# Changes

## Ellipsoid in three-dimensional Euclidean space

, 12:49, 12 August 2011
no edit summary
| 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 $\R^3$.

==Fundamental forms and curvatures==
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==Sections==

For the discussion here, we assume the ellipsoid is given in the form:

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

Below we describe the nature of sections:

{| class="sortable" border="1"
! Plane with which we're intersecting !! Description of intersection
|-
| $x = x_0$ || empty if $|x_0| > a$<br>a single point $(x_0,0,0)$ if $|x_0| = a$.<br>an ellipse of the form $\frac{y^2}{b^2(1 - (x_0/a))^2} + \frac{z^2}{c^2(1 - (x_0/a))^2} = 1, x = x_0$ in the plane $x = x_0$.
|-
| $y = y_0$ || {{fillin}}
|-
| $z = z_0$ ||
|}
==Automorphisms==