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diff --git a/info/pstricks_calcnotes/For_Ps_Output/ThreeAppsPS.tex b/info/pstricks_calcnotes/For_Ps_Output/ThreeAppsPS.tex new file mode 100644 index 0000000000..b1cfddca66 --- /dev/null +++ b/info/pstricks_calcnotes/For_Ps_Output/ThreeAppsPS.tex @@ -0,0 +1,1165 @@ +\documentclass[11pt,a4paper,oneside]{article} +\usepackage{calculator} +\usepackage{calculus} +\usepackage{amsthm} +\usepackage{amsmath} +\usepackage{cmtt} +\usepackage[dvips]{geometry} +\usepackage{pstricks} +\usepackage{graphicx} +\usepackage{graphics} +\usepackage{pst-plot} +\usepackage{pst-node} +\usepackage{multido} +\usepackage{pst-xkey} +\usepackage{pst-func} +\usepackage{pstricks-add} +\usepackage[dvips,colorlinks,linktocpage]{hyperref} +\def\hantt{\^e}\def\accentcircflx{\hskip-.3em\raisebox{0.32ex}{\'{}}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\RiemannSum#1#2#3#4#5#6#7#8#9{% +\psplot[linecolor=blue]{#1}{#2}{#3} +\pscustom[linecolor=red]{% +\psline{-}(#1,0)(#1,0) +\multido{\ni=#5,\ne=#6}{#4} +{\psline(*{\ni} {#8})(*{\ne} {#9})}} +\multido{\ne=#6,\nc=#7}{#4} +{\psdot(*{\nc} {#3}) +\psline[linestyle=dotted,dotsep=1.5pt](\nc,0)(*{\nc} {#3}) +\psline[linecolor=red](\ne,0)(*{\ne} {#9})} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcount\Fa\newcount\Fct\newcount\tempA +\def\Factor{\ifnum\Fa=1\relax\else\advance\Fa by -1\multiply\Fct by \Fa\Factor\fi} +\def\FACTORIAL#1{\Fa=#1 \ifnum\Fa=0 \Fct=1\relax\else\Fct=\Fa \Factor\fi\global\tempA=\Fct} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcount\BINOM +\newcount\temp +\newcount\tmp +\def\BINOMIAL#1#2{% +\temp=#1\advance\temp by -#2 +\FACTORIAL{#1} +\tmp=\tempA +\FACTORIAL{\temp} +\temp=\tempA +\divide\tmp by \temp +\FACTORIAL{#2} +\temp=\tempA +\divide\tmp by \temp\global\BINOM=\tmp} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newdimen\Xa\newdimen\Yb\newcount\kc +\newdimen\BSTemp +\def\xmult#1{\ifnum\kc<#1\advance\kc by 1\Yb=\decimal\Xa\Yb\xmult{#1}\else\relax\fi} +\def\BERNSTEIN#1#2#3{\Xa=#3pt\kc=0\Yb=1pt\xmult{#1}\kc=0\Xa=-\Xa\advance\Xa by 1pt\xmult{#2} +\BINOMIAL{6}{#1}\global\BSTemp=\BINOM\Yb} +\newdimen\Xrf\newdimen\Yrf +\newdimen\XBST\newdimen\YBST +\def\XBC#1#2#3{% +\BERNSTEIN{0}{6}{#2} +\Xrf=#1pt +\XBST=\decimal\BSTemp\Xrf +\BERNSTEIN{1}{5}{#2} +\Xrf=#1pt\advance\Xrf by -1.5pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{2}{4}{#2} +\Xrf=#1pt\advance\Xrf by 2pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{3}{3}{#2} +\Xrf=#1pt\advance\Xrf by 6pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{4}{2}{#2} +\Xrf=#1pt\advance\Xrf by 5pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{5}{1}{#2} +\Xrf=#1pt\advance\Xrf by 1.5pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{6}{0}{#2} +\Xrf=#1pt +\advance\XBST by \decimal\BSTemp\Xrf +\global\XBST=#3\XBST} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\YBC#1#2#3{% +\BERNSTEIN{0}{6}{#2} +\Yrf=#1pt +\YBST=\decimal\BSTemp\Yrf +\BERNSTEIN{1}{5}{#2} +\Yrf=#1pt\advance\Yrf by 2pt +\advance\YBST by \decimal\BSTemp\Yrf +\BERNSTEIN{2}{4}{#2} +\Yrf=#1pt\advance\Yrf by 5pt +\advance\YBST by \decimal\BSTemp\Yrf +\BERNSTEIN{3}{3}{#2} +\Yrf=#1pt\advance\Yrf by 3pt +\advance\YBST by \decimal\BSTemp\Yrf +\BERNSTEIN{4}{2}{#2} +\Yrf=#1pt\advance\Yrf by -1pt +\advance\YBST by \decimal\BSTemp\Yrf +\BERNSTEIN{5}{1}{#2} +\Yrf=#1pt\advance\Yrf by -2pt +\advance\YBST by \decimal\BSTemp\Yrf +\BERNSTEIN{6}{0}{#2} +\Yrf=#1pt +\advance\YBST by \decimal\BSTemp\Yrf +\global\YBST=#3\YBST} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\XBsix#1#2{% +(1-t)^6*(#1)*(#2)+6*t*(1-t)^5*(#1-1.5)*(#2)+15*t^2*(1-t)^4*(#1+2)*(#2)+ +20*t^3*(1-t)^3*(#1+6)*(#2)+15*t^4*(1-t)^2*(#1+5)*(#2)+6*t^5*(1-t)*(#1+1.5)*(#2)+t^6*(#1)*(#2)} +\def\YBsix#1#2{% +(1-t)^6*(#1)*(#2)+6*t*(1-t)^5*(#1+2)*(#2)+15*t^2*(1-t)^4*(#1+5)*(#2)+ +20*t^3*(1-t)^3*(#1+3)*(#2)+15*t^4*(1-t)^2*(#1-1)*(#2)+6*t^5*(1-t)*(#1-2)*(#2)+t^6*(#1)*(#2)} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\NetDraw#1#2#3#4#5{% +\newcount\CellNum +\newdimen\XRef +\newdimen\YRef +\newdimen\Xfact +\newdimen\Yfact +\newdimen\XMin +\newdimen\XMax +\newdimen\YMin +\newdimen\YMax +\newdimen\Xsize +\newdimen\Ysize +\newdimen\tempx +\newdimen\tempy +\CellNum=#1 +\XRef=#2pt +\YRef=#3pt +\Xfact=#4pt +\Yfact=#5pt +\XMin=\decimal\Xfact\XRef \advance\XMin by -1.50\Xfact +\XMax=\decimal\Xfact\XRef \advance\XMax by 5.00\Xfact +\YMin=\decimal\Yfact\YRef \advance\YMin by -6.00\Yfact +\YMax=\decimal\Yfact\YRef \advance\YMax by \Yfact +\Xsize=\XMax \advance\Xsize by -\XMin \divide\Xsize by \CellNum +\Ysize=\YMax \advance\Ysize by -\YMin \divide\Ysize by \CellNum +\parametricplot[algebraic,fillstyle=solid,fillcolor=yellow!30,plotpoints=200,linewidth=0.5pt]{0}{0.5} +{\XBeight{\decimal\XRef}{\decimal\Xfact}|\YBeight{\decimal\YRef}{\decimal\Yfact}} +\parametricplot[algebraic,fillstyle=solid,fillcolor=yellow!30,plotpoints=200,linewidth=0.5pt]{0.5}{1} +{\XBeight{\decimal\XRef}{\decimal\Xfact}|\YBeight{\decimal\YRef}{\decimal\Yfact}} +\multido{\nz=0.00+0.05}{200}{\XBC{\decimal\XRef}{\nz}{\decimal\Xfact}\YBC{\decimal\YRef}{\nz}{\decimal\Yfact} +\multido{\nx=\decimal\XMin+\decimal\Xsize}{\the\CellNum} +{\tempx=\nx pt\multido{\ny=\decimal\YMin+\decimal\Ysize}{\the\CellNum} +{\tempy=\ny pt\ifdim\YBST<\tempy\relax\else\advance\tempy by \Ysize% + \ifdim\YBST>\tempy\relax\else + \ifdim\XBST<\tempx\relax\else\advance\tempx by \Xsize + \ifdim\XBST>\tempx\relax\else + \pspolygon[fillstyle=solid,fillcolor=red!90,linecolor=black,linewidth=0.2pt] + (\nx,\ny)(\decimal\tempx,\ny)(\decimal\tempx,\decimal\tempy)(\nx,\decimal\tempy)(\nx,\ny)\relax\fi\fi\fi\fi} +}} +\advance\CellNum by 1 +\multido{\nx=\decimal\XMin+\decimal\Xsize}{\the\CellNum} +{\psline[linewidth=0.2pt](\nx,\decimal\YMax)(\nx,\decimal\YMin)} +\multido{\ny=\decimal\YMin+\decimal\Ysize}{\the\CellNum} +{\psline[linewidth=0.2pt](\decimal\XMin,\ny)(\decimal\XMax,\ny)} +\parametricplot[algebraic,linecolor=white,plotpoints=200,linewidth=0.5pt]{0}{0.5} +{\XBeight{\decimal\XRef}{\decimal\Xfact}|\YBeight{\decimal\YRef}{\decimal\Yfact}} +\parametricplot[algebraic,linecolor=white,plotpoints=200,linewidth=0.5pt]{0.5}{1} +{\XBeight{\decimal\XRef}{\decimal\Xfact}|\YBeight{\decimal\YRef}{\decimal\Yfact}} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\avecfld#1#2#3#4#5#6{% +\multido{#2}{#4} +{\multido{#1}{#3} +{\parametricplot[algebraic,arrows=->,linecolor=red]{0}{1} +{\nx+((#5)*t)*(1/sqrt(1+(#6)^2))|\ny+((#5)*t)*(1/sqrt(1+(#6)^2))*(#6)}}}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\anothervecfld#1#2#3#4#5#6{% +\multido{#2}{#4} +{\multido{#1}{#3} +{\curvepnodes[algebraic,plotpoints=2]{0}{1} +{\nx+((#5)*t)*(1/sqrt(1+(#6)^2))|\ny+((#5)*t)*(1/sqrt(1+(#6)^2))*(#6)}{P} +\psline[linewidth=0.7pt,linecolor=red]{->}(P0)(P1)}}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\xch{\catcode`\p=12 \catcode`\t=12}\def\ych{\catcode`\p=11 \catcode`\t=11} +\xch \def\dec#1pt{#1}\ych \def\decimal#1{\expandafter\dec \the#1} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\def\vecfldnew#1#2#3#4#5#6#7#8{% +\newcount\intg \newdimen\fx \newdimen\fy \newdimen\slope \newdimen\interm +\def\fintg{\interm=#8 \interm=\intg\interm \ifdim\ifdim\slope<0pt-\fi\slope<\interm\advance\intg by -1\relax\else\advance\intg by 1\fintg\fi} +\multido{#2}{#4} +{\multido{#1}{#3} +{\curvepnodes[algebraic,plotpoints=2]{0}{1}{\nx+((#5)*t)*(1/sqrt(1+(#6)^2))|\ny+((#5)*t)*(1/sqrt(1+(#6)^2))*(#6)}{P} +#7 \slope=10\slope\fintg +\ifnum\intg>10\psline[linecolor=red]{->}(P0)(P1)\else\ifnum\intg=0\psline[linecolor=red!5]{->}(P0)(P1)\else\multiply\intg by 10 +\psline[linecolor=red!\the\intg]{->}(P0)(P1)\fi\fi +\intg=0\slope=0pt +}}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\pagestyle{headings} +\topmargin=-0.6cm +\textwidth=16.7cm +\textheight=23cm +\headheight=2.5ex +\headsep=0.6cm +\oddsidemargin=.cm +\evensidemargin=-.4cm +\parskip=0.7ex plus0.5ex minus 0.5ex +\baselineskip=17pt plus2pt minus2pt +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\catcode`@=11 +\renewcommand\section{\@startsection {section}{1}{\z@}% + {-3.5ex \@plus -1ex \@minus -.2ex}% + {2.3ex \@plus.2ex}% + {\normalfont\large\bfseries}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-3.25ex\@plus -1ex \@minus -.2ex}% + {1.5ex \@plus .2ex}% + {\normalfont\normalsize\bfseries}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\gdef\acknw{\section*{% +{\acknwname}\markright{\protect\textsl{\acknwname}}}% +\addcontentsline{toc}{section}{\acknwname}} +\gdef\acknwname{Acknowledgment} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\renewcommand\sectionmark[1]{\markright{\thesection. #1}} +\newcounter{lk} +\newenvironment{listof}{\begin{list}{\rm(\roman{lk})}{\usecounter{lk}% +\setlength{\topsep}{0ex plus0.1ex}% +\setlength{\labelwidth}{1cm}% +\setlength{\itemsep}{0ex plus0.1ex}% +\setlength{\itemindent}{0.5cm}% +}}{\end{list}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\title{Three applications of macros in \texttt{PSTricks}\thanks{PSTricks is the original work of Timothy Van Zandt (email address: \texttt{tvz@econ.insead.fr}). +It is currently edited by Herbert Vo\ss\ (\texttt{hvoss@tug.org}).}} +\author{Le Phuong Quan\\ +\small{(Cantho University, Vietnam)}\\ +\small{\texttt{lpquan@ctu.edu.vn}}} +\begin{document} +\maketitle +\tableofcontents +\section{Drawing approximations to the area under a graph by rectangles} +\subsection{Description} + +We recall here an operation in Calculus. Let $f(x)$ be a function, defined and bounded on +an interval $[a,b]$. If $f$ is integrable (in Riemann sense) on $[a,b]$, then its definite integral over this interval +is +$$\int_a^bf(x)dx=\lim_{\|P\|\to 0}\sum_{i=1}^nf(\xi_i)\Delta x_i,$$ +where $P\colon a=x_0<x_1<\cdots<x_n=b$, $\Delta x_i=x_i-x_{i-1}$ is a partition of $[a,b]$, $\xi_i\in[x_{i-1},x_i]$, $i=1,2,\ldots,n$, +and $\|P\|=\max\{\Delta x_i\colon i=1,2,\ldots,n\}$. Hence, when $\|P\|$ is small enough, we may have an +approximation +\begin{equation}\label{eqn1} +I=\int_a^bf(x)dx\approx\sum_{i=1}^nf(\xi_i)\Delta x_i. +\end{equation} +Because $I$ is independent to the choice of $P$ and $\xi_i$, we may +divide $[a,b]$ into $n$ subintervals with equal length and choose $\xi_i=(x_i+x_{i-1})/2$. +Then, $I$ can be approximately seen as the sum of areas of the rectangles with sides +$f(\xi_i)$ and $\Delta x_i$. + +We will make a drawing procedure to illustrate the approximation (\ref{eqn1}). Firstly, we establish +commands to draw the ``sum'' of rectangles, like the area under piecewise-constant functions +(called \textsl{step shape\/}, for brevity). The chosen procedure here +obviously includes a combination of the macros \texttt{\symbol{92}pscustom} (to \textsl{join\/} horizontal segments, automatically) +and \texttt{\symbol{92}multido}. In particular, horizontal segments are depicted within the loop +\texttt{\symbol{92}multido} by +$$\textmtt{\symbol{92}psplot[{\it settings}]\{$x_{i-1}$\}\{$x_i$\}\{$f(\xi_i)$\}}$$ +Then, \texttt{\symbol{92}pscustom} will join these segments altogether with the ending points +$(a,0)$, $(b,0)$ to make the boundary of the step shape. Next, we draw the points $(\xi_i,f(\xi_i))$, $i=1,2,\ldots,n$, +and the dotted segments between these points and the points $(\xi_i,0)$, $i=1,2,\ldots,n$, by +\begin{align*} +&\textmtt{\symbol{92}psdot[algebraic,\dots](*\{$\xi_i$\} \{$f(x)$\})}\\ +&\textmtt{\symbol{92}psline[algebraic,linestyle=dotted,\dots]($\xi_i$,$0$)(*\{$\xi_i$\} \{$f(x)$\})} +\end{align*} +where we use the structure \textmtt{(*\{{\it value}\} \{$f(x)$\})} to obtain $(\xi_i,f(\xi_i))$. Finally, we draw +vertical segments to split