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\begin{document}

\input{213-Lecture-Macros}		%% 	macros, settings, et.c


\title{Math 213 - Quadric Surfaces}			%%	title page
\author{Peter A. Perry}
\institute{University of Kentucky}
\date{September 11, 2019}
\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	
%		Due Date Reminders
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Reminders}

\begin{itemize}

\item	Access your WebWork account \emph{only through Canvas}!

\medskip

\item	Homework A3 on section 12.5 is due tonight
\item	Homework A4 on section 12.6 is due Friday
\item	Quiz \# 2 on sections 12.3-12.4 takes place in tomorrow's  recitation

\medskip
		
	
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Unit Outline
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Unit I: Geometry and Motion in Space}
\begin{itemize}

\item[{12.1}]		Lecture 1: Three-Dimensional Coordinate Systems
\item[{12.2}] 		Lecture 2: Vectors in the Plane and in Space
\item[{12.3}]		Lecture 3:The Dot Product
\item[{12.4}]		Lecture 4:The Cross Product
\item[{12.5}]		Lecture 5:	Equations of Lines and Planes, I
\item[{12.5}]		Lecture 6: Equations of  Lines and Planes, II
\begingroup
\color{red}
\item[{12.6}]		Lecture 7: Surfaces in Space
\endgroup
\item[{13.1}]		Lecture 8: Vector Functions and Space Curves
\item[{13.2}]		Lecture 9 Derivatives and Integrals of Vector Functions
\item[{}]				Lecture 10:  Exam I Review
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%	Learning Goals
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Learning Goals}
\begin{itemize}
\item		Recall quadratic curves from MA 114
\item		Identify and graph cylinders
\item		Find traces of quadric surfaces
\item		Identify and graph quadric surfaces
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%		New material begins here...
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%		Make font size on all slides small
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\begingroup
\small


\section{Quadratic Curves}

\begin{frame}
\begin{minipage}{0.25\textwidth}
\begin{tikzpicture}[scale=0.25]
%% Ellipse
\drawxy{4}
\draw[thick,red,smooth,domain=0:6.2832]	plot	({3*cos(\x r)},{2*sin(\x r)});
\draw[black,fill=black]	(-2.236,0)	circle(0.1cm);
\draw[black,fill=black]	(2.236,0)	circle(0.1cm);
\draw[dashed]			(-2.236,0)	--	(0,2)	--	(2.236,0);
\end{tikzpicture}

\medskip

\begin{tikzpicture}[scale=0.25]
\drawxy{4}
\begin{scope}
\clip	(-4,-4)	rectangle 	(4,4);
\draw[thick,red,smooth,domain=-2:2]	plot	({\x},{\x*\x});
\end{scope}
\draw[black,fill=black]	(0,0.5)	circle(0.1cm);
\draw[cyan,dashed]	(-4,-0.5)	--	(4,-0.5);
\end{tikzpicture}

\medskip

\begin{tikzpicture}[scale=0.25]
\drawxy{4}
\begin{scope}
\clip (-4,-4)		rectangle 	(4,4);
\draw[thick,red,smooth,domain=-3:3]	plot({cosh(\x)},{sinh(\x)});
\draw[thick,red,smooth,domain=-3:3]	plot({-cosh(\x)},{sinh(\x)});
%% asymptotes
\draw[dashed,cyan]	(-4,-4)	--	(4,4);
\draw[dashed,cyan]	(-4,4)		--	(4,-4);
%% foci
\draw[black,fill=black]	(-1.414,0)	circle(0.1cm);
\draw[black,fill=black]	(1.414,0)	circle(0.1cm);
\end{scope}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.65\textwidth}
\footnotesize
A \emph{quadratic curve} is the graph of a second-degree equation in two variables taking one of the forms
$$ Ax^2 + By^2 +J =0, \quad Ax^2 +By^2 + Jy = 0 $$

\medskip

In MA 114 you learned about the following \emph{quadratic curves}:

\medskip
\begin{itemize}
\item		The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $ with foci
			$(\pm c,0)$, where $a^2 = b^2 + c^2$
			
			\medskip
			
\item		The parabola $x^2 = 4p y$ with focus at $(0,p)$ and directrix at $y=-p$

			\medskip
			
\item		The hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$
			with foci at $(\pm c,0)$ where $c^2=a^2+b^2$
\end{itemize}

