ellipsoid

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

   $$\parallel v\parallel =v^T\mathrm{diag}(\frac{1}{a},\frac{1}{b},\frac{1}{c})v,v={(x,y,z)}^T.$$