# ellipsoid

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

 $\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}+\left(\frac{z}{c}% \right)^{2}=1$

for some $a,b,c>0$.

## Properties

1. 1.

If $a=b=c$, the ellipsoid reduces to a sphere.

2. 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. 3.

The ellipse determined by $a,b,c$ is the unit sphere of the norm

 $\|v\|=v^{T}\operatorname{diag}(\frac{1}{a},\frac{1}{b},\frac{1}{c})v,\quad v=(% x,y,z)^{T}.$
Title ellipsoid Ellipsoid 2013-03-22 14:56:45 2013-03-22 14:56:45 matte (1858) matte (1858) 6 matte (1858) Definition msc 51M05 Sphere QuadraticSurfaces Ellipse2 VolumeOfEllipsoid