\end{minipage}

\end{frame}

\begin{frame}
\begin{minipage}{0.40\textwidth}
\begin{tikzpicture}[scale=0.2]
\drawxyz{6}
\begin{scope}
%% clip to coordinate box
\clip	(0,0,5)	--	++(5,0,0)	--	++(0,0,-10)	--	++	(0,5,0)	--	
						++(-5,0,0)	--	++(0,0,10)		--	++(0,-5,0);
%% left bounding curve
\draw[cyan,smooth]	plot[domain=-5:5]	({0},{\x*\x},{\x});
%% right bounding curve
\draw[cyan,smooth]	plot[domain=-2.236:2.236]	(5,{\x*\x},{\x});
%% rulins for \pm \sqrt{i}, i=1:5
\foreach \x in {1,-1,1.414,-1.414,1.732,-1.732,2,-2,2.236,-2.236}
	\draw[black!50]		(0,{\x*\x},{\x})	--	++(5,0,0);
%% shade sides of parabolic cylinder
\shade[top color=blue!10,bottom color=blue,opacity=0.5]
	plot[domain=0:2.236]	({0},{\x*\x},{\x})	--	++(5,0,0)	--	
	plot[domain=2.236:0]	({5},{\x*\x},{\x})	--	++(-5,0,0);
\end{scope}
\end{tikzpicture}

\medskip

\begin{tikzpicture}[scale=0.2]
\drawxyz{6}
%% plot ellipses at $z=i, i=0,1,2,3,4,5$
\foreach		\i	in	{0.707,1,1.225,1.414,1.5811,1.722,1.87,2} 
	{
		\draw[gray!50] 	plot[domain=0:6.2832]	({2*\i*sin(\x r)}, {\i*\i},  {\i*cos(\x r)} ) ;
	}
%% plot parabolas by stepping in y; z = x^2 + y^2/2
\foreach 	\y	in	{-3.5,-3,-2.5,-2,-1.5,-1,-0.5,0,0.5,1,1.5,2,2.5,3,3.5}
	{
		\begin{scope}
		\clip	((\y,0,6)	--	++(0,4,0)	--	++(0,0,-12)	--	++(0,-4,0)	--	++(0,0,12);
		\draw[gray!50]	plot[domain=-2:2]	({\y},{\x*\x + 0.25*\y*\y},{\x});
		\end{scope}
	}
%% actually the results are a bit gruesome, so shade in the front part
\shade[left color=blue!90,right color=blue!10,opacity=0.8]
	plot[domain=-1.57:1.57,smooth]		({4*sin(\x r)}, {4},  {2*cos(\x r)})		--
	plot[domain=4:-4,smooth]				({\x}  ,{0.25*\x*\x}  , {0});
\end{tikzpicture}

\medskip

\begin{tikzpicture}[scale=0.15]
\drawxyz{9}
%% plot downward parabolas
\begin{scope}
\clip	(-5,4,4)	--	++(0,0,-8)	--	++(8,0,0)	--	++(0,-8,0)	--	++(0,0,8)	--	++(-8,0,0)	--	++(0,8,0);
\foreach \y in {-5,-4,-3,-2,-1,0,1,2,3,4} 
	{
		\draw	[gray!50]	plot[domain=-4:4]	({\y}, {0.1*(\y*\y-\x*\x)}, {\x}   );
	}
\foreach	\y	in	{-4,-3,-2,-1,0,1,2,3,4}
	{
		\draw[gray!50]	plot[domain=-5:4]	({\x}	,{0.1*(\x*\x-\y*\y)},	{\y});
	}
\end{scope}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.50\textwidth}
\footnotesize
In MA 213 we'll study \emph{quadric surfaces} in three dimensional space, such as:

\medskip

\begin{itemize}
\item		\emph{Cylinders} which consist of all lines that are parallel to a given line and pass through a given plane curve

\medskip

\item		\emph{Elliptic Paraboloids} which will model functions with local maxima or minima

\medskip

\item		\emph{Hyperbolic Paraboloids} (``saddles'') which model a new kind of critical point, called a \emph{saddle point},  for functions of two variables

%\medskip

%\item		\emph{Ellipsoids}

%\medskip

%\item		\emph{Hyperboloids}
\end{itemize}

\end{minipage}
\end{frame}

\section{Cylinders}

\begin{frame}
\frametitle{Cylinders}
\footnotesize
\fcolorbox{cyan}{white}
{
	\begin{minipage}{0.8\textwidth}
		A \textbf{cylinder} is a surface that consists of all lines (called \textbf{rulings}) that are parallel 			to a given line and pass through a given plane curve.
	\end{minipage}
}