the step shape into rectangular cells by +$$\textmtt{\symbol{92}psline[algebraic,\dots]($x_i$,$0$)(*\{$x_i$\} \{$f(x-\Delta x_i/2)$\})}$$ +The process of approximation is depicted in Figure \ref{Fig1}. +\begin{figure}[htbp] +\centering\begin{pspicture}(-2,-2)(3,3.5) +\psset{yunit=0.2} +\psaxes[labelFontSize=$\footnotesize$,Dy=2,ticksize=2.2pt,labelsep=4pt]{->}(0,0)(-2.5,-10)(3.5,15.5) +\psplot[plotpoints=500,algebraic,linecolor=red]{-2}{3}{x^3-2*x^2+6} +\pscustom{ +\psline{-}(-2,0)(-2,0) +\multido{\nx=-2.000+0.500,\ny=-1.750+0.500,\ni=-1.500+0.500}{10} +{\psplot[algebraic]{\nx}{\ni}{(\ny)^3-2*(\ny)^2+6}} +\psline{-}(3,0)} +\end{pspicture} +\hskip3em +\begin{pspicture}(-2,-2)(3,3.5) +\psset{yunit=0.2} +\psaxes[labelFontSize=$\footnotesize$,Dy=2,ticksize=2.2pt,labelsep=4pt]{->}(0,0)(-2.5,-10)(3.5,15.5) +\psplot[plotpoints=500,algebraic,linecolor=red]{-2}{3}{x^3-2*x^2+6} +\pscustom{ +\psline{-}(-2,0)(-2,0) +\multido{\nx=-2.000+0.500,\ny=-1.750+0.500,\ni=-1.500+0.500}{10} +{\psplot[algebraic]{\nx}{\ni}{(\ny)^3-2*(\ny)^2+6}} +\psline{-}(3,0)} +\multido{\nx=-2.000+0.500,\ny=-1.750+0.500,\ni=-1.500+0.500}{10} +{\psline[linestyle=dotted,dotsep=1.5pt,dotstyle=o,linecolor=gray](\ny,0)(*{\ny} {x^3-2*x^2+6}) +\psdot[dotsize=1.2pt 1,dotstyle=Bo](*{\ny} {x^3-2*x^2+6})} +\end{pspicture}\\[3ex] +\centering\begin{pspicture}(-2,-2)(3,3) +\psset{yunit=0.2} +\psaxes[labelFontSize=$\footnotesize$,Dy=2,ticksize=2.2pt,labelsep=4pt]{->}(0,0)(-2.5,-10)(3.5,15.5) +\psplot[plotpoints=500,algebraic,linecolor=red]{-2}{3}{x^3-2*x^2+6} +\pscustom{ +\psline{-}(-2,0)(-2,0) +\multido{\nx=-2.000+0.500,\ny=-1.750+0.500,\ni=-1.500+0.500}{10} +{\psplot[algebraic]{\nx}{\ni}{(\ny)^3-2*(\ny)^2+6}} +\psline{-}(3,0)} +\multido{\nx=-2.000+0.500,\ny=-1.750+0.500,\ni=-1.500+0.500}{10} +{\psline[linestyle=dotted,dotsep=1.5pt,dotstyle=o,linecolor=gray](\ny,0)(*{\ny} {x^3-2*x^2+6}) +\psdot[dotsize=1.2pt 1,dotstyle=Bo](*{\ny} {x^3-2*x^2+6})} +\multido{\nx=-2.000+0.500,\ny=-1.750+0.500,\ni=-1.500+0.500}{9} +{\psline(\ni,0)(*{\ni} {(x-0.25)^3-2*(x-0.25)^2+6})} +\end{pspicture} +\caption{Steps to make the drawing procedure.}\label{Fig1} +\end{figure} + +We now combine the above steps to make a procedure whose calling sequence consists of main parameters +$a$, $b$, $f$ and $n$, and dependent parameters $x_{i-1}$, $x_i$, $\xi_i$, $f(\xi_i)$ and +$f(x\pm\Delta x_i/2)$. For instance, let us consider approximations to the integral of $f(x)=\sin x-\cos x$ +over $[-2,3]$ in cases of $n=5$ and $n=20$. They are given in Figure \ref{Fig2}. +\begin{figure}[htbp] +\centering\begin{pspicture}(-2.5,-3)(3.5,3.01) +\psset{plotpoints=500,algebraic,dotsize=1pt 2,yunit=2} +\psaxes[labelFontSize=$\footnotesize$,Dy=1]{->}(0,0)(-2.5,-1.5)(3.5,1.5) +\RiemannSum{-2}{3}{sin(x)-cos(x)}{5} +{-2.0+1.0}{-1.0+1.0}{-1.5+1.0} +{sin(x+0.5)-cos(x+0.5)}{sin(x-0.5)-cos(x-0.5)} +\end{pspicture} +\hskip4em +\begin{pspicture}(-2.5,-3)(3.5,3.01) +\psset{plotpoints=500,algebraic,dotsize=1pt 2,yunit=2} +\psaxes[labelFontSize=$\footnotesize$,Dy=1]{->}(0,0)(-2.5,-1.5)(3.5,1.5) +\RiemannSum{-2}{3}{sin(x)-cos(x)}{20} +{-2.000+0.250}{-1.750+0.250}{-1.875+0.250} +{sin(x+0.125)-cos(x+0.125)}{sin(x-0.125)-cos(x-0.125)} +\end{pspicture} +\caption{Approximations to the integral of $f(x)=\sin x-\cos x$ over $[-2,3]$.}\label{Fig2} +\end{figure} + +In summary, we can make a procedure to illustrate the approximation (\ref{eqn1}), say \texttt{RiemannSum}, whose calling sequence has the form of +$$\textmtt{\symbol{92}RiemannSum\{$a$\}\{$b$\}\{$f(x)$\}\{$n$\}\{$x_{\rm ini}$\}\{$x_{\rm end}$\}\{$x_{\rm choice}$\}\{$f(x+\Delta x_i/2)$\}\{$f(x-\Delta x_i/2)$\}},$$ +where $x_0=a$ and for each $i=1,2\ldots,n$: +\begin{align*} +x_i&=a+\dfrac{b-a}{n}i,\quad\Delta x_i=x_i-x_{i-1}=\dfrac{b-a}{n},\\ +x_{\rm ini}&=x_0+\Delta x_i,\quad x_{\rm end}=x_1+\Delta x_i,\quad x_{\rm choice}=\dfrac{x_{\rm ini}+x_{\rm end}}{2}=\dfrac{x_0+x_1}{2}+\Delta x_i. +\end{align*} +Note that $x_{\rm ini}$, $x_{\rm end}$ and $x_{\rm choice}$ are given in such forms to be +suitable to variable declaration in \texttt{\symbol{92}multido}. They are nothing but +$x_{i-1}$, $x_i$ and $\xi_i$, respectively, at the step $i$-th in the loop. + +Tentatively, in \texttt{PSTricks} language, the definition of \texttt{RiemannSum} is suggested to be +\bigskip\hrule +\noindent\begin{tabular}{@{}l} +\verb!\def\RiemannSum#1#2#3#4#5#6#7#8#9{%!\\ +\verb!\psplot[linecolor=blue]{#1}{#2}{#3}!\\ +\verb!\pscustom[linecolor=red]{%!\\ +\verb!\psline{-}(#1,0)(#1,0)!\\ +\verb!\multido{\ni=#5,\ne=#6}{#4}!\\ +\verb!{\psline(*{\ni} {#8})(*{\ne} {#9})}}!\\ +\verb!\multido{\ne=#6,\nc=#7}{#4}!\\ +\verb!{\psdot(*{\nc} {#3})!\\ +\verb!\psline[linestyle=dotted,dotsep=1.5pt](\nc,0)(*{\nc} {#3})!\\ +\verb!\psline[linecolor=red](\ne,0)(*{\ne} {#9})}}! +\end{tabular}\hrule +\subsection{Examples} +We give here two more examples just to see that using the drawing procedure is very easy. In the first example, we approximate +the area under the graph of the function $f(x)=x-(x/2)\cos x+2$ on the interval $[0,8]$. To draw the approximation, we try +the case $n=16$; thus $x_0=0$ and for each $i=1,\ldots,16$, we have +$x_i=0.5\,i$, $\Delta x_i=0.5$, $x_{\rm ini}=0.00+0.50$, $x_{\rm end}=0.50+0.50$ and $x_{\rm choice}=0.25+0.50$. +\begin{figure}[htbp] +\centering\begin{pspicture}(0,0)(5.1,6.6) +\psset{plotpoints=500,algebraic,dotsize=1pt 2,unit=0.6} +\RiemannSum{0}{8}{x-(x/2)*cos(x)+2}{16}{0.00+0.50}{0.50+0.50}{0.25+0.50} +{x+0.25-((x+0.25)/2)*cos(x+0.25)+2}{x-0.25-((x-0.25)/2)*cos(x-0.25)+2} +\psaxes[labelFontSize=$\footnotesize$,Dy=1,ticksize=2.2pt,labelsep=4pt]{->}(0,0)(8.5,11) +\end{pspicture} +\vskip0.5ex +\caption{An approximation to the area under the graph of $f(x)=x-(x/2)\cos x+2$ on $[0,8]$.