# Changes

## Ellipsoid in three-dimensional Euclidean space

, 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 || $\frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} + \frac{(z - z_0)^2}{c^2} = 1$ || || $x = x_0 + a\cos u \sin v, y = y_0 + b \sin u \sin v, z = z_0 + c \cos v$|| || 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) || $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ || $a$, $b$, $c$ 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 $a \ge b \ge c$. || $x = a\cos u \sin v, y = b \sin u \sin v, z = c \cos v$ || Here, $u$ and $v$ are ellipsoidal equivalents of the angles $\theta, \phi$ used in [[spherical polar coordinates]]. || This version is centered at the origin and oriented parallel to the axes.