\bigskip

\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}[scale=0.3]
%% shade interior of cylinder before drawing axes!
\shade[left color=blue!10,right color=blue!80,opacity=0.8]
	(0,0,8)	circle(2cm);	
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,15);
\draw[thick,->,>=stealth]	(0,0,0)	--	(6,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,6,0);
%%
\node[below]	at	(0,0,15)		{\scriptsize{$x$}};
\node[right]	at	(6,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,6,0)		{\scriptsize{$z$}};
%%
%%		bounding circles
\draw	[cyan]	(0,0,0)	circle(2cm);
\draw[cyan]	(0,0,8)	circle(2cm);

%% shade outside
\shade[left color=blue!80,right color=cyan!10]
	plot[domain=-40:130,variable=\theta]	({2*cos(\theta)}, {2*sin(\theta)}, {8})	--	++(0,0,-8)	--
	plot[domain=130:-40,variable=\theta]	({2*cos(\theta)}, {2*sin(\theta)}, {0})	--	++(0,0,8);
%% rulings
\foreach \theta in {-40,0,30,60,90,130}
	\draw		({2*cos(\theta)},{2*sin(\theta)},{0})	--	++(0,0,8);
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}
\footnotesize
\textbf{Example 1}: $y^2+z^2 =1$

\begin{itemize}
\item	What is the given curve?
\item	What is the given line?
\end{itemize}
\end{minipage}

\end{frame}


\begin{frame}
\frametitle{Cylinders}
\footnotesize
\fcolorbox{cyan}{white}
{
	\begin{minipage}{0.8\textwidth}
		A \textbf{cylinder} is a surface that consists of all lines (called \textbf{rulings}) that are parallel 			to a given line and pass through a given plane curve.
	\end{minipage}
}

\bigskip

\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}[scale=0.3]
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,15);
\draw[thick,->,>=stealth]	(0,0,0)	--	(6,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,6,0);
%%
\node[below]	at	(0,0,15)		{\scriptsize{$x$}};
\node[right]	at	(6,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,6,0)		{\scriptsize{$z$}};
%% bounding parabolas
\draw[cyan]	plot[domain=-4:4]	({\x},{0.25*\x*\x},{0});
\draw[cyan]	plot[domain=-4:4]	({\x},{0.25*\x*\x},{9.5});
%% top and bottom rulings
\foreach \y in {-4,4}
	\draw[cyan]		({\y}, {0.25*\y*\y}, {0})	--	++(0,0,9.5);
%% rulings
\foreach \y in {-3,-2,-1,2,3}
	\draw		({\y}, {0.25*\y*\y}, {0})	--	++(0,0,9.5);
%% shade visible interior with cheap clipping trick
\shade[left color=blue!10,right color=blue!90,opacity=0.5]
	plot[domain=-4:4]	({\x},{0.25*\x*\x},9.5)	--	plot[domain=0:-4]	({\x},{0.25*\x*\x},0)	--
		++(0,0,9.5);
%% color visible exterior uniformly dark
\shade[blue,opacity=0.8]	(2,1,9.5)	--	(2,1,0)	--	plot[domain=2:4]	({\x},{0.25*\x*\x},0)	--	
++(0,0,9.5)	-- plot[domain=4:2]	({\x},{0.25*\x*\x},9.5);
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}
\footnotesize
\textbf{Example 2}: $z=y^2$

\begin{itemize}
\item	What is the given curve?
\item	What is the given line?
\end{itemize}
\end{minipage}

\end{frame}

\section{Quadric Surfaces}

\begin{frame}
\frametitle{Quadric Surfaces}
\footnotesize
\begin{center}
\fcolorbox{cyan}{white}
{
\begin{minipage}{0.8\textwidth}
A \textbf{quadric surface} is the graph of a second-degree equation in $x$, $y$, and $z$
taking one of the standard forms
$$ Ax^2 + By^2 + Cz^2 + J = 0, \quad Ax^2 +By^2 + Iz  = 0 $$
\end{minipage}
}
\end{center}

\medskip
We can graph a quadric surface by studying its \emph{traces} in planes parallel to the $x$, $y$, and $z$ axes.  The traces are always quadratic curves!

\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}[scale=0.4]
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,5);
\draw[thick,->,>=stealth]	(0,0,0)	--	(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,5,0);
%%
\node[below]	at	(0,0,5)		{\scriptsize{$x$}};
\node[right]	at	(5,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,5,0)		{\scriptsize{$z$}};
%%
\draw[cyan] 				plot[domain=0:360,smooth]	({4*cos(\x)},{1.414*sin(\x)},{0});
\draw[cyan,dashed] 	plot[domain=0:360,smooth]	({4*cos(\x)},0,{sin(\x)});
\shade[ball color=blue!60,opacity=0.5]	plot[domain=0:360,smooth]	({4*cos(\x)},{1.414*sin(\x)},{0});
\end{tikzpicture}

