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Ellipsoid in three-dimensional Euclidean space

893 bytes added, 12:49, 12 August 2011
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| Two of the parameter values are equal, and they are both less than the third || [[prolate spheroid]] || This has full rotational symmetry about the third axis, and can be obtained by taking the [[surface of revolution]] obtained from an [[ellipse]] in a plane through that axis. The ellipse is being revolved about its major axis.
==Basic topology==
The ellipsoid is topologically homeomorphic, and in fact, diffeomorphic, to the [[2-sphere]]. It is a compact regular surface and the interior region it bounds is diffeomorphic to an open unit disk in <math>\R^3</math>.
==Fundamental forms and curvatures==
For the discussion here, we assume the ellipsoid is given in the form:
<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1</math>
Below we describe the nature of sections:
{| class="sortable" border="1"
! Plane with which we're intersecting !! Description of intersection
| <math>x = x_0</math> || empty if <math>|x_0| > a</math><br>a single point <math>(x_0,0,0)</math> if <math>|x_0| = a</math>.<br>an ellipse of the form <math>\frac{y^2}{b^2(1 - (x_0/a))^2} + \frac{z^2}{c^2(1 - (x_0/a))^2} = 1, x = x_0</math> in the plane <math>x = x_0</math>.
| <math>y = y_0</math> || {{fillin}}
| <math>z = z_0</math> ||
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