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ellipsoid (Definition)

An ellipsoid is a subset of $ \mathbbmss{R}^3$ consisting of points $ (x,y,z)\in \mathbbmss{R}^3$ such that

$\displaystyle \left(\frac{x}{a}\right)^2+ \left(\frac{y}{b}\right)^2+ \left(\frac{z}{c}\right)^2=1 $
for some $ a,b,c>0$.

Properties

  1. If $ a=b=c$, the ellipsoid reduces to a sphere.
  2. If we fix the value of any of $ x,y,z$ to some constant, say $ x=C$, we obtain an ellipse in the plane $ (C,y,z)$.
  3. The ellipse determined by $ a,b,c$ is the unit sphere of the norm
    $\displaystyle \Vert v \Vert = v^T \operatorname{diag} (\frac{1}{a}, \frac{1}{b}, \frac{1}{c}) v, \quad v=(x,y,z)^T. $



"ellipsoid" is owned by matte.
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See Also: sphere, quadratic surfaces, ellipse, volume of ellipsoid


Attachments:
spheroid (Definition) by rspuzio
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Cross-references: norm, unit sphere, plane, ellipse, fix, sphere, points, subset
There are 10 references to this entry.

This is version 3 of ellipsoid, born on 2005-01-12, modified 2005-02-19.
Object id is 6637, canonical name is Ellipsoid.
Accessed 3226 times total.

Classification:
AMS MSC51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)

Pending Errata and Addenda
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