\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}
In what follows we'll use the method of traces to graph the \emph{ellipsoid}
$$ x^2 + \frac{y^2}{4}  + \frac{z^2}{2} = 1 $$
\end{minipage}

\end{frame}

\begin{frame}
\frametitle{Interlude: A Buddhist Parable}
\scriptsize
\begin{center}
\includegraphics[scale=1.7]{Images/1024px-Blind_monks_examining_an_elephant.jpg}
\end{center}

%% Image credit: Wikipedia Commons
%% https://commons.wikimedia.org/wiki/File:Blind_monks_examining_an_elephant.jpg

\begin{quotation}
A group of blind men heard that a strange animal, called an elephant, had been brought to the town, but none of them were aware of its shape and form. Out of curiosity, they said: "We must inspect and know it by touch, of which we are capable". So, they sought it out, and when they found it they groped about it. In the case of the first person, whose hand landed on the trunk, said "This being is like a thick snake". For another one whose hand reached its ear, it seemed like a kind of fan. As for another person, whose hand was upon its leg, said, the elephant is a pillar like a tree-trunk. The blind man who placed his hand upon its side said, "elephant is a wall". Another who felt its tail, described it as a rope. The last felt its tusk, stating the elephant is that which is hard, smooth and like a spear.
\end{quotation}
\end{frame}

\begin{frame}
\frametitle{Graphing the Ellipsoid by Traces}
\footnotesize
\begin{center}
$$ x^2 + \frac{y^2}{4} + \frac{z^2}{2} = 1 $$
\end{center}
\begin{minipage}{0.35\textwidth}
\begin{tikzpicture}[scale=0.4]
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,5);
\draw[thick,->,>=stealth]	(0,0,0)	--	(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,5,0);
%%
\node[below]	at	(0,0,5)		{\scriptsize{$x$}};
\node[right]	at	(5,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,5,0)		{\scriptsize{$z$}};
%%
\uncover<2->
{
\draw[cyan,dashed] 	plot[domain=0:360,smooth]	({4*cos(\x)},0,{sin(\x)});
}
\only<2>
{
\shade[opacity=0.5]		(0,0,0)	--	++(0,0,5)	--	++(5,0,0)	--	++(0,0,-5)	--	++(-5,0,0);
}
%%
\only<3>
{
\shade[opacity=0.5]		(0,0,0)	--	++(5,0,0)	--	++(0,5,0)	--	++(-5,0,0)	--	++(0,-5,0);
}
\uncover<3->
{
\draw[cyan,dashed] 				plot[domain=0:360,smooth]	({4*cos(\x)},{1.414*sin(\x)},{0});
}
\only<4>
{
\shade[opacity=0.5]	(0,0,0)	--	++(0,0,5)	--	++(0,5,0)	--	++(0,0,-5)	--	++(0,-5,0);
}
\uncover<4->
{
\draw[dashed,cyan]				plot[domain=0:360,smooth]	(0,{1.414*sin(\x)},{cos(\x)});
}
\uncover<5->
{
\shade[ball color=blue!60,opacity=0.5]	plot[domain=0:360,smooth]	({4*cos(\x)},{1.414*sin(\x)},{0});
}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.55\textwidth}
\footnotesize
\uncover<2->
{
Trace in the  $xy$-plane ($z=0$):
$$ x^2 + \frac{y^2}{4} = 1 $$
}

\smallskip

\uncover<3->
{
Trace in the $yz$-plane ($x=0$):
$$ \frac{y^2}{4} + \frac{z^2}{2} = 1 $$
}

\smallskip

\uncover<4->
{
Trace in the $xz$-plane ($y=0$):
$$ x^2 + \frac{z^2}{2} = 0 $$
}
\end{minipage}
\end{frame}


%% Traces parallel to xy plane

\begin{frame}
\footnotesize
\frametitle{More on Traces}
\footnotesize
It often helps to find the traces in planes \emph{parallel} to the $xy$, $xz$, or $yz$ planes. 
$$ x^2 + \frac{y^2}{4} + \frac{z^2}{2} = 1 $$

