%% 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="The View  Box for 3D Plots">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:04:27 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="DM6-6.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM6.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM6-4.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}

\input{tcilatex}
\begin{document}


\section{The View\label{Plot 3D, view} Box for 3D Plots}

The environment for plotting curves and surfaces in space is similar\label%
{Plot 3D} to the environment for plotting in the plane. The view 
\index{View@View!3D plots@3D plots}is a box, a rectangular solid determined
by inequalities of the form $x_{0}\leq x\leq x_{1}$, $y_{0}\leq y\leq y_{1}$%
, and $z_{0}\leq z\leq z_{1}$. The frame is a rectangular region of the
computer screen, just as before.

\textsl{Scientific Notebook} chooses a default view with $-5\leq x\leq 5$, $%
-5\leq y\leq 5$, and with the $z$-coordinates determined automatically. If
you use other variable names, the order is determined alphabetically.

\begin{quote}
$\blacktriangleright $ To plot an expression involving two variables
\end{quote}

\begin{itemize}
\begin{enumerate}
\item[1.]  Place the insertion point in the expression.

\item[2.]  Click the \textsf{3D Plot} button on the Compute toolbar; or from
the \textsf{Plot 3D} submenu, choose \textsf{Rectangular}.\medskip 
\end{enumerate}
\end{itemize}

The following plot shows the surface $z=x^{3}-3xy^{2}$ with the default 
\textsf{Domain Intervals} for $x$ and $y$, and the default \textsf{View
Intervals} for $x$, $y$ and $z$. To make this plot, leave the insertion
point in the expression $x^{3}-3xy^{2}$, and click the \textsf{3D Plot}
button on the Compute toolbar, or from the \textsf{Plot 3D} submenu, choose 
\textsf{Rectangular}.

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

\begin{quotation}
$x^{3}-3xy^{2}$
\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{$x^{3}-3xy^{2}$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-5";xmax
"5";ymin "-5";ymax "5";xviewmin "-5";xviewmax "5";yviewmin "-5.2";yviewmax
"5.204";zviewmin "-260";zviewmax "260.2";rangeset"X";phi 45;theta
45;plottype 5;num-x-gridlines 25;num-y-gridlines 25;plotstyle
"wireframe";axesstyle "normal";plotshading "Z";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};}}

\section{\label{3D Plotting Tools}Plotting Tools for 3D Plots}

\begin{quote}
$\blacktriangleright $ To display 3D plotting tools
\end{quote}

\begin{itemize}
\item  Select the view by double-clicking the plot. 
\index{View@View!select@select}\medskip
\end{itemize}

The plotting tools appear at the upper-right edge of the frame. The frame
also has eight gray handles, and the mouse pointer has the shape of a hand
when it is over the view.

\subsection{\ Changing the Plot Orientation\label{Plot 3D, orientation}}

The plot orientation determines the angle from which you view 
\index{View@View!3D orientation@3D orientation}the 3D space.

\begin{quote}
$\blacktriangleright $ To change the orientation
\end{quote}

\begin{enumerate}
\item[1.]  Double-click to select the view.

\item[2.]  Move the mouse pointer over the plot, so that the pointer changes
to a hand.

\item[3.]  Press the left mouse button and move the mouse. As you move the
mouse while holding the button down, a 3-dimensional box indicates the
orientation of the axes.\medskip When you release the button, the plot is
redrawn with the new orientation.
\end{enumerate}

\subsection{\ Scaling\label{3D Plot Properties} and Translating the Plot}

You can change the scale and translate\label{Translating 3D plot}\label%
{Scaling 3D plot} the plot using the \textsf{Plot Properties} tabbed dialog.%
\FRAME{dtbpF}{303.125pt}{372.375pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
303.125pt;height 372.375pt;depth 0pt;original-width
301.125pt;original-height 370.375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'PlotPrp4.wmf';file-properties "XNPEU";}}

\begin{quote}
$\blacktriangleright $ To bring up the \textsf{Plot Properties} tabbed
dialog 
\index{Plot 3D@Plot 3D!dialog box@dialog box}for a 3D plot
\end{quote}

\begin{itemize}
\begin{enumerate}
\item[1.]  Select the frame by clicking the frame or the view by
double-clicking the frame.

\item[2.]  Click the plot-tool icon depicting a miniature dialog box; or
from the \textsf{Edit} menu, choose \textsf{Properties}; or click 
\FRAME{itbpF}{0.3001in}{0.3001in}{0.0701in}{}{}{property.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.3001in;height 0.3001in;depth
0.0701in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'Property.wmf';file-properties "XNPEU";}}.\medskip
\end{enumerate}
\end{itemize}

Make your changes in the dialog box, and then choose the \textsf{OK} button
to save your changes and to refresh the function plot, or choose the \textsf{%
Cancel} button to exit the dialog box without taking any action.

\subsubsection{Item List\label{Item number}}

The box in the upper-left corner of the \textsf{Plot Properties + Plot
Components } tabbed dialog shows one of the expressions or functions that
you are plotting in the current view. These expressions are referred to as
``items'' 
\index{Plot 3D@Plot 3D!item list@item list}\label{Plot 3D, items}and are
listed by \textsf{Item Number}. You can view the current functions or
expressions by clicking the up-down arrow buttons. To change the curves or
surfaces that are plotted, you can change, delete, or add expressions or
functions in the box. To change an item, edit it in the box. To add an item,
click the \textsf{Add Item} button, and then type the item in the empty box
that comes up. To delete an item from the list, click the arrow buttons
until the item is displayed in the box, and then click the \textsf{Delete
Item} button.

