ellipsoidEllipsoid2005-01-12 11:56:482005-02-19 01:49:50Definition% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{mathrsfs}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
\usepackage{bbm}
\newcommand{\Z}{\mathbbmss{Z}}
\newcommand{\C}{\mathbbmss{C}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\Q}{\mathbbmss{Q}}
\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |}
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}An \emph{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 $a,b,c>0$.
\subsubsection*{Properties}
\begin{enumerate}
\item If $a=b=c$, the ellipsoid reduces to a sphere.
\item If we fix the value of any of $x,y,z$ to some constant, say $x=C$,
we obtain an ellipse in the plane $(C,y,z)$.
\item The ellipse determined by $a,b,c$ is the unit sphere of the
norm
$$
\Vert v \Vert = v^T \operatorname{diag} (\frac{1}{a}, \frac{1}{b}, \frac{1}{c}) v, \quad v=(x,y,z)^T.
$$
\end{enumerate}