}\label{Fig3} +\end{figure} + +To get Figure \ref{Fig3}, we have used the following \LaTeX\ code: +\bigskip\hrule +\noindent\begin{tabular}{@{}l} +\verb!\begin{pspicture}(0,0)(4.125,5.5)!\\ +\verb!\psset{plotpoints=500,algebraic,dotsize=2.5pt,unit=0.5}!\\ +\verb!\RiemannSum{0}{8}{x-(x/2)*cos(x)+2}{16}{0.00+0.50}{0.50+0.50}{0.25+0.50}!\\ +\verb!{x+0.25-((x+0.25)/2)*cos(x+0.25)+2}{x-0.25-((x-0.25)/2)*cos(x-0.25)+2}!\\ +\verb!\psaxes[ticksize=2.2pt,labelsep=4pt]{->}(0,0)(8.5,11)!\\ +\verb!\end{pspicture}! +\end{tabular}\hrule\bigskip + +In the second example below, we will draw an approximation to the integral of $f(x)=x\sin x$ over $[1,9]$. +Choosing $n=10$ and computing parameters needed, we get Figure \ref{Fig4}, mainly by +the command +\begin{align*} +&\textmtt{\symbol{92}RiemannSum\{$1$\}\{$9$\}\{$x\sin x$\}\{$10$\}\{$1.00+0.80$\}\{$1.80+0.80$\}\{$1.40+0.80$\}}\\ +&\textmtt{\{$(x+0.4)\sin(x+0.4)$\}\{$(x-0.4)\sin(x-0.4)$\}} +\end{align*} +in the drawing procedure. +\begin{figure}[htbp] +\centering\begin{pspicture}(0,-2.5)(4.75,4.25) +\psset{plotpoints=500,algebraic,dotsize=1pt 2,unit=0.5} +\RiemannSum{1}{9}{x*sin(x)}{10}% +{1.00+0.80}{1.80+0.80}{1.40+0.80}% +{(x+0.4)*sin(x+0.4)}{(x-0.4)*sin(x-0.4)} +\psaxes[labelFontSize=$\footnotesize$,Dy=1,ticksize=2.2pt,labelsep=4pt]{->}(0,0)(0,-5)(9.5,8.5) +\end{pspicture} +\caption{An approximation to the integral of $f(x)=x\sin x$ over $[1,9]$.}\label{Fig4} +\end{figure} +\section{Drawing the vector field of an ordinary differential equation of order one} +\subsection{Description}\label{sect1} + +Let us consider the differential equation +\begin{equation}\label{eqn2} +\frac{dy}{dx}=f(x,y). +\end{equation} +At each point $(x_0,y_0)$ in the domain $D$ of $f$, we will put a vector $\mathbf{v}$ with slope +$k=f(x_0,y_0)$. If $y(x_0)=y_0$, then $k$ is the slope of the tangent to the solution curve $y=y(x)$ +of (\ref{eqn2}) at $(x_0,y_0)$. The $\mathbf{v}$'s make a \textsl{vector field\/} and the picture +of this field would give us information about the shape of solution curves of (\ref{eqn2}), even +we have not found yet any solution of (\ref{eqn2}). + +The vector field of (\ref{eqn2}) will be depicted on a finite grid of points in $D$. This grid is made of +lines, paralell to the axes $Ox$ and $Oy$. The intersectional points of those lines are called \textsl{grid points\/} +and often indexed by $(x_i,y_j)$, $i=0,\ldots,N_x$, $j=0,\ldots,N_y$. For convenience, we will use +polar coordinates to locate the terminal point $(x,y)$ of a field vector, with the initial point at +grid point $(x_i,y_j)$. Then, we can write +\begin{align*} +x&=x_i+r\cos\varphi,\\ +y&=y_j+r\sin\varphi. +\end{align*} +Because $k=f(x_i,y_j)=\tan\varphi$ is finite, we may take $-\pi/2<\varphi<\pi/2$. +From $\sin^2\varphi+\cos^2\varphi=1$ and $\sin\varphi=k\cos\varphi$, we derive +$$\cos\varphi=\frac{1}{\sqrt{1+k^2}},\quad\sin\varphi=\frac{k}{\sqrt{1+k^2}}.$$ +\begin{figure}[htbp] +\centering\begin{pspicture}(0,0)(5,5) +\psset{unit=2} +\psaxes[labelFontSize=$\footnotesize$,Dx=0.5,Dy=0.5,labels=none,ticksize=2pt,labelsep=2pt,linewidth=0.5pt] +{->}(0,0)(2.5,2.5) +\psdots[dotstyle=*,dotsize=3pt,linecolor=red](1,1)(1.5,1)(1,1.5) +\psline[linewidth=0.3pt](1.5,0.5)(1.5,2)(1,2)(1,0.5)(2,0.5)(2,1.5) +\psline[linewidth=0.3pt](1,1.5)(2,1.5)\psline[linewidth=0.3pt](1,1)(2,1) +\psarc[linewidth=0.5pt,linestyle=dotted,dotsep=1.5pt](1,1){0.5}{-90}{90} +\psarc[linewidth=0.5pt,linestyle=dotted,dotsep=1.5pt](1.5,1){0.5}{-90}{90} +\psarc[linewidth=0.5pt,linestyle=dotted,dotsep=1.5pt](1,1.5){0.5}{-90}{90} +\psline[linewidth=0.8pt,linecolor=red]{->}(1,1.5)(1.27,1.92) +\psline[linewidth=0.8pt,linecolor=red]{->}(1,1)(1.382,1.322) +\psline[linewidth=0.8pt,linecolor=red]{->}(1.5,1)(1.977,1.147) +\rput(1,-0.1){$x_i$}\rput(1.5,-0.1){$x_{i+1}$}\rput(-0.1,1){$y_j$}\rput(-0.2,1.5){$y_{j+1}$} +\end{pspicture} +\caption{Field vectors on a grid.}\label{Fig5} +\end{figure} +The field vectors should all have the same magnitude and we choose here that length to be +$1/2$, that means $r=1/2$. Thus, vectors on the grid have their initial and terminal points as +$$(x_i,y_j),\quad \Big(x_i+\frac{1}{2}\cos\varphi,y_j+\frac{1}{2}\sin\varphi\Big),$$ +respectively. Hence, we easily get the parametrization of a vector at grid point +$(x_i,y_j)$: +\begin{align*} +x&=x_i+\frac{t}{2}\cos\varphi=x_i+\frac{t}{2\sqrt{1+k^2}},\\ +y&=y_j+\frac{t}{2}\sin\varphi=y_j+\frac{tk}{2\sqrt{1+k^2}}, +\end{align*} +where $t$ goes from $0$ to $1$, as along the direction of the vector. + +Of macros in \texttt{PSTricks} to draw lines from their parametrization, we select \texttt{\symbol{92}parametricplot}\footnote{\footnotesize +This macro is of ones, often added and updated in the package \texttt{pstricks-add}, the authors: +Dominique Rodriguez (\texttt{dominique.rodriguez@waika9.com}), Herbert Vo\ss\ (\texttt{voss@pstricks.de}).} +for its fitness. The macro has the syntax +$$\textmtt{\symbol{92}parametricplot[{\it settings}]\{$t_{\rm min}$\}\{$t_{\rm max}$\}\{$x(t)$|$y(t)$\}},$$ +where we may use the option \texttt{algebraic} to make the declaration of $x(t)$ and $y(t)$ simpler +in \texttt{ASCII} code. + +From the above description of one field vector, we now construct the vector field +on a grid within a domain $R=\{(x,y)\colon a\le x\le b,\,c\le y\le d\}$. To determine the grid, we confine grid points to the range +\begin{equation}\label{eqn3} +a\le x_i\le b,\quad c\le y_j\le d. +\end{equation} +We start with initial values $x_0=a$ and $y_0=c$ to have points $(x_i,y_j)$ with increments $\Delta x=\Delta y=\delta$, corresponding to the length of vectors and the distance +between grid points as indicated in Figure \ref{Fig5}. Thus, to draw vectors at grid points +$(x_i,y_j)$, we need two loops for indices $i$ and $j$, with $0\le i\le\lfloor m/\delta\rfloor$, $0\le j\le\lfloor n/\delta\rfloor$, where +$m=b-a$, $n=d-c$. Apparently, these two loops are nested \texttt{\symbol{92}multido}s, with variable declaration +for each loop as follows +\begin{align*} +\texttt{\symbol{92}nx}&=\text{initial value}+\text{increment}=x_0+\Delta x,\\ +\texttt{\symbol{92}ny}&=\text{initial value}+\text{increment}=y_0+\Delta y. +\end{align*} +Finally, we will replace \texttt{\symbol{92}nx}, \texttt{\symbol{92}ny} by $x_i$, $y_j$ in the +below calling sequence for simplicity. + +Thus, the main command to draw the vector field of the equation (\ref{eqn2}) on the grid (\ref{eqn3}) +is +\begin{align*} +&\textmtt{\symbol{92}multido\big\{$y_j=y_0+\Delta y$\big\}\big\{$\lfloor n/\delta\rfloor$\big\}}\textmtt{\bigg\{\symbol{92}multido\big\{$x_i=x_0+\Delta x$\big\}\big\{$\lfloor m/\delta\rfloor$\big\}}\\ +&\quad\textmtt{\Big\{\symbol{92}parametricplot[{\it settings}]\{$0$\}\{$1$\}\Big\{$x_i+\frac{t}{2\sqrt{1+\big[f(x_i,y_j)\big]^2}}$\Big| +$y_j+\frac{tf(x_i,y_j)}{2\sqrt{1+\big[f(x_i,y_j)\big]^2}}$\Big\}\bigg\}} +\end{align*} +where we at least use \texttt{arrows=->} and \texttt{algebraic} for \textit{settings}. + +We can combine the steps mentioned above to define a drawing procedure, say \texttt{\symbol{92}avecfld}, +that consists of $6$ parameters in the order as +\texttt{\symbol{92}nx=}$x_0+\Delta x$, \texttt{\symbol{92}ny=}$y_0+\Delta y$, $\lfloor m/\delta\rfloor$, $\lfloor n/\delta\rfloor$, $\delta$ +and $f(\texttt{\symbol{92}nx},\texttt{\symbol{92}ny})$. We may change these values to modify +the vector field or to avoid the vector intersection. Such a procedure is suggested to be +\bigskip\hrule +\noindent\begin{tabular}{@{}l} +\verb!\def\avecfld#1#2#3#4#5#6{%!\\ +\verb!\multido{#2}{#4}{\multido{#1}{#3}!\\ +\verb!{\parametricplot[algebraic,arrows=->,linecolor=red]{0}{1}!\\ +\verb!{\nx+((#5)*t)*(1/sqrt(1+(#6)^2))|\ny+((#5)*t)*(1/sqrt(1+(#6)^2))*(#6)}}}}! +\end{tabular}\hrule\bigskip + +Actually, the procedure \texttt{\symbol{92}parametricplot} is used here only to draw a vector by its parametrization $(x(t),y(t))$, so we can use the structure \texttt{\symbol{92}curvepnodes} in the package \texttt{pst-node}\footnote{\footnotesize +Package authors: Timothy Van Zandt (\texttt{tvz@econ.insead.fr}), Michael Sharpe (\texttt{msharpe@euclid.ucsd.edu}) and Herbert Vo\ss\ (\texttt{hvoss@tug.org}).} to +extract the two ending points of the curve $(x(t),y(t))$ by the command +$$\textmtt{\symbol{92}curvepnodes[algebraic,plotpoints=2]\{0\}\{1\}\{$x(t)$|$y(t)$\}\{P\}},$$ +where \texttt{P} is a name of the root of nodes and we just get the two nodes \texttt{P0}, \texttt{P1} when executing this command. Then, the corresponding +vector is drawn by the command +$$\textmtt{\symbol{92}psline[linecolor={\it settings}]\{->\}(P0)(P1)}$$ + +Therefore, another procedure to draw a vector field may be defined as +\bigskip\hrule +\noindent\begin{tabular}{@{}l} +\verb!\def\anothervecfld#1#2#3#4#5#6{%!\\ +\verb!\multido{#2}{#4}{\multido{#1}{#3}!\\ +\verb!{\curvepnodes[algebraic,plotpoints=2]{0}{1}!\\ +\verb!{\nx+((#5)*t)*(1/sqrt(1+(#6)^2))|\ny+((#5)*t)*(1/sqrt(1+(#6)^2))*(#6)}{P}!\\ +\verb!\psline[linecolor=red]{->}(P0)(P1)}}}! +\end{tabular}\hrule +\subsection{Examples} +Firstly, we consider the equation that describes an object falling in a resistive medium: +\begin{equation}\label{eqn4} +\frac{dv}{dt}=9.8-\frac{v}{5}, +\end{equation} +where $v=v(t)$ is the speed of the object in time $t$. In Figure \ref{Fig6}, the vector field of (\ref{eqn4}) is given +on the grid $R=\{(t,y)\colon 0\le t\le 9,\,46\le v\le 52\}$, together with the graph of the equilibrium solution +$v=49$. +\begin{figure}[htbp] +\centering\begin{pspicture}(0,41.4)(8.55,47.25) +\psset{xunit=0.9,yunit=0.9} +\multido{\ny=46.25+0.50}{13} +{\multido{\nx=0.25+0.50}{18} +{\curvepnodes[algebraic,plotpoints=2]{0}{1}{\nx+(t/2)*(1/sqrt(1+(9.8-0.2*(\ny))^2))|\ny+(t/2)*(1/sqrt(1+(9.8-(0.2)*(\ny))^2))*(9.8-(0.2)*(\ny))}{P} +\psline[linewidth=0.7pt,linecolor=red]{->}(P0)(P1)}} +\psplot[algebraic,linewidth=1.2pt]{0}{9.25}{49} +\psaxes[labelFontSize=$\footnotesize$,ticksize=2.2pt,labelsep=4pt,Dy=1,Dx=1,Oy=46,linewidth=0.7pt]{->}(0,46)(0,46)(9.5,52.5) +\rput(9.5,45.8){$t$}\rput(-0.2,52.5){$y$} +\end{pspicture} +\caption{The vector field of (\ref{eqn4}).}\label{Fig6} +\end{figure} + +Figure \ref{Fig6} is made of the following \LaTeX\ code: +\bigskip\hrule +\noindent\begin{tabular}{@{}l} +\verb!\begin{pspicture}(0,46)(9.5,52.5)!\\ +\verb!\anothervecfld{\nx=0.25+0.50}{\ny=46.25+0.50}{18}{12}{0.5}{9.8-0.2*\ny}!\\ +\verb!\psplot[algebraic,linewidth=1.2pt]{0}{9}{49}!\\ +\verb!\psaxes[Dy=1,Dx=1,Oy=46]{->}(0,46)(0,46)(9.5,52.5)!\\ +\verb!\rput(9.5,45.8){$t$}\rput(-0.2,52.5){$y$}!\\ +\verb!\end{pspicture}! +\end{tabular}\hrule\bigskip + +Let us next consider the problem +\begin{equation}\label{eqn5} +\frac{dy}{dx}=x+y,\quad y(0)=0. +\end{equation} +It is easy to check that $y=e^x-x-1$ is the unique solution to (\ref{eqn5}). We now draw +the vector field of (\ref{eqn5}) and the solution curve\footnote{\footnotesize +We have used ${\rm ch}(1)+{\rm sh}(1)$ for the declaration of $e$, natural base of logarithmic function.} on the grid $R=\{(x,y)\colon 0\le x\le 3,\,0\le y\le 5\}$ in +Figure \ref{Fig7}. +\begin{figure}[htbp] +\centering\begin{pspicture}(0,0)(3.25,5.5) +\psset{unit=1} +\multido{\ny=0.25+0.50}{10} +{\multido{\nx=0.25+0.50}{6} +{\curvepnodes[algebraic,plotpoints=2]{0}{1}{\nx+(t/2)*(1/sqrt(1+(\nx+\ny)^2))|\ny+(t/2)*(1/sqrt(1+(\nx+\ny)^2))*(\nx+\ny)}{P} +\psline[linewidth=0.7pt,linecolor=red]{->}(P0)(P1)}} +\psplot[algebraic,linewidth=1.2pt]{0}{2.15}{(sh(1)+ch(1))^x-x-1} +\psaxes[labelFontSize=$\footnotesize$,Dy=1,Dx=1,ticksize=2.2pt,labelsep=4pt,linewidth=0.7pt]{->}(0,0)(3.5,5.5) +\rput(3.5,-0.2){$x$}\rput(-0.25,5.5){$y$} +\end{pspicture} +\caption{The vector field of (\ref{eqn5}).