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


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{sw20jart}

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

%TCIDATA{<META NAME="ViewSettings" CONTENT="31">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="3D Plots of Functions and Expressions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:04:50 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{<LINK REL="begin" HREF="DM6-1.tex">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="end" HREF="DM6-7.tex">}
%TCIDATA{<LINK REL="next" HREF="DM6-7.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM6.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM6-5.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}
%TCIDATA{MapleDefs=
%$\rho  \left( \phi ,\theta \right) =\left(  1.2\right)  ^{\phi }\sin \theta  $
%}


\input{tcilatex}
\begin{document}


\section{3D Plots\label{3D Plots} of Functions and Expressions}

You can plot a wide variety of surfaces with \textsf{Plot 3D}. This section
provides examples of \textsf{Rectangular}, \textsf{Cylindrical}, \textsf{%
Spherical}, and \textsf{Tube} plots. The details of \textsf{Gradient} and 
\textsf{Vector Field} plots are provided under \hyperref{Vector Calculus}{}{%
}{DM9.tex}.

The \textsf{Maple Settings} default in \textsl{Scientific Notebook} displays
a reduced \textsf{Maple} menu that shows only \textsf{Rectangular} on the 
\textsf{Plot 3D} submenu. To use the other 3D plotting techniques, go to 
\textsf{Maple Settings}, \textsf{General} page, and check \textsf{Display
Full Maple Menu}.

\strut

\subsection{\ Expressions}

\begin{quote}
$\blacktriangleright $ To plot one or more expressions involving two
variables
\end{quote}

\begin{enumerate}
\item[1.]  Enter the expression(s) in your document. If you want to plot
several expressions at once, enter them separated by (red) commas.

\item[2.]  With the insertion point in the expression, from the \textsf{Plot
3D} submenu 
\index{Plot 3D@Plot 3D!rectangular@rectangular}\label{Plot 3D, rectangular}%
choose \textsf{Rectangular}.\medskip 
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To add 
\index{Add plot@Add plot}\label{Add expression to Plot 3D} \label%
{Plot 3D, multiple expressions}an expression involving two variables to a 3D
plot
\end{quote}

\begin{enumerate}
\item[1.]  Select the expression.

\item[2.]  Drag the expression onto the plot.\medskip 
\end{enumerate}

or\medskip

\begin{enumerate}
\item[1.]  Open the \textsf{Plot Properties + Plot Components }tabbed dialog.

\item[2.]  Click \textsf{Add Item}.

\item[3.]  Type or \textsf{Paste} the expression in the \textsf{Item}
box.\medskip
\end{enumerate}

\subsection{\ Defined Functions\label{Defined functions}}

You can plot a defined function of two variables in two different ways.
Recall that you define a function such as 
\[
f(x,y)=%
\dfrac{xy}{\left( x^{2}+y^{2}\right) ^{2}} 
\]
by placing the insertion point in the expression and choosing \textsf{New
Definition} from the \textsf{Define} submenu.

\begin{quote}
$\blacktriangleright $ To plot a defined function $f$ of two variables
\end{quote}

\begin{enumerate}
\item[1.]  Select the function name $f$ or select the expression $f(x,y)$.

\item[2.]  From the \textsf{Plot 3D} submenu, choose \textsf{Rectangular}%
.\medskip
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To add a defined function $g$ of two variables to a
3D plot
\end{quote}

\begin{enumerate}
\item[1.]  Select the function name $g$ or select the expression $g(x,y)$.