\medskip

\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}[scale=0.35]
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,5);
\draw[thick,->,>=stealth]	(0,0,0)	--	(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,5,0);
%%
\node[below]	at	(0,0,5)		{\scriptsize{$x$}};
\node[right]	at	(5,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,5,0)		{\scriptsize{$z$}};
%%
%%  Traces in xy plane
\foreach \k in { -1.4,-1, -0.5, 0, 0.5, 1.0,1.4}
	{
		\draw[gray!50] 
			plot[domain=0:360,smooth] ({(4-2*\k*\k)*cos(\x)},{\k},{(1-0.5*\k*\k)*sin(\x)});
	}
%%
%% Bottom bounding plane
\only<2>
{
\draw[dashed]	(0,0,0)	--	(0,-1.414,0);
\draw[dashed]			(0,-1.414,5)		--	++(0,0,-10);
\draw[dashed]			(-5,-1.414,0)	--	++(10,0,0);
\shade[opacity=0.3]	(-5,-1.414,5)	--	++(0,0,-10)	--	++(10,0,0)	--	++(0,0,10)	--	++(-10,0,0);
}
%% Top bounding plane
\only<3>
{
\shade[opacity=0.3]	(-5,1.414,5)	--	++(0,0,-10)	--	++(10,0,0)	--	++(0,0,10)	--	++(-10,0,0);
\draw[dashed]			(0,1.414,5)		--	++(0,0,-10);
\draw[dashed]			(-5,1.414,0)	--	++(10,0,0);
}

\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}
\footnotesize
Trace in planes $z=k$:
\begin{align*}
x^2 + \frac{y^2}{4} + \frac{k^2}{2}&=1\\
x^2 + \frac{y^2}{4}	&=	1-	\frac{k^2}{2}\\
\frac{x^2}{1-k^2/2} + \frac{y^2}{4-2k^2} &=	1
\end{align*}

\medskip

\begin{itemize}
\item		Traces are ellipses
\item		No trace if $|k| > \sqrt{2}$	
\end{itemize}

\end{minipage}
\end{frame}

%% Traces parallel to yz plane

\begin{frame}
\footnotesize
\frametitle{More on Traces}
\footnotesize
It often helps to find the traces in planes \emph{parallel} to the $xy$, $xz$, or $yz$ planes. 
$$ x^2 + \frac{y^2}{4} + \frac{z^2}{2} = 1 $$

\medskip

\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}[scale=0.35]
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,5);
\draw[thick,->,>=stealth]	(0,0,0)	--	(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,5,0);
%%
%
%%
\node[below]	at	(0,0,5)		{\scriptsize{$x$}};
\node[right]	at	(5,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,5,0)		{\scriptsize{$z$}};
%%
%%  Traces in yz plane
\foreach \k in { -1,-0.75, 0,0.75,1.0}
	{
		\draw[gray!50] 
			plot[domain=0:360,smooth] ({(4-4*\k*\k)*cos(\x)},{(2-2*\k*\k)*sin(\x)},{\k});
	}
%%
%% Back bounding plane - redraw traces as darker and color ellipsoid
\only<2>
{
\shade[opacity=0.5,top color=black,bottom color=white]	(-5,5,-1)	--	++(10,0,0)	--	++(0,-10,0)	--	++(-10,0,0)	--	++(0,10,0);
\draw[dashed]			(-5,0,-1)	--	++(10,0,0);
\draw[dashed]			(0,-5,-1)	--	++(0,10,0);
\draw[dashed]			(0,0,0)	--	(0,0,-5);
%%
\foreach \k in { -1,-0.75, 0,0.75,1.0}
	{
		\draw[blue!70] 
			plot[domain=0:360,smooth] ({(4-4*\k*\k)*cos(\x)},{(2-2*\k*\k)*sin(\x)},{\k});
	}
\shade[ball color=cyan,opacity=0.3]	
			plot[domain=0:360,smooth]	({3.99*cos(\x)}, {1.99*sin(\x)},{0});
}

%% Front bounding plane
\only<3>
{
\shade[ball color=blue,opacity=0.3]	
			plot[domain=0:360,smooth]	({3.99*cos(\x)}, {1.99*sin(\x)},{0});
\shade[opacity=0.5]	(-5,5,1)	--	++(10,0,0)	--	++(0,-10,0)	--	++(-10,0,0)	--	++(0,10,0);
\draw[dashed]			(-5,0,1)	--	++(10,0,0);
\draw[dashed]			(0,-5,1)	--	++(0,10,0);
\draw[dashed]			(0,0,0)	--	(0,0,-5);
}
\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}
\footnotesize
Trace in planes $x=k$:
\begin{align*}
k^2 + \frac{y^2}{4} +\frac{z^2}{2} &=	1\\
\frac{y^2}{4}+\frac{z^2}{2}			&=	1-k^2\\
\frac{y^2}{4-4k^2} + \frac{z^2}{2-2k^2} 	&=	1
\end{align*}

\medskip

\begin{itemize}
\item		Traces are ellipses
\item		No trace if $|k| > 1$	
\end{itemize}

