math>, which we can avoid by stipulating that <math>a \ge b</math>.

The surface, however, ''is'' unique up to affine transformations, which include transformations that do not preserve the affine structure.

===Implicit and parametric descriptions===

{| class="sortable" border="1"
! Degree of generality !! Implicit description !! What the parameters mean !! Parametric description !! What the additional parameters mean !! Comment
|-
| Arbitrary || {{fillin}} || || || || This version need not be centered at the origin and need not be oriented parallel to the axes.
|-
| Up to rotations || <math>\frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} - \frac{(z - z_0)^2}{c^2} = 1</math> || <math>a,b,c</math> are positive numbers representing the semi-axis lengths. || <math>x = x_0 + a\cos u \cosh v, y = y_0 + b \sin u \cosh v, z = z_0 + c \sinh v</math>|| || This version need not be centered at the origin but is oriented parallel to the axes.
|-
| Up to rigid motions (rotations, translations, reflections) || <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1</math> || || <math>x = a\cos u \cosh v, y = b \sin u \cosh v, z = c \sinh v</math> || ||
|-
| Up to similarity transformations || <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} - z^2 = 1</math> || We ca normalize <math>c</math> to 1 using a similarity transformation. || <math>x = a\cos u \cosh v, y = b \sin u \cosh v, z = \sinh v</math> || ||
|-
| Up to all affine transformations (''not permissible if we want to study geometric structure'') || <math>x^2 + y^2 - z^2 = 1</math> || ||
|}
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