}\label{Fig7} +\end{figure} + +We then go to the logistic equation, which is chosen to be a model for the dependence +of the population size $P$ on time $t$ in Biology: +\begin{equation}\label{eqn6} +\frac{dP}{dt}=kP\Big(1-\frac{P}{M}\Big), +\end{equation} +where $k$ and $M$ are constants, respectively various to selected species and environment. +For specification, we take, for instance, $k=0.5$ and $M=100$. The right hand side of +(\ref{eqn6}) then becomes $f(t,P)=0.5\,P(1-0.01\,P)$. In Figure \ref{Fig8}, we draw the vector field +of (\ref{eqn6}) on the grid $R=\{(t,P)\colon 0\le t\le 10,\,95\le P\le 100\}$ and the equilibrium +solution curve $P=100$. Furthermore, with the initial condition $P(0)=95$, the equation (\ref{eqn6}) +has the unique solution $P=1900(e^{-0.5t}+19)^{-1}$. This solution curve is also given in Figure \ref{Fig8}. +\begin{figure}[htbp] +\centering\begin{pspicture}(0,76)(8.4,80.4) +\psset{xunit=0.8,yunit=0.8} +\multido{\ny=95.25+0.50}{10} +{\multido{\nx=0.25+0.50}{20} +{\curvepnodes[algebraic,plotpoints=2]{0}{1}{\nx+(t/2)*(1/sqrt(1+((0.5)*(\ny)*(1-(0.01)*(\ny)))^2))|\ny+(t/2)*(1/sqrt(1+((0.5)*(\ny)*(1-(0.01)*(\ny)))^2))*((0.5)*(\ny)*(1-(0.01)*(\ny)))}{P} +\psline[linewidth=0.7pt,linecolor=red]{->}(P0)(P1)}} +\psplot[algebraic,linewidth=1.2pt]{0}{10.25}{100} +\psplot[algebraic,linewidth=1.2pt]{0}{10.25}{1900/((ch(1)+sh(1))^(-0.5*x)+19)} +\psaxes[labelFontSize=$\footnotesize$,Dy=1,Dx=1,Oy=95,ticksize=2pt,labelsep=4pt,linewidth=0.7pt]{->}(0,95)(0,95)(10.5,100.5) +\rput(10.5,94.8){$t$}\rput(-0.25,100.5){$P$} +\end{pspicture} +\caption{The vector field of (\ref{eqn6}) with $k=0.5$ and $M=100$.}\label{Fig8} +\end{figure} + +The previous differential equations are all of seperated variable or linear cases that +can be solved for closed-form solutions by some simple integration formulas. We will consider one more +equation of the non-linear case whose solution can only be approximated by numerical methods. +The vector field of such an equation is so useful and we will use the Runge-Kutta curves (of order $4$) +to add more information about the behaviour of solution curves. Here, those Runge-Kutta curves are depicted by the procedure +\texttt{\symbol{92}psplotDiffEqn}, also updated from the package \texttt{pstricks-add}. + +The vector field of the non-linear differential equation +\begin{equation}\label{eqn7} +\frac{dy}{dx}=y^2-xy+1 +\end{equation} +will be depicted on the grid $R=\{(x,y)\colon -3\le x\le 3,\,-3\le y\le 3\}$ and the solutions +of Cauchy problems for (\ref{eqn7}), corresponding to initial conditions +\begin{listof} +\item $y(-3)=-1$, +\item $y(-2)=-3$, +\item $y(-3)=-0.4$, +\end{listof} +will be approximated by the method of Runge-Kutta, with the grid size $h=0.2$. It is very easy +to recognize approximate curves, respective to (i), (ii) and (iii) in Figure \ref{Fig9} below. +\begin{figure}[htbp] +\centering\begin{pspicture}(-3.6,-3.6)(4.2,4.2) +\psset{unit=1.2,dotsize=2.6pt} +\anothervecfld{\nx=-3.00+0.4}{\ny=-3.00+0.4}{16}{16}{0.35}{(\ny)^2-(\nx)*(\ny)+1} +\psplotDiffEqn[linewidth=1.2pt,algebraic,showpoints=true,plotpoints=24,method=rk4]{-3}{1.9}{-1}{(y[0])^2-x*y[0]+1} +\psplotDiffEqn[linewidth=1.2pt,algebraic,showpoints=true,plotpoints=25,method=rk4]{-2}{3}{-3}{(y[0])^2-x*y[0]+1} +\psplotDiffEqn[linewidth=1.2pt,algebraic,showpoints=true,plotpoints=10,method=rk4]{-3}{-0.96}{-0.4}{(y[0])^2-x*y[0]+1} +\psaxes[labelFontSize=$\footnotesize$,Dy=1,Dx=1,ticksize=2.2pt,labelsep=4pt,linewidth=0.7pt]{->}(0,0)(-3,-3)(3.5,3.5) +\rput(3.5,-0.2){$x$}\rput(-0.25,3.5){$y$} +\end{pspicture} +\caption{The vector field of (\ref{eqn7}) and the Runge-Kutta curves.}\label{Fig9} +\end{figure} +\subsection{Remarks on how to color arrows properly for a vector field} +There remains a problem in drawing a vector field. That is coloring arrows. +Obviously, their color shade should vary according to their slope and this would give us the picture of domains containing +increase or decrease solutions of a differential equation. In some cases, we even know how large the rate of change of those +solutions is in a specific domain. + +In Subsection \ref{sect1}, we know for the equation (\ref{eqn2}) that $f(x_i,y_j)$ is right the slope of +field vectors at grid points $(x_i,y_j)$, and we will divide these slopes into some number of scales, corresponding to the +degree of color shades. Here, we confine our interest to a continuous function $f(x,y)$ in two independent variables on a domain +$R=\{(x,y)\colon a\le x\le b,\,c\le y\le d\}$ and choose the scale of $10$ degrees. This number of degrees can be changed to any positive integer. + +According to the input data from the differential equation (\ref{eqn2}), the set $R$ and the grid points on it and the value $M=\max\{|f(x_i,y_j)|\colon +0\le i\le\lfloor m/\Delta x\rfloor,\,0\le j\le\lfloor n/\Delta y\rfloor\}$, where $m=b-a$ and $n=d-c$, +we can now define the degree of color shade for each arrow in our vector field. It should be an integer $n_{ij}$ such that +$n_{ij}=\lfloor 10|f(x_i,y_j)|/M\rfloor$, that is +\begin{equation}\label{eqn8}n_{ij}M\le 10|f(x_i,y_j)|<(n_{ij}+1)M.\end{equation} +For finding such an integer, in \TeX\ codes, we need one \texttt{\symbol{92}newcount} for it and two \texttt{\symbol{92}newdimen} for +$f(x_i,y_j)$ and intermediate values to be compared with $|f(x_i,y_j)|$. For more explanation, let us begin with settings +\texttt{\symbol{92}newcount\symbol{92}intg} (referring (ref.) to ``integer''), \texttt{\symbol{92}newdimen\symbol{92}slope} (ref. to ``slope'') and \texttt{\symbol{92}newdimen\symbol{92}interm} +(ref. to ``intermediate values''). Then, the integer $n_{ij}$ at stage $(i,j)$ within the two \texttt{\symbol{92}multido} loops can be defined by the recursive macro \texttt{\symbol{92}fintg} (ref. to ``find the integer'') as follows +\begin{verbatim} + \def\fintg{\interm=Mpt \interm=\intg\interm% + \ifdim\ifdim\slope<0pt -\fi\slope<\interm\advance\intg by -1\relax + \else\advance\intg by 1\fintg\fi} +\end{verbatim} +where \texttt{M} and \texttt{\symbol{92}slope} are holding the values $M$ and $f(x_i,y_j)$, respectively. Note that, before running our macro, \verb!