%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Quadric Surfaces">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Feb 12 22:07:52 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=Reference.cst}
%TCIDATA{PageSetup=72,72,72,72,0}
%TCIDATA{<LINK REL="contents" HREF="..\Reference\ReferenceIndex.tex#Reference Index">}
%TCIDATA{<LINK REL="next" HREF="cal10_07.tex">}
%TCIDATA{<LINK REL="parent" HREF="CAL_contents.tex">}
%TCIDATA{<LINK REL="previous" HREF="cal10_05.tex">}
%TCIDATA{AllPages=
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \thepage}
%}


\input{tcilatex}
\begin{document}


\section{Quadric Surfaces\label{Quadric surface}}

A \textbf{quadric surface} is the graph of a second-degree equation in three
variables $x$, $y$ and $z$. The most general form of such an equation is  
\[
Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0
\]
where $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$, and $J$ are constants. By
various translations and rotations this equation can be rewritten in one of
two standard forms: 
\[
\begin{array}{lll}
Ax^{2}+By^{2}+Cz^{2}+J=0 & \text{or} & Ax^{2}+By^{2}+Iz=0
\end{array}
\]

In order to sketch the graph of a quadric surface (or any surface), it is
useful to look at the curves that intersect the surface with planes parallel
to the coordinate planes. \label{Cross-section}These curves are called 
\textbf{traces} (or \textbf{cross-sections}) of the surface.

\subsubsection{Ellipsoids\label{Ellipsoid}}

A quadric surface with equation 
\[
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1
\]
where $a,b\neq 0$ is called an \textbf{ellipsoid }because its traces are
ellipses. For example, the horizontal plane $z=k$ where $\left| k\right|
<\left| c\right| $ intersects the surface in the ellipse 
\[
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1-\frac{k^{2}}{c^{2}}
\]

\subsubsection{Hyperboloids\label{Hyperboloid}}

The quadric surface 
\[
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1
\]
\vspace{1pt}like the ellipsoid, is symmetric with respect to the coordinate
planes. The trace in any horizontal plane $z=k$ where $\left| k\right|
<\left| c\right| $ is an ellipse 
\[
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1+\frac{k^{2}}{c^{2}}
\]
but the traces in the $xz-p$lanes and $yz$-planes are hyperbolas. For
example, when $y=0$, 
\[
\frac{x^{2}}{a^{2}}-\frac{z^{2}}{c^{2}}=1
\]
Furthermore, the surface consists of just one piece, so it is called a 
\textbf{hyperboloid of one sheet.} The $z$ -axis is called the \textbf{axis}
of the hyperboloid. Similarly, quadric surfaces of the form $-\frac{x^{2}}{%
a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ and $\frac{x^{2}}{a^{2}}-%
\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ are hyperboloids of one sheet.

Next consider quadric surfaces of the form 
\[
-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1
\]
The traces in the $xz$-planes and $yz$-planes are hyperbolas. If $\left|
k\right| >\left| c\right| $, the horizontal plane $z=k$ intersects the
surface in an ellipse. If $\left| k\right| <\left| c\right| $ the plane $z=k$
does not intersect the surface. Thus the surface consist of two parts, one
above the plane $z=\left| c\right| $ and one below the plane $z=-\,\left|
c\right| $. This type of quadric surface is called a \textbf{hyperboloid of
two sheets} with axis the $z$-axis. Similarly 
\[
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1
\]
and 
\[
-\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1
\]
are hyperboloids of two sheets.

\subsubsection{Cones\label{Cone}}

Quadric surfaces of the form 
\[
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\frac{z^{2}}{c^{2}}
\]
are called \textbf{cones}. They have the property that if $P$ is any point
on the surface, then the line $OP$ lies entirely on the cone.

\noindent \hrulefill

\begin{center}
\begin{tabular}{ccc}
{\small \hyperref{\FRAME{itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Computing Techniques}{}{}{%
../../Help/DMindex.tex}} & {\small \hyperref{\FRAME{itbpF}{13.9375pt}{%
14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
13.9375pt;height 14.4375pt;depth 3pt;original-width
24.0625pt;original-height 24.8125pt;cropleft "0";croptop "0.9959";cropright
"0.9937";cropbottom "0";filename 'Nblogo.wmf';file-properties "XNPEU";}}%
General Information}{}{}{../../Help/Index.tex}} & {\small \hyperref{\FRAME{%
itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Reference Library}{}{}{%
../ReferenceIndex.tex}}
\end{tabular}
\end{center}

\noindent \hrulefill

\end{document}