\end{minipage}
\end{frame}

%%	Paraboloid

\begin{frame}
\frametitle{Elliptic Paraboloid}
\footnotesize
We'll use the method of traces to graph the surface
$$ z = x^2 + y^2 $$

\medskip
\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}[scale=0.4]
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,5);
\draw[thick,->,>=stealth]	(0,0,0)	--	(5,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,6,0);
%%
\node[below]	at	(0,0,5)		{\scriptsize{$x$}};
\node[right]	at	(5,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,6,0)		{\scriptsize{$z$}};
%%
\only<2>
{
\foreach \z in {0, 0.707, 1.414, 2 }
	{
		\draw[gray!50] 	plot[domain=0:360]		({\z*cos(\x) },{\z*\z},{\z*sin(\x)});
	}
\shade[opacity=0.7]	(0,0,0)	--	++(0,0,5)	--	++(5,0,0)	--	++(0,0,-5)	--	++(-5,0,0);
}
\only<3>
{
\foreach \y in {-2,-1,0,1,2}
	{
		\draw[gray!50]	plot[domain=-2:2]	({\y},{\y*\y+\x*\x},{\x});
		\draw[cyan,fill=cyan]	({\y},{\y*\y},0) circle(0.1cm);
	}
\draw[cyan,dashed]	plot[domain=-2.5:2.5]	({\x},{\x*\x},{0});
}
\only<4>
{
%% top bounding circle
\draw[cyan]	plot[domain=0:360,smooth]		({2*cos(\x)},4,{2*sin(\x)});
%% trace in plane y=0
\draw[cyan,dashed]	plot[domain=-2:2,smooth]			(0,{\x*\x},{\x});
%% trace in plane z=sqrt{2}
\draw[cyan,dashed]	plot[domain=0:360,smooth]			({1.414*cos(\x)},2,{1.414*sin(\x)});
%% bounding visible curves by eyeball
\draw[cyan]	plot[domain=0:2]	(-{\x*0.819 },{\x*\x}, {0.574*\x });	%% -55^o
\draw[cyan]	plot[domain=0:2]	({0.913*\x},{\x*\x}, {-0.407*\x });	%%	114^o
%% shade exterior
\shade[left color=blue,right color=white,opacity=0.5]
	plot[domain=-55:114]	({2*sin(\x)},4,{2*cos(\x)})	--	
	plot[domain=2:0,red]	({0.913*\x},{\x*\x}, {-0.407*\x })	--
	plot[domain=0:2,red]	(-{\x*0.819 },{\x*\x}, {0.574*\x });
%% shade interior a bit lighter
\shade[left color=blue,right color=white,opacity=0.2]
	plot[domain=0:360]		({2*cos(\x)},{4},{2*sin(\x)});
}

\end{tikzpicture}
\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}
\footnotesize
\only<2>
{
Traces in planes $z=k$:
$$ x^2 + y^2 = k $$

\medskip

\begin{itemize}
\item	No traces below $z=0$
\item	Traces are circles of radius $\sqrt{k}$ and center $(0,0,k)$
\end{itemize}
}
\only<3>
{
Traces in planes $y=k$:
$$ x^2 + k^2= z $$

\medskip

\begin{itemize}
\item	No restriction on traces
\item	Traces are upward parabolas with vertex $(0,k,k^2)$
\end{itemize}
}
\only<4>
{Putting it all together$\ldots$

\medskip

\begin{itemize}
\item		Traces parallel to $xy$ plane are circles
\item		Traces parallel $yz$ or $xz$ plane are parabolas
\end{itemize}
}
\end{minipage}

\end{frame}

%%  	Saddle

\begin{frame}
\frametitle{Hyperbolic Paraboloid (Saddle)}
\footnotesize
We'll use the method of traces to graph the surface
$$ z= y^2 - x^2 $$