\subsubsection{Domain Intervals\label{Plot 3D, domain intervals}}

You set the \textsf{Domain Intervals} in the \textsf{Plot 
\index{Plot 3D@Plot 3D!domain intervals@domain intervals}Properties + Plot
Components} tabbed dialog. These intervals determine the region where the
sampling points are chosen that determine the plot.

\subsubsection{View Intervals\label{Plot 3D, view intervals}}

The default \textsf{View Intervals} are determined by the \textsf{Domain
Intervals} and are shown in the \textsf{Axes \& View} page of the \textsf{%
Plot Properties }dialog. To change the \textsf{View Intervals}, remove the
checkmark from \textsf{Default} and set the \textsf{View Intervals} by 
\index{Plot 3D@Plot 3D!view@view}editing the numbers in the boxes. The $z$%
-axis can have quite a different scale from the $x$- and $y$-axes, and a
bounded surface is generally not truncated at the top or bottom. You can
check \label{Equal scaling for Plot 3D} 
\index{Equal scaling@Equal scaling} 
\index{Scaling@Scaling}\textsf{Equal Scaling Along Each Axis} to force the
same scale on all three axes.

\subsubsection{Sample Size}

The plot is determined by the data points computed. You can increase the
quality of the graph by increasing the \textsf{Sample Size. } \label%
{Plot 3D, sample size} 
\index{Plot 3D@Plot 3D!sample size@sample size}Increasing the \textsf{Sample
Size} also slows down the plotting process, of course. This is especially
critical for Plot 3D because the number of data points computed is the
produce of the two numbers entered under\textsf{\ Sample Size}. Thus the
default number of data points computed is $25^{2}=\allowbreak 625$. So you
may want to specify a relatively small sample size for real-time
demonstrations and a relatively large sample size for printed documents.

\subsubsection{Style}

In a plot, 
\index{Plot 3D@Plot 3D!style@style}\label{Plot 3D, style}the grid points
computed are connected by polynomial surfaces or curves, or are simply
displayed as points, depending on your choice of \textsf{Style}. The default
style is \textsf{WireFrame}, which is generated relatively quickly on the
screen. You may wish to switch to \textsf{Patch}, which takes much longer
but can be much prettier, for a final printed document.

\begin{itemize}
\item  \label{Patch}\textsf{Patch} connects the points by small patches of
polynomial surfaces and a wire frame grid. 
\index{Patch@Patch}

\item  \textsf{Patch [no grid]} is the same as \textsf{Patch} but without
the grid.

\item  \textsf{Patch \& Contour }adds level curves to the patches.

\item  \textsf{Contours} 
\index{Contour@Contour}\label{Contour}is a set of level curves on a
transparent surface.

\item  \textsf{WireFrame} 
\index{Wireframe@Wireframe}\label{Wireframe}is the default setting; it
connects the points by straight line segments on a transparent surface.

\item  \textsf{Hidden Line} 
\index{Hidden line@Hidden line}\label{Hidden line}is the same as \textsf{%
WireFrame,} except that the surface is opaque.

\item  \textsf{Points} 
\index{Points@Points}displays only the computed points.\medskip
\end{itemize}

If you do not see all these choices in the dialog, use the up arrow and down
arrow to reach hidden choices.

\subsubsection{Lighting and Color\label{Lighting}}

You can change the \textsf{Lighting} 
\index{Lighting@Lighting}from the 
\index{Plot 3D@Plot 3D!lighting and color@lighting and color}default \textsf{%
0} to \textsf{1--4}. \textsf{Color} 
\index{Color@Color}\label{Plot 3D, color}\label{Color for Plot 3D}choices
include \textsf{XYZ}, \textsf{XY}, \textsf{Z}, \textsf{Z} \textsf{Hue} 
\index{Hue@Hue}\label{Hue}, \textsf{Z Grayscale} 
\index{Grayscale@Grayscale}\label{Grayscale}, or \textsf{No Coloring}. The
color choice \textsf{Z} means that the color choice depends on the $z$%
-coordinate only, whereas \textsf{XYZ} means that color is a function of all
three coordinates.

\subsubsection{Axes\label{Axes for Plot 3D}}

Finally, you can specify the appearance of the 
\index{Plot 3D@Plot 3D!axes@axes} 
\index{Axes@Axes!3D plots@3D plots}\label{Plot 3D, axes}axes as \textsf{%
Normal} 
\index{Normal@Normal}\textsf{, None, Boxed} 
\index{Boxed@Boxed}\label{Boxed axes}\textsf{, }or\textsf{\ Frame}\label%
{Frame axes}.

\begin{itemize}
\item  \textsf{Normal} gives the usual $x$-, $y$-, and $z$-axes if the
origin is contained inside the view box; otherwise, it draws three axes as
close as possible to the origin.

\item  \textsf{Boxed} shows the plot inside a box frame.

\item  \textsf{Frame} displays the left edge and two lower edges of the
box.\medskip
\end{itemize}

All these choices (except \textsf{None}) display numerical labels on the
axes or box edges.

\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{3D Plots of Functions and Expressions}{}{}{DM6-6.tex#3D
Plots}

\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{Plot properties}{}{}{DMIndex.tex#Plot properties}, \hyperref{%
Plotting tools}{}{}{DMIndex.tex#Plotting tools}}

\end{document}
