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.

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

$\|v\|=v^{T}\operatorname{diag}(\frac{1}{a},\frac{1}{b},\frac{1}{c})v,\quad v=(% x,y,z)^{T}.$