\bigskip

\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}[scale=0.25]
%%
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,10);
\draw[thick,->,>=stealth]	(0,0,0)	--	(10,0,0);
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,6,0);
%%
\node[below]	at	(0,0,10)		{\scriptsize{$x$}};
\node[right]	at	(10,0,0)		{\scriptsize{$y$}};
\node[above]	at	(0,6,0)		{\scriptsize{$z$}};
%%
\only<2-3>
{
%% hyperbolas above
\uncover<2-3>
{
\foreach \z in {1,2,3}
{
	\begin{scope}
		\clip	(-7,\z,5)	--	++(0,0,-10)	--	++(14,0,0)	--	++(0,0,10)	--	++(-14,0,0);
		\draw[gray!50]	plot[domain=-3:3]	({(sqrt(\z)*cosh(\x)},\z,{sqrt(\z)*sinh(\x)});
		\draw[gray!50]	plot[domain=-3:3]	({-sqrt(\z)*cosh(\x)},\z,{sqrt(\z)*sinh(\x)});
	\end{scope}
}
\draw	[cyan,dashed]	plot[domain=-2:2]	({\x},{\x*\x},0);
%% lines on xy plane
\draw[gray!50]	(5,0,-5)	 --	++	(-10,0,10);
\draw[gray!50]	(5,0,5)	--	++(-10,0,-10);
}
\uncover<3-3>
{
%% hyperbolas below xy plane
\foreach \z in {1,2,3}
{
	\begin{scope}
		\clip	(-7,-\z,5)	--	++(0,0,-10)	--	++(14,0,0)	--	++(0,0,10)	--	++(-14,0,0);
		\draw[gray!50]	plot[domain=-3:3]	({(sqrt(\z)*sinh(\x)},-\z,{sqrt(\z)*cosh(\x)});
%% crude hidden line removal!
%		\draw[gray!50]	plot[domain=-3:3]	({sqrt(\z)*sinh(\x)},-\z,{-sqrt(\z)*cosh(\x)});
	\end{scope}
}
\draw[dashed]	(0,0,0)	--	++(0,-5,0);
\draw[cyan,dashed]	plot[domain=-2:2]	(0,{-\x*\x},{\x});
}
}
\only<4>
{
\begin{scope}
\foreach \y in {-3,-2,-1,1,2,3}
	\draw[gray!50]	plot[domain=-2.5:2.5]		({\y},{\y*\y - \x*\x},{\x});
\end{scope}
\draw[cyan,dashed]	plot[domain=-3:3]		({\x} ,{\x*\x}  , 0);
}
\only<5>
{
%%%%%%%%%%%%%%%%%%%%%%%%
%
%		This code replicates 2-3-4
%		There is likely a smarter way to do this
%
%%%%%%%%%%%%%%%%%%%%%%%%
%% hyperbolas above
\foreach \z in {1,2,3,4}
{
	\begin{scope}
		\clip	(-7,\z,5)	--	++(0,0,-10)	--	++(14,0,0)	--	++(0,0,10)	--	++(-14,0,0);
		\draw[gray!50]	plot[domain=-3:3]	({(sqrt(\z)*cosh(\x)},\z,{sqrt(\z)*sinh(\x)});
		\draw[gray!50]	plot[domain=-3:3]	({-sqrt(\z)*cosh(\x)},\z,{sqrt(\z)*sinh(\x)});
	\end{scope}
}
\draw	[cyan,dashed]	plot[domain=-2:2]	({\x},{\x*\x},0);
%% lines on xy plane
\draw[gray!50]	(5,0,-5)	 --	++	(-10,0,10);
\draw[gray!50]	(5,0,5)	--	++(-10,0,-10);
%% hyperbolas below xy plane
\foreach \z in {1,2,3}
{
	\begin{scope}
		\clip	(-7,-\z,5)	--	++(0,0,-10)	--	++(14,0,0)	--	++(0,0,10)	--	++(-14,0,0);
		\draw[gray!50]	plot[domain=-3:3]	({(sqrt(\z)*sinh(\x)},-\z,{sqrt(\z)*cosh(\x)});
%% crude hidden line removal!
%		\draw[gray!50]	plot[domain=-3:3]	({sqrt(\z)*sinh(\x)},-\z,{-sqrt(\z)*cosh(\x)});
	\end{scope}
}
\draw[dashed]	(0,0,0)	--	++(0,-5,0);
\draw[cyan,dashed]	plot[domain=-2:2]	(0,{-\x*\x},{\x});
%% parabolas  - only show visible side
\foreach \y in {-2.5,-2,-1.5,-1,-0.5,0.5,1,1.5,2,2.5}
	\draw[gray!50]	plot[domain=0:2.5]		({\y},{\y*\y - \x*\x},{\x});
}
\end{tikzpicture}

\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}
\only<2-3>
{
Traces in plane $z=k$:
$$ y^2 - x^2 = k$$

\medskip

Hyperbolas:
\begin{itemize}
\uncover<2-3>
{
\item		Opening in the $y$ direction, $k>0$
}
\uncover<3-3>
{
\item		Opening in the $x$ direction, $k<0$
}
\end{itemize}
}

\only<4>
{
Traces in plane $y=k$:
$$ z = k^2 - x^2 $$

\medskip

Downward parabolas,
vertex at $(k^2,0)$
}
\only<5>
{
Putting it all together$\ldots$

\medskip

\begin{itemize}
\item	Traces parallel to $xy$ plane are hyperbolas
\item	Traces parallel to $xz$ plane are downward parabolas
\end{itemize}

\vfill
}
\end{minipage}
\end{frame}

\begin{frame}
\frametitle{Hyperboloid of One Sheet}
\footnotesize
We'll use the method of traces to graph the surface
$ x^2 + y^2 - z^2 = 1 $.

