# Difference between revisions of "Ellipsoid in three-dimensional Euclidean space"

From Diffgeom

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! Degree of generality !! Implicit description !! What the parameters mean !! Parametric description !! What the additional parameters mean !! Comment | ! Degree of generality !! Implicit description !! What the parameters mean !! Parametric description !! What the additional parameters mean !! Comment | ||

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− | | Arbitrary || {{fillin}} || || || || | + | | Arbitrary || {{fillin}} || || || || This version need not be centered at the origin and need not be oriented parallel to the axes. |

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− | | 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> || || || || | + | | 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> || || || || This version need not be centered along the origin but is oriented parallel to the axes. |

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− | | 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> || || || || | + | | 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> || || || || This version is centered at the origin and oriented parallel to the axes. |

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− | | Up to similarity transformations || <math>x^2 + \frac{y^2}{q^2} + \frac{z^2}{r^2} = 1</math> || Here, <math>q = b/a, r = c/a</math> || || || | + | | Up to similarity transformations || <math>x^2 + \frac{y^2}{q^2} + \frac{z^2}{r^2} = 1</math> || Here, <math>q = b/a, r = c/a</math> || || || For this version, we can normalize one of the three values <math>a,b,c</math>. In the form given here, we chose to normalize <math>a</math>. |

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## Revision as of 01:39, 6 August 2011

## Definition

This surface type is *not* unique up to isometry or even up to similarity transformations, but rather, depends upon multiple nonzero parameters . If we're considering the surface up to rigid isometries, the parameters are unique up to permutations. If we're considering the surface up to similarity transformations, the parameters are unique up to the action of permutations *and* projective equivalence.

### 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 | This version need not be centered along the origin but is oriented parallel to the axes. | ||||

Up to rigid motions (rotations, translations, reflections) | This version is centered at the origin and oriented parallel to the axes. | ||||

Up to similarity transformations | Here, | For this version, we can normalize one of the three values . In the form given here, we chose to normalize . |