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Ellipsoid in three-dimensional Euclidean space

1,225 bytes added, 01:53, 6 August 2011
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| 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 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 = y \sin u \sin v, z = c = z \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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| 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>. If we chose <math>a \ge b \ge c</math>, we would get <math>0 < q,r \le 1</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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==Fundamental forms and curvatures==
 
For the table below, we consider the parametric description that is valid for
 
<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1</math>
 
namely the description:
 
<math>x = a\cos u \sin v, b = y \sin u \sin v, c = z \cos v</math>
 
{| class="sortable" border="1"
! Quantity !! Meaning in general !! Value
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| <math>E,F,G</math> for [[first fundamental form]] using this parametrization || The Riemannian metric is given by <math>ds^2 = E \, du^2 + 2F \, du \, dv + G \, dv^2</math> || {{fillin}}
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| <math>e,f,g</math> 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 <math>\begin{pmatrix} E & F \\ F & G \\\end{pmatrix}</math> || {{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 = <math>EG - F^2</math> || {{fillin}}
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==Verification of theorems==
 
===Gauss-Bonnet theorem===
 
{{fillin}}
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