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Elliptic hyperboloid of one sheet

, 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 $a,b,c$...."
==Definition==

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

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 || $\frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} - \frac{(z - z_0)^2}{c^2} = 1$ || $a,b,c$ are positive numbers representing the semi-axis lengths. || $x = x_0 + a\cos u \cosh v, y = y_0 + b \sin u \cosh v, z = z_0 + c \sinh v$|| || This version need not be centered at the origin but is oriented parallel to the axes.
|-
| Up to rigid motions (rotations, translations, reflections) || $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ || || $x = a\cos u \cosh v, y = b \sin u \cosh v, z = c \sinh v$ || ||
|-
| Up to similarity transformations || $\frac{x^2}{a^2} + \frac{y^2}{b^2} - z^2 = 1$ || We ca normalize $c$ to 1 using a similarity transformation. || $x = a\cos u \cosh v, y = b \sin u \cosh v, z = \sinh v$ || ||
|-
| Up to all affine transformations (''not permissible if we want to study geometric structure'') || $x^2 + y^2 - z^2 = 1$ || ||
|}
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