\slope! should be multiplied +by $10$ with the assignment \texttt{\symbol{92}slope=10\symbol{92}slope}, as defined in (\ref{eqn8}). Besides, by simulating the expression of $f(x,y)$, the calculation of $f(x_i,y_j)$ +should be declared with operations on \texttt{\symbol{92}newcount}s and \texttt{\symbol{92}newdimen}s. Then, the integer $n_{ij}$, which is found at stage $(i,j)$, should take its +degree, say $k$, from $0$ to $10$ by its value, suitably associated to the command \textmtt{\symbol{92}psline[linecolor=red!case-k]\{->\}(P0)(P1)}. +Here, we choose \texttt{red} for the main color (it can be changed, of course), and \texttt{case-k} will be replaced with an appropriate percentage of \texttt{red}. Finally, +such a color scale is local and relative, so we can use one more parameter in the procedure to adjust color shades. +The old procedures take $6$ parameters and the new one will take two more parameters: one for declaration of computing $f(x_i,y_j)$ and the other +for adjusting color shades. + +Let us take some examples on how to compute $f(x_i,y_j)$ by \TeX\ codes or by commands from the package \texttt{calculator}\footnote{\footnotesize +Package author: Robert Fuster (\texttt{rfuster@mat.upv.es}).}. For a simple polynomial $f(x,y)$, +computing $f(x_i,y_j)$ by \TeX\ codes might be facile. Because \verb!\nx! and \verb!\ny! are respectively holding values of +$x_i$ and $y_j$, we need the two corresponding variables \verb!\newdimen\fx! and \verb!\newdimen\fy! to take these values. By assigning \verb!\fx=\nx pt\fy=\ny pt!, +we compute $f(\verb!\nx!,\verb!\ny!)$ and assign its value to \verb!\slope!. The declaration of calculations for some cases of $f(x,y)$ is given in the following table. + +\begin{table}[htbp] +\centering\begin{tabular}{c|l} +$f(x,y)$&\multicolumn{1}{c}{\TeX\ codes for computing $f(\texttt{\symbol{92}nx},\texttt{\symbol{92}ny})$} \\ \hline +$x+y$&\verb!\advance\slope by \fx \advance\slope by \fy!\\ \hline +$1-xy$&\verb!\advance\slope by -\decimal\fx\fy \advance\slope by 1pt!\\ \hline +$y(3-y)$&\verb!\advance\slope by -\decimal\fy\fy \advance\slope by 3\fy!\\ \hline +$y^2-xy$&\verb!\advance\slope by \decimal\fy\fy \advance\slope by -\decimal\fx\fy!\\ \hline +\end{tabular} +\end{table} +In the table, the command \verb!\decimal!, which is quotative from \cite{five} for producing decimal numbers from dimensions, is put in the preamble using a definition as +\begin{verbatim} + \def\xch{\catcode`\p=12 \catcode`\t=12}\def\ych{\catcode`\p=11 \catcode`\t=11} + \xch \def\dec#1pt{#1}\ych \def\decimal#1{\expandafter\dec \the#1} +\end{verbatim} + +For a transcendental or rational function $f(x,y)$, we may use the package \texttt{calculator} for +computing $f(x_i,y_j)$. The following table shows how to perform calculations. +\begin{table}[htbp] +\centering\begin{tabular}{c|l} +$f(x,y)$&\multicolumn{1}{c}{The commands from the package \texttt{calculator} for computing $f(\texttt{\symbol{92}nx},\texttt{\symbol{92}ny})$} \\ \hline +$\sin(y-x)$&\verb!\SUBTRACT{\ny}{\nx}{\sola}\SIN{\sola}{\solb}\slope=\solb pt!\\ \hline +\raisebox{-2ex}[0pt][0pt]{$2xy/(1+y^2)$}&\verb!\SUMfunction{\ONEfunction}{\SQUAREfunction}{\Fncty}!\\ +&\verb!\Fncty{\ny}{\soly}{\Dsoly}\DIVIDE{\Dsoly}{\soly}{\tempa}!\\ +&\verb!\MULTIPLY{\nx}{\tempa}{\tempb}\slope=\tempb pt!\\ \hline +\end{tabular} +\end{table} + +From the old macros \verb!\avecfld! or \verb!\anothervecfld!, we will construct the new one \verb!\vecfldnew! by adding up to the former the two parameters as described above. According to +the description of new parameters and of known ones, the calling sequence of \verb!\vecfldnew! may have the form of +$$\textmtt{\symbol{92}vecfldnew\{\symbol{92}nx$=x_0+\Delta x$\}\{\symbol{92}ny$=y_0+\Delta y$\}\{$n_x$\}\{$n_y$\}\{$\ell$\}\{$f(\texttt{\symbol{92}nx},\texttt{\symbol{92}ny})$\}\{{\rm\TeX\ codes}\}\{$n_a$\}}$$ +where $n_a$ is an estimate value for $M$ and can be adjusted to be greater or less than $M$. This flexible mechanism might be to increase or decrease the degree of +color shades. Finally, \verb!\intg! and \verb!\slope! should be reset to +zero at the end of each stage. Now, all materials to make the new macro are ready, and its definition is suggested to be +\bigskip\hrule +\noindent\begin{tabular}{@{}l} +\verb!\def\vecfldnew#1#2#3#4#5#6#7#8{%!\\ +\verb!\newcount\intg \newdimen\slope \newdimen\interm \newdimen\fx \newdimen\fy!\\ +\verb!\def\fintg{\interm=#8 \interm=\intg\interm%!\\ +\verb! \ifdim\ifdim\slope<0pt -\fi\slope>\interm \advance\intg by 1\fintg\fi}!\\ +\verb!\multido{#2}{#4}!\\ +\verb!{\multido{#1}{#3}!\\ +\verb!{\curvepnodes[algebraic,plotpoints=2]{0}{1}!\\ +\verb!{\nx+((#5)*t)*(1/sqrt(1+(#6)^2))|\ny+((#5)*t)*(1/sqrt(1+(#6)^2))*(#6)}{P}!\\ +\verb!#7\slope=10\slope \fintg \ifnum\intg>10\psline[linecolor=red]{->}(P0)(P1)!\\ +\verb+\else\ifnum\intg=0\psline[linecolor=red!5]{->}(P0)(P1)+\\ +\verb+\else\multiply\intg by 10\psline[linecolor=red!\the\intg]{->}(P0)(P1)\fi\fi+\\ +\verb+\intg=0\slope=0pt+\\ +\verb+}}}+ +\end{tabular}\hrule\bigskip + +If we predefine some scale of degrees, instead of the code $\verb!\ifnum\intg>10!\ldots\verb!\fi\fi!$, the structure \verb!\ifcase! can be used as +$$\begin{array}{c} +\verb!\ifcase\intg!\\ +\verb+\psline[linecolor=red!5]{->}(P0)(P1)\or+\\ +\verb+\psline[linecolor=red!10]{->}(P0)(P1)\or+\\ +\vdots\\ +\verb!