\item[2.]  Drag your selection onto the plot.\medskip
\end{enumerate}

For the example that follows, define $f(x,y)=\frac{xy}{\left(
x^{2}+y^{2}\right) ^{2}}$ and $g(x,y)=-5$. This example shows 3D rectangular
plots of $f(x,y)$ and of both $f(x,y)$ and $g(x,y)$, with \textsf{Ranges} $%
-1\leq x\leq 1$ and $-1\leq y\leq 1$, and \textsf{View Intervals} $-1\leq
x\leq 1$, $-1\leq y\leq 1$, and $-5\leq z\leq 5.$ The default \textsf{View
Intervals} for $z$ produces a plot that contains only a single vertical
line. In the \textsf{Plot Properties + Axes \& View} tabbed dialog,
deactivate\textsf{\ Default} under \textsf{View Intervals}, and change the
interval for $z$ to get the plots shown in the example.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Rectangular}
\end{quote}

\begin{quotation}
$f(x,y)$ \qquad Add $g(x,y)$ for the second plot.
\end{quotation}

\begin{center}
\FRAME{itbpF}{1.9995in}{1.6994in}{0in}{}{}{Figure }{\special{language
"Scientific Word";type "MAPLEPLOT";width 1.9995in;height 1.6994in;depth
0in;display "PICT";plot-snapshots TRUE;function \TEXUX{$\dfrac{xy}{\left(
x^{2}+y^{2}\right) ^{2}}$};linecolor "black";linestyle 1;linethickness
1;pointstyle "point";xmin "-1";xmax "1";ymin "-1";ymax "1";xviewmin
"-1";xviewmax "1";yviewmin "-1";yviewmax "1";zviewmin "-5";zviewmax
"5";viewset"XYZ";rangeset"XYZ";phi 60;theta 30;plottype 5;num-x-gridlines
25;num-y-gridlines 25;plotstyle "hidden";axesstyle "none";plotshading
"Z";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};}} 
\FRAME{itbpF}{1.9977in}{1.676in}{0in}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 1.9977in;height 1.676in;depth 0in;display
"PICT";maintain-aspect-ratio TRUE;plot-snapshots TRUE;function
\TEXUX{$\frac{xy}{\left( x^{2}+y^{2}\right) ^{2}}$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$-5$};linecolor "black";linestyle 1;linethickness 1;pointstyle
"point";xmin "-1";xmax "1";ymin "-1";ymax "1";xviewmin "-1";xviewmax
"1";yviewmin "-1";yviewmax "1";zviewmin "-5";zviewmax
"5";viewset"XYZ";rangeset"XYZ";phi 60;theta 30;plottype 5;num-x-gridlines
25;num-y-gridlines 25;plotstyle "hidden";axesstyle "none";plotshading
"Z";xis \TEXUX{x};yis \TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};}}
\end{center}

\subsection{\ \label{Parameterized surfaces, rectangular}Parametric Plots}

\textsl{Parameterized surfaces} in rectangular coordinates are given by 
\index{Parameterized surfaces@Parameterized surfaces!rectangular coordinates@rectangular coordinates}%
equations of the form $x=f(s,t)$, $y=g(s,t)$ and $z=h(s,t)$. These are very
general and allow you to generate a wide variety of interesting plots.

\begin{quote}
$\blacktriangleright $ To plot a parameterized surface
\end{quote}

\begin{enumerate}
\item[1.]  Enter expressions in a vector, making each expression a separate
component.

\item[2.]  With the insertion point in the vector, from the \textsf{Plot 3D}
submenu, choose \textsf{Rectangular.}
\end{enumerate}

\strut

The parameterized surface defined by the equations

\begin{eqnarray*}
x &=&s\sin s\cos t \\
y &=&s\cos s\cos t \\
z &=&s\sin t
\end{eqnarray*}
can be created as the 3D rectangular plot of the vector $\left[ s\sin s\cos
t,s\cos s\cos t,s\sin t\right] $. In the following plots, $0\leq s\leq 2\pi $
and $0\leq t\leq \pi $.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Rectangular}
\end{quote}

\begin{quotation}
$\left[ s\sin s\cos t,s\cos s\cos t,s\sin t\right] $
\end{quotation}

