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

From Diffgeom

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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> || || || || |

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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> || || || || |

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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> || || || |

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## Revision as of 01:37, 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 |
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Up to rotations | |||||

Up to rigid motions (rotations, translations, reflections) | |||||

Up to similarity transformations | Here, |