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Ellipsoid in three-dimensional Euclidean space

3 bytes removed, 02:03, 6 August 2011
Implicit and parametric descriptions
| 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> || || <math>x = x_0 + a\cos u \sin v, y = y_0 + b \sin u \sin v, z = z_0 + c \cos v</math>|| || This version need not be centered along at 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> || <math>a</math>, <math>b</math>, <math>c</math> are positive numbers that represent the lengths of the three ellipsoidal semi-axes. Note that because we allow rigid motions, we can permute and rearrange to force <math>a \ge b \ge c</math>. || <math>x = a\cos u \sin v, y = b \sin u \sin v, z = c \cos v</math> || Here, <math>u</math> and <math>v</math> are ellipsoidal equivalents of the angles <math>\theta, \phi</math> used in [[spherical polar coordinates]]. || This version is centered at the origin and oriented parallel to the axes.
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