\psline[linecolor=red]{->}(P0)(P1)\fi! +\end{array}$$ + +The first example is given with two values of $n_a$ to see how different the color shades are between the two cases. The left vector field in Figure \ref{Fig10} +is made of the calling sequence +\begin{verbatim} +\vecfldnew{\nx=-2.00+0.3}{\ny=-2.00+0.3}{14}{14}{0.3}{(\nx)-2*(\ny)} +{\fy=\ny pt \fx=\nx pt \advance\slope by -2\fy \advance\slope by \fx}{7pt} +\end{verbatim} +\begin{figure}[htbp] +\centering\includegraphics[width=4.6cm]{vecb5} +\hskip1cm\includegraphics[width=4.6cm]{vecb6} +\caption{The vector fields of the equation $y'=x-2y$ with $n_a=\texttt{7pt}$ (the left) and $n_a=\texttt{4pt}$ (the right)}\label{Fig10} +\end{figure} + +In Figure \ref{Fig11}, the vector fields of the equations $y'=y-x$ and $y'=x(2-y)$ are respectively drawn by the calling sequences +\begin{verbatim} +\vecfldnew{\nx=-3.00+0.4}{\ny=-3.00+0.4}{15}{15}{0.35}{(\ny)-(\nx)} +{\fy=\ny pt \fx=\nx pt \advance\slope by -\fx \advance\slope by \fy}{5pt} +\end{verbatim} +and +\begin{verbatim} +\vecfldnew{\nx=-3.00+0.4}{\ny=-3.00+0.4}{15}{15}{0.35}{(\nx)*(2-(\ny))} +{\fy=\ny pt \fx=\nx pt \advance\slope by -\decimal\fx\fy +\advance\slope by 2\fx}{6pt} +\end{verbatim} +\begin{figure}[htbp] +\centering\includegraphics[width=6.2cm]{vecb3} +\hskip1cm\includegraphics[width=6.2cm]{vecb4} +\caption{The vector fields of the equation $y'=y-x$ (the left) and $y'=x(2-y)$ (the right).}\label{Fig11} +\end{figure} + +Finally, we consider two more examples on vector fields of differential equations $y'=f(x,y)$ containing trigonometric or rational functions on their right side. The calling sequences +\begin{verbatim} +\vecfldnew{\nx=-3.00+0.4}{\ny=-3.00+0.4}{15}{15}{0.35}{sin(\nx)*cos(\ny)} +{\SIN{\nx}{\tmpa}\COS{\ny}{\tmpb}\MULTIPLY{\tmpa}{\tmpb}{\tmpc} +\slope=\tmpc pt}{0.6pt} +\end{verbatim} +and +\begin{verbatim} +\vecfldnew{\nx=-3.00+0.3}{\ny=-3.00+0.3}{20}{20}{0.3}{2*(\nx)*(\ny)/(1+(\ny)^2)} +{\SUMfunction{\ONEfunction}{\SQUAREfunction}{\Fncty}\Fncty{\ny}{\soly}{\Dsoly} +\DIVIDE{\Dsoly}{\soly}{\tempa}\MULTIPLY{\nx}{\tempa}{\tempb} +\slope=\tempb pt}{2.5pt} +\end{verbatim} +respectively result in the vector fields on the left and on the right in Figure \ref{Fig12}. + +\begin{figure}[htbp] +\centering\includegraphics[width=6.2cm]{vecb1} +\hskip1cm\includegraphics[width=6.2cm]{vecb2} +\caption{The vector fields of the equation $y'=\sin(x)\cos(y)$ (the left) and $y'=2xy/(1+y^2)$ (the right).}\label{Fig12} +\end{figure} +\section{Drawing partitions of a simply connected plane domain} +\subsection{Description} +B\'{e}zier curves have been used in many different aspects, but mostly in computer graphics to model smooth curves designed by computer +programs. We just recall here the vector function of a B\'{e}zier curve of degree $n$ with $n+1$ control points whose position +vectors are $\mathbf{r}_0$, $\mathbf{r}_1$, \dots, $\mathbf{r}_n$. It is the vector function $\mathbf{r}(t)$ in a real variable $t\in[0,1]$ that +is defined as +\begin{equation}\label{eqn9} +\mathbf{r}(t)=\sum_{i=0}^n\binom{n}{i}t^i(1-t)^{n-i}\mathbf{r}_i. +\end{equation} +Since $\mathbf{r}(0)=\mathbf{r}_0$, $\mathbf{r}(1)=\mathbf{r}_n$ and $\mathbf{r}'(0)=n(\mathbf{r}_1-\mathbf{r}_0)$, $\mathbf{r}'(1)=n(\mathbf{r}_n-\mathbf{r}_{n-1})$, +we can take a curve such that it is closed and tangent to a line $(\Delta)$ at the point whose position vector is $\mathbf{r}_0$, by +letting $\mathbf{r}_n=\mathbf{r}_0$ and the points whose position vectors are $\mathbf{r}_0$, $\mathbf{r}_1$ and $\mathbf{r}_{n-1}$ +all be on the line $(\Delta)$. + +Taking a closed B\'{e}zier curve to be the boundary of a plane domain has the great advantage. That is because coordinates of points +on the curve are evaluated by polynomials, and the curve has necessary smoothness, beautiful enough for graphic illustrations. +\texttt{PSTricks} has the procedure \verb!\parametricplot! to draw such a curve. But, an important problem here is how to control +the coordinates of its points to construct a procedure for drawing a partition of a simply connected plane domain and coloring +(or marking) its cells having common points with the boundary of the domain. We particularly need such a procedure for illustrations +of essential notions in measure theory or double integral definition. Because of the limitation in accuracy imposed +by the \TeX\ arithmetic, we should mention about $2$-dimension vector functions of a B\'{e}zier curve of small degree (in fact, as small as possible). +According to the above requirements for the closed boundary of a simply connected domain, we will design a family of closed B\'{e}zier curves of +degree $6$, although we may extend their degree to $12$. + +From the expression of $\mathbf{r}(t)$ in (\ref{eqn9}), we will construct the following functions and procedures: +\begin{itemize} +\item The factorial function \texttt{FACTORIAL}. +\item The binomial function \texttt{BINOMIAL}. +\item Polynomials $t^m(1-t)^n$. +\item A procedure to compute a sum of $7$ terms: values of $\mathbf{r}(t)$ as linear combinations +of the Bernstein polynomials +$$\binom{6}{k}t^k(1-t)^{6-k},\quad k=0,\ldots,6.$$ +\end{itemize} +These functions and procedures are macros given by definitions with or without parameters. In the following subsections we will construct them step by step and +show how to make connections between them. +\subsubsection{Factorial functions and binomial coefficients} +To obtain the factorial function, a procedure for multiplying consecutively an integer $n$ with its diminished values will be constructed. For the declared +integer variables \verb!\Fa! and \verb!\Fct!, the latter will hold values of the function, and the former will hold factors that are multiplied consecutively +and reduced by $1$. Namely, if \verb!\Fa! is $0$ or $1$ then \verb!\Fct! takes $1$; else if $\texttt{\symbol{92}Fa}