\begin{center}
\FRAME{itbpFU}{2.0903in}{1.9977in}{0in}{\Qcb{Plot Style: Patch\protect%
\smallskip }}{}{Figure }{\special{language "Scientific Word";type
"MAPLEPLOT";width 2.0903in;height 1.9977in;depth 0in;display "PICT";function
\TEXUX{$\left[ s\sin s\cos t,s\cos s\cos t,s\sin t\right] $};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax
"6.28";ymin "0";ymax "3.14";xviewmin "-4.998";xviewmax "5.002";yviewmin
"-6.531";yviewmax "6.536";zviewmin "-0.1256";zviewmax
"6.408";rangeset"X";phi 45;theta 45;plottype 5;num-x-gridlines
25;num-y-gridlines 25;plotstyle "patch";axesstyle "none";plotshading "Z";xis
\TEXUX{s};yis \TEXUX{t};var1name \TEXUX{$s$};var2name \TEXUX{$t$};}}\ \FRAME{%
itbpFU}{2.092in}{1.9995in}{0in}{\Qcb{Plot Style: Wire Frame}}{}{Figure }{%
\special{language "Scientific Word";type "MAPLEPLOT";width 2.092in;height
1.9995in;depth 0in;display "PICT";function \TEXUX{$\left[ s\sin s\cos
t,s\cos s\cos t,s\sin t\right] $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "0";xmax "6.28";ymin "0";ymax
"3.14";xviewmin "-4.998";xviewmax "5.002";yviewmin "-6.531";yviewmax
"6.536";zviewmin "-0.1256";zviewmax "6.408";rangeset"X";phi 45;theta
45;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{s};yis
\TEXUX{t};var1name \TEXUX{$s$};var2name \TEXUX{$t$};}}
\end{center}

\strut

Although \textsf{Patch} produces high-quality graphics, certain surfaces,
such as the one here, can be easier to visualize in \textsf{WireFrame}%
.\smallskip

\subsection{\ \label{Curves in space}Curves in Space}

A space curve is defined by three functions $x=f(t)$, $y=g(t)$, $z=h(t)$ of
a single variable. You can create a ``fat curve'' by specifying a radius for
the curve in the \textsf{Plot Properties} dialog box. This radius can be
constant or can be a function of $t$. The \textsf{Sample Size} is the number
of computed points along the curve; the \textsf{Number of Tube Points} is
the number of computed points in a cross section of the tube. \textsf{Ranges}
refers to the range of computed values for the parameter $t$. The \textsf{%
View Intervals} include intervals for $x$, $y$, and $z$ of the form $%
x_{0}\leq x\leq x_{1}$, $y_{0}\leq y\leq y_{1}$, $z_{0}\leq z\leq z_{1}$.

\begin{quote}
$\blacktriangleright $ To plot a space curve\label{Space curve}
\end{quote}

\begin{enumerate}
\item[1.]  Enter the three defining expressions as the components of a
three-element vector.

\item[2.]  With the insertion point in the vector, from the \textsf{Plot 3D}
submenu 
\index{Plot 3D@Plot 3D!tube@tube} 
\index{Tube plots@Tube plots}\label{Tube plots} 
\index{Fat curves}choose \label{Fat curves}\textsf{Tube}. \label%
{Plot 3D, tube}

\item[3.]  To change the radius or the view, double-click the plot, click
revise, and change the settings.\medskip
\end{enumerate}

The ``fat curve'' is designed to show which parts of the curve are close to
the observer and which are far away. Otherwise, a curve in space is
difficult to visualize. In the following example, the radius is set to $1$
and $0\leq t\leq 6.28$ ($\approx 2\pi $).

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Tube}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{c}
-10\cos t-2\cos (5t)+15\sin (2t) \\ 
-15\cos (2t)+10\sin t-2\sin (5t) \\ 
10\cos (3t)
\end{array}
\right] $
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\left[
\MATRIX{1,3}{c}\VR{,,c,,,}{,,,,,}\HR{,,,}\CELL{-10\cos t-2\cos (5t)+15\sin
(2t)}\CELL{-15\cos (2t)+10\sin t-2\sin (5t)}\CELL{10\cos (3t)}\right]
$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";radius
\TEXUX{$1$};xmin "0";xmax "6.28";xviewmin "-22.51";xviewmax "22.48";yviewmin
"-16.76";yviewmax "24.78";zviewmin "-11.42";zviewmax "11.42";rangeset"X";phi
45;theta 45;plottype 13;num-x-gridlines 50;num-y-gridlines 10;plotstyle
"wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{t};}}You can draw a
``thin curve,'' by setting the radius to 0 in the dialog box.