\medskip

\begin{minipage}{0.45\textwidth}
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords,scale=0.4]
\draw[thick,->,>=stealth]	(0,0,0)	--	(7,0,0) 	node[anchor = north east] 	{\footnotesize{$x$}};
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,5,0)	node[anchor = west]			{\footnotesize{$y$}};
\draw[thick,->,>=stealth]	(0,0,0)	--	(0,0,5)	node[anchor = south]			{\footnotesize{$z$}};
%% uncover a single circle. Note we scale down the radius by 0.5 to fit the picture
\only<2>
{
\draw[red!50,fill=red!50,opacity=0.5]	(-4,-4,2)	--	(-4,4,2)	--	(4,4,2)	--	(4,-4,2)	--	cycle;
\draw[red,thick]	plot[domain=0:360,variable=\t]		({0.5*sqrt{17}*cos(\t)},{0.5*sqrt(17)*sin(\t)},2);
}
\only<3>
{
%% Aargh! Hard-coded because I couldn't get for loop to work
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({3.64*cos(\t)},{3.64*sin(\t)},-3.5);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({3.16*cos(\t)},{3.16*sin(\t)},-3);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({2.69*cos(\t)},{2.69*sin(\t)},-2.5);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({2.23*cos(\t)},{2.23*sin(\t)},-2);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({1.81*cos(\t)},{1.81*sin(\t)},-1.5);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({1.41*cos(\t)},{1.41*sin(\t)},-1);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({1.12*cos(\t)},{1.12*sin(\t)},-0.5);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({cos(\t)},{sin(\t)},0);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({3.64*cos(\t)},{3.64*sin(\t)},3.5);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({3.16*cos(\t)},{3.16*sin(\t)},3);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({2.69*cos(\t)},{2.69*sin(\t)},2.5);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({2.23*cos(\t)},{2.23*sin(\t)},2);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({1.81*cos(\t)},{1.81*sin(\t)},1.5);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({1.41*cos(\t)},{1.41*sin(\t)},1);
\draw[red!50,opacity=0.5]
	plot[domain=0:360,variable=\t]		({1.12*cos(\t)},{1.12*sin(\t)},0.5);
}

\end{tikzpicture}

\end{minipage}
\qquad
\begin{minipage}{0.45\textwidth}

Rewrite the equation as:
$$ x^2 + y^2 = 1 + z^2$$

\medskip
What are the traces in the plane $z=k$?

\medskip

\uncover<2->
{
	Equation:  $x^2 + y^2 = 1 + k^2$
	
	\medskip
	
	Curve: Circles with center at $(0,0,k)$ of radius $\sqrt{1+k^2}$
	
	\medskip
}

\uncover<3->
{
	Surface:  \emph{hyperboloid of one sheet}
	whose axis of symmetry is the $z$ axis
}

\end{minipage}

\end{frame}


\section{Summary}

\begin{frame}
\frametitle{Summary}
\footnotesize
\begin{itemize}
\item	We recalled the equations and graphs of ellipses, parabolas, and hyperbolas from MA 114
\medskip
\item	We learned how to graph equations of cylinders like $z=y^2$ or $x^2+z^2=1$
\medskip
\item	We learned how to find \emph{traces} of quadric surfaces by restricting to planes
$x=h$, $y=k$, or $z=\ell$
\medskip
\item	We learned how to visualize a quadric surface given its traces
\medskip
\item	We discussed four important quadric surfaces:
\medskip
		\begin{itemize}
		\item 	The ellipsoid (example: $x^2/9 + y^2/4 + z^2 = 1$)
		\smallskip
		\item	The elliptic paraboloid (example: $z= x^2 + y^2$)
		\smallskip
		\item	The ``saddle" (example: $z=x^2-y^2$)
		\smallskip
		\item	The hyperboloid of one sheet (example: $x^2 + y^2 - z^2 = 1$)
		\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Homework}
\begin{itemize}
\item	Re-read section 12.6 and continue work on webwork A4 (due Friday)
\item Read and study section 13.1 on vector functions and space curves
\item	Prepare for tomorrow's recitation on section 12.6
\item	Prepare for tomorrow's quiz on sections 12.3-12.4
\end{itemize}
\end{frame}

\endgroup

\end{document}

%% hiding place

