# Changes

## Ellipsoid in three-dimensional Euclidean space

, 01:53, 6 August 2011
no edit summary
| 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 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 $a \ge b \ge c$. || $x = a\cos u \sin v, y = b = y \sin u \sin v, z = c = z \cos v$ || 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.
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| Up to similarity transformations || $x^2 + \frac{y^2}{q^2} + \frac{z^2}{r^2} = 1$ || Here, $q = b/a, r = c/a$. If we chose $a \ge b \ge c$, we would get $0 < q,r \le 1$. || || || For this version, we can normalize one of the three values $a,b,c$. In the form given here, we chose to normalize $a$.
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==Fundamental forms and curvatures==

For the table below, we consider the parametric description that is valid for

$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$

namely the description:

$x = a\cos u \sin v, b = y \sin u \sin v, c = z \cos v$

{| class="sortable" border="1"
! Quantity !! Meaning in general !! Value
|-
| $E,F,G$ for [[first fundamental form]] using this parametrization || The Riemannian metric is given by $ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2$ || {{fillin}}
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| $e,f,g$ for [[second fundamental form]] using this parametrization || {{fillin}} || {{fillin}}
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| [[principal curvature]]s using this parametrization || eigenvalues of the shape operator, which is the matrix $\begin{pmatrix} E & F \\ F & G \\\end{pmatrix}$ || {{fillin}}
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| [[mean curvature]] || arithmetic mean of the principal curvatures || {{fillin}}
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| [[Gaussian curvature]] || product of the principal curvatures = determinant of shape operator = $EG - F^2$ || {{fillin}}
|}

==Verification of theorems==

===Gauss-Bonnet theorem===

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