\label{Surfaces of revolution}By typing an expression in $t$ for the radius
and choosing the curve to be a straight line, you can get surfaces of
revolution. 
\index{Surfaces of revolution@Surfaces of revolution}In the following
example, the radius is set to $1-\sin t$, the range for $t$ is $-2\pi \leq
t\leq 2\pi $, and the number of tube points is 30.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Tube}
\end{quote}

\begin{quotation}
$\left[ t,0,0\right] $
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\left[ t,0,0\right] $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";radius \TEXUX{$1-\sin t$};xmin
"-6.28";xmax "6.28";xviewmin "-6.531";xviewmax "6.536";yviewmin
"-2.068";yviewmax "2.081";zviewmin "-2.076";zviewmax "2.078";rangeset"X";phi
45;theta 45;plottype 13;num-x-gridlines 50;num-y-gridlines 30;plotstyle
"wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{t};}}

The spine of the surface of revolution can be any line, as illustrated by
the next example plotted with \textsf{Radius:} $4+\sin 3t+2\cos 5t$, \textsf{%
Axes: Frame}, \textsf{Style: Hidden Line}, and \textsf{t Range: } $-5\leq
t\leq 5$.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Tube}
\end{quote}

\begin{quotation}
$\left( 2t,-3t,t\right) $
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";plot-snapshots TRUE;function \TEXUX{$\left( 2t,-3t,t\right)
$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";radius
\TEXUX{$4+\sin 3t+2\cos 5t$};xmin "-5";xmax "5";xviewmin "-14.83";xviewmax
"16.21";yviewmin "-19.62";yviewmax "18.91";zviewmin "-10.9";zviewmax
"11.45";rangeset"X";phi 45;theta 45;plottype 13;num-x-gridlines
50;num-y-gridlines 10;plotstyle "hidden";axesstyle "frame";plotshading
"Z";xis \TEXUX{t};}}

\subsection{\ \label{Cylindrical coordinates}Cylindrical Coordinates}

\begin{quotation}
In the 
\index{Cylindrical coordinates@Cylindrical coordinates}\textsl{cylindrical
coordinate system}, a 
\index{Plot 3D@Plot 3D!cylindrical@cylindrical}point $P$ is \label%
{Plot 3D, cylindrical}represented by a triple $\left( r,\theta ,z\right) $,
where $\left( r,\theta \right) $ represents a point in polar coordinates and 
$z$ is the usual rectangular third coordinate. Thus, to convert from
cylindrical to rectangular coordinates, we use the equations 
\[
x=r\cos \theta \qquad y=r\sin \theta \qquad z=z 
\]
To go from rectangular to cylindrical coordinates, we use the equations 
\[
r^{2}=x^{2}+y^{2}\qquad \tan \theta =%
\dfrac{y}{x}\qquad z=z 
\]
\end{quotation}

The default assumption is that $r$ is a function of $\theta $ and $z$. As
usual, you can plot several surfaces on the same axes by dragging
expressions onto a plot.

\subsubsection{Expressions}

\begin{quote}
$\blacktriangleright $ To make a cylindrical plot of an expression
\end{quote}

\begin{enumerate}
\item[1.]  Enter the expression in your document.

\item[2.]  With the insertion point in the expression, from the \textsf{Plot
3D} submenu choose \textsf{Cylindrical}.\medskip
\end{enumerate}

The following example shows a plot of the cylinder $r=1$ obtained as the 3D
cylindrical plot of the expression $1$, with $0\leq \theta \leq 2\pi $, and $%
0\leq z\leq 1$.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Cylindrical}
\end{quote}

\begin{quotation}
$1$
\end{quotation}

\FRAME{dtbpF}{2.9992in}{1.9995in}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.9992in;height 1.9995in;depth
0pt;display "PICT";plot-snapshots TRUE;function \TEXUX{$1$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax
"6.28";ymin "0";ymax "1";xviewmin "-1.04";xviewmax "1.041";yviewmin
"-1.04";yviewmax "1.041";zviewmin "-0.02";zviewmax "1.02";phi 67;theta
39;plottype 14;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"hidden";axesstyle "none";plotshading "Z";xis \TEXUX{v627};yis
\TEXUX{z};var1name \TEXUX{$\theta $};var2name \TEXUX{$z$};}}

To get the following plot, use the ranges $0\leq \theta \leq 2\pi $ and $%
0\leq z\leq 1$.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Cylindrical}
\end{quote}

\begin{quotation}
$1-z$
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";plot-snapshots TRUE;function \TEXUX{$1-z$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax
"6.28";ymin "0";ymax "1";xviewmin "-1.04";xviewmax "1.041";yviewmin
"-1.04";yviewmax "1.041";zviewmin "-0.02";zviewmax "1.02";rangeset"XY";phi
45;theta 45;plottype 14;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"hidden";axesstyle "none";plotshading "Z";xis \TEXUX{v952};yis
\TEXUX{z};var1name \TEXUX{$\theta $};var2name \TEXUX{$z$};}}

You can plot more than one expression at once by separating the expressions
with a (red) comma. The following example shows a plot of the cylinders $r=1$
and $r=1-z$ obtained as the 3D cylindrical plot of the expressions $1$ and $%
1-z$, with $0\leq \theta \leq 2\pi $, and $0\leq z\leq 1$. To get this plot,
either plot both expressions together (as indicated), or drag one of the
expressions onto a plot of the other.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Cylindrical}
\end{quote}

\begin{quotation}
$1,1-z$
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";plot-snapshots TRUE;function \TEXUX{$1,1-z$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax
"6.28";ymin "0";ymax "1";xviewmin "-1.04";xviewmax "1.041";yviewmin
"-1.04";yviewmax "1.041";zviewmin "-0.02";zviewmax "1.02";rangeset"XY";phi
45;theta 45;plottype 14;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"wireframe";axesstyle "none";plotshading "Z";xis \TEXUX{v952};yis
\TEXUX{z};var1name \TEXUX{$\theta $};var2name \TEXUX{$z$};}}

\subsubsection{Defined Functions}

For the following cylindrical plot, define $r(\theta ,z)=\left( z+\sin
\theta \right) ^{2}\ $with \textsf{New Definition} from the \textsf{Define}
submenu. Use the limits $-3.1416\leq \theta \leq 3.1416$ and $-5\leq z\leq 5$%
.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Cylindrical}
\end{quote}

\begin{quotation}
$r^{2}$
\end{quotation}

\FRAME{dtbpF}{7.6179cm}{5.0786cm}{0pt}{}{}{Plot }{\special{language
"Scientific Word";type "MAPLEPLOT";width 7.6179cm;height 5.0786cm;depth
0pt;display "PICT";function \TEXUX{$\left( z+\sin \theta \right) ^{4}\
$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin
"-3.1416";xmax "3.1416";ymin "-5";ymax "5";xviewmin "-824.2";xviewmax
"824.8";yviewmin "-1348";yviewmax "1349";zviewmin "-5.2";zviewmax
"5.204";rangeset"X";recompute TRUE;phi 85;theta 37;plottype
14;num-x-gridlines 25;num-y-gridlines 25;plotstyle "patch";axesstyle
"none";plotshading "Z";xis \TEXUX{v57683};yis \TEXUX{z};var1name
\TEXUX{$\theta $};var2name \TEXUX{$z$};}}

\subsubsection{Parameterized Surfaces in Cylindrical Coordinates}

You can create cylindrical plots of parameterized surfaces defined by
functions such as $z(r,\theta )=r+\cos \theta $. You plot the parameterized
surface $r=f(s,t)$, $\theta =g(s,t)$, $z=h(s,t)$ in cylindrical coordinates
by entering the expressions for $r$, $\theta $, and $z$ into a vector $%
\left( f(s,t),g(s,t),h(s,t)\right) $ and choosing \textsf{Cylindrical} from 
\index{Plot 3D@Plot 3D!cylindrical@cylindrical}the \textsf{Plot 3D} submenu.

\begin{quote}
$\blacktriangleright $ To create a parameterized 
\index{Parameterized surfaces@Parameterized surfaces!cylindrical coordinates@cylindrical coordinates}%
cylindrical plot
\end{quote}

\begin{enumerate}
\item[1.]  Enter the three defining expressions for $r$, $\theta $, and $z$
as the components of a vector.\medskip

\item[2.]  With the insertion point in the vector, from the \textsf{Plot 3D}
submenu choose \textsf{Cylindrical}.\medskip
\end{enumerate}

The following example shows the ``spiral staircase'' $z=\theta $, a 3D
cylindrical plot of the vector $\left[ r,\theta ,\theta \right] $, with $%
0\leq r\leq 1$, $0\leq \theta \leq 4\pi $, and.\textsf{Plot Style} set to 
\textsf{Hidden Line}.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Cylindrical}
\end{quote}

\begin{quotation}
$\left[ r,\theta ,\theta \right] $
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";plot-snapshots TRUE;function \TEXUX{$\left[ r,\theta ,\theta \right]
$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin
"-0";xmax "12.6";ymin "0";ymax "1";xviewmin "-1.04";xviewmax
"1.041";yviewmin "-1.04";yviewmax "1.041";zviewmin "-0.252";zviewmax
"12.86";rangeset"XY";phi 45;theta 45;plottype 14;num-x-gridlines
25;num-y-gridlines 25;plotstyle "hidden";axesstyle "none";plotshading
"Z";xis \TEXUX{v952};yis \TEXUX{r};var1name \TEXUX{$\theta $};var2name
\TEXUX{$r$};}}

\subsection{\ \label{Spherical coordinates}Spherical Coordinates}

\begin{quotation}
The 
\index{Spherical coordinates@Spherical coordinates}\textsl{spherical
coordinates} $\left( \rho ,\theta ,\phi \right) $ locate 
\index{Plot 3D@Plot 3D!spherical@spherical}a \label{Plot 3D, spherical}point 
$P$ in space by giving the distance $\rho $ from the origin, the angle $%
\theta $ projected onto the $xy$-plane (the \textsl{polar angle}), and the
angle $\phi $ with the positive $z$-axis (the \textsl{vertical angle}). The
conversion into rectangular coordinates is given by 
\[
x=\rho \sin \phi \cos \theta 
\text{\qquad \quad }y=\rho \sin \phi \sin \theta \text{\qquad \quad }z=\rho
\cos \phi 
\]
and the distance formula implies 
\[
\rho ^{2}=x^{2}+y^{2}+z^{2} 
\]
\end{quotation}

The default assumption is that $\rho $ is a function of $\phi $ and $\theta $%
. You can use other names for the polar and vertical angles. Any two
variables you give will be interpreted as the polar and vertical angles.
Even when you use the standard notation, however, the roles of the variables
may be reversed in the default interpretation from what you intended. You
can correct this interpretation with the \textsf{Switch Variables} option in
the \textsf{Plot Properties} dialog box.

You can plot more than one surface on the same axes in the usual way.

\subsubsection{Expressions}

\begin{quote}
$\blacktriangleright $ To make a spherical plot
\end{quote}

\begin{enumerate}
\item[1.]  Enter an expression involving $\theta $ and $\phi $ in your
document.

\item[2.]  With the insertion point in the expression, from the \textsf{Plot
3D} submenu, choose \textsf{Spherical}.\medskip
\end{enumerate}

A sphere can be plotted as a function of its radius. Take $0\leq \theta \leq
2\pi $ and $0\leq z\leq 2\pi .$ Set \textsf{Equal Scaling on Each Axis} and
choose \textsf{Hidden Line} for \textsf{Plot Style}.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Spherical}
\end{quote}

\begin{quotation}
$2$
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$2$};linecolor "black";linestyle 1;linethickness
1;pointstyle "point";xmin "0";xmax "6.2832";ymin "0";ymax "6.2832";xviewmin
"-2.08";xviewmax "2.082";yviewmin "-2.08";yviewmax "2.082";zviewmin
"-2.08";zviewmax "2.082";rangeset"XY";phi 45;theta 45;plottype
15;constrained TRUE;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"hidden";axesstyle "none";plotshading "Z";xis \TEXUX{v952};yis
\TEXUX{v966};var1name \TEXUX{$\theta $};var2name \TEXUX{$\phi $};}}

\subsubsection{Defined Functions}

You can create a plot of a function defined in spherical coordinates $\rho
=\rho (\theta ,\phi )$.

\begin{quote}
$\blacktriangleright $ To make a spherical plot of a defined function $\rho $
of $\theta $ and $\phi $
\end{quote}

\begin{enumerate}
\item[1.]  Define $\rho $ as a function of $\theta $ and $\phi $ using 
\textsf{New Definition} on the \textsf{Define} submenu.

\item[2.]  Select the function name $\rho $ or select the expression $\rho
(\theta ,\phi )$.

\item[3.]  From the \textsf{Plot 3D} submenu, choose \textsf{Spherical}%
.\medskip
\end{enumerate}

\begin{example}
\label{Switch variables}\label{Switch variables (example)}\label{Nautilus}To
plot the nautilus determined by the expression $(1.2)^{\phi }\sin (\theta )$
you can do any one of the following:

\begin{itemize}
\item  Plot the expression $(1.2)^{\phi }\sin (\theta )$ and then choose 
\textsf{Switch Variables} on the \textsf{Plot Components} page of the 
\textsf{Plot Properties} dialog. Use the ranges $-1\leq \phi \leq 2\pi $ and 
$0\leq \theta \leq \pi $,\ to get the view of the nautilus shown below.

\item  Define $\rho (\theta ,\phi )=(1.2)^{\phi }\sin (\theta )$, plot the
expression $\rho (\theta ,\phi )$ and then choose \textsf{Switch Variables}
on the \textsf{Plot Components} page of the \textsf{Plot Properties} dialog.
Use the ranges $-1\leq \phi \leq 2\pi $ and $0\leq \theta \leq \pi $,\ to
get the view of the nautilus shown below.

\item  Define the function $\rho (\phi ,\theta )=(1.2)^{\phi }\sin (\theta )$
and plot the function name $\rho $. (Note the variables are already switched
here.) Use the ranges $-1\leq \theta \leq 2\pi $ and $0\leq \phi \leq \pi $
to get the view of the nautilus shown below.\bigskip
\end{itemize}
\end{example}

\begin{quotation}
\emph{Note }The \textsf{Switch Variables} option is not available for a plot
that depends only on the function name.
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Spherical}
\end{quote}

\begin{quotation}
$\rho $
\end{quotation}

\FRAME{dtbpF}{2.9992in}{1.9995in}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.9992in;height 1.9995in;depth
0pt;display "PICT";function \TEXUX{$(1.2)^{\phi }\sin (\theta )$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-1";xmax
"6.28";ymin "0";ymax "3.14";xviewmin "-1.896";xviewmax "3.243";yviewmin
"-2.455";yviewmax "1.43";zviewmin "-1.634";zviewmax "1.635";rangeset"X";phi
45;theta 45;plottype 15;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"patch";axesstyle "none";plotshading "Z";xis \TEXUX{v966};yis
\TEXUX{v952};var1name \TEXUX{$\phi $};var2name \TEXUX{$\theta $};}}

\subsubsection{Parameterized Surfaces in Spherical Coordinates}

\textsl{Parameterized surfaces} in spherical coordinates are 
\index{Parameterized surfaces@Parameterized surfaces!spherical coordinates@spherical coordinates}%
given by equations of the form $\rho =f(s,t)$, $\theta =g(s,t)$, and $%
\varphi =h(s,t)$. These\label{Parameterized surfaces, spherical} are very
general and allow you to generate a wide variety of interesting plots.

\begin{quote}
$\blacktriangleright $ To plot a parameterized surface
\end{quote}

\begin{enumerate}
\item[1.]  Enter the defining expressions as the three components of a
vector.

\item[2.]  With the insertion point in the vector, from the \textsf{Plot 3D}
submenu choose \textsf{Spherical}.\medskip
\end{enumerate}

The 3D spherical plot of the vector $\left[ \rho ,\theta ,1\right] $ gives
the cone $\varphi =1$. For the following plot, the view is set with $-1\leq
\rho \leq 1$ and $0\leq \theta \leq 2\pi $.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Spherical}
\end{quote}

\begin{quotation}
$\left[ \rho ,\theta ,1\right] $
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{Figure }{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\left( x,y,1\right) $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "0";xmax "6.28";ymin "-1";ymax
"1";xviewmin "-0.8751";xviewmax "0.8758";yviewmin "-0.8751";yviewmax
"0.8758";zviewmin "-0.5619";zviewmax "0.5623";rangeset"X";recompute TRUE;phi
69;theta 55;plottype 15;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"hidden";axesstyle "none";plotshading "Z";xis \TEXUX{y};yis
\TEXUX{x};var1name \TEXUX{$y$};var2name \TEXUX{$x$};}}\medskip

You can plot the surface defined by $\rho =s$, $\theta =s^2+t^2$, $\varphi
=t $ by entering the three expressions as coordinates of a vector. For the
following plot, take $0\leq s\leq 1$ and $-1\leq t\leq 1$.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 3D + Spherical}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{c}
s \\ 
s^{2}+t^{2} \\ 
t
\end{array}
\right) $
\end{quotation}

\FRAME{dtbpF}{2.9992in}{1.9995in}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.9992in;height 1.9995in;depth
0pt;display "PICT";function \TEXUX{$\left(
\MATRIX{1,3}{c}\VR{,,c,,,}{,,,,,}\HR{,,,}\CELL{s}\CELL{s^{2}+t^{2}}\CELL{t}%
\right) $};linecolor "black";linestyle 1;linethickness 1;pointstyle
"point";xmin "0";xmax "1";ymin "-1";ymax "1";xviewmin "-0.3642";xviewmax
"0.3645";yviewmin "-0.7958";yviewmax "0.7964";zviewmin "-0.02";zviewmax
"1.02";rangeset"X";recompute TRUE;phi 45;theta 45;plottype
15;num-x-gridlines 25;num-y-gridlines 25;plotstyle "patch";axesstyle
"none";plotshading "XY";xis \TEXUX{s};yis \TEXUX{t};var1name
\TEXUX{$s$};var2name \TEXUX{$t$};}}

\label{Printing Plots}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Plotting Curves and Surfaces}{}{}{DM6-1.tex#Plotting}

\item  \hyperref{Frame Properties}{}{}{DM6-2.tex#Frame Properties}

\item  \hyperref{The View for 2D Plots}{}{}{DM6-3.tex#View intervals}

\item  \hyperref{2D Plots of Functions and Expressions}{}{}{DM6-4.tex#Plot
expression}

\item  \hyperref{The View Box for 3D Plots}{}{}{DM6-5.tex#Plot 3D, view}

\item  \hyperref{Exercises and Solutions}{}{}{DM6-7.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{38.875pt}{24.5pt}{6pt}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid-file "F";width 38.875pt;height 24.5pt;depth
6pt;original-width 36.125pt;original-height 22.5625pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'index.wmf';file-properties
"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Curves in space}{}{}{DMIndex.tex#Curves in space}, \hyperref{%
Cylindrical coordinates}{}{}{DMIndex.tex#Cylindrical coordinates}, \hyperref{%
Plot properties}{}{}{DMIndex.tex#Plot properties}, \hyperref{Plotting tools}{%
}{}{DMIndex.tex#Plotting tools}, \hyperref{Surfaces of revolution}{}{}{%
DMIndex.tex#Surfaces of revolution}, \hyperref{Tube plots}{}{}{%
DMIndex.tex#Tube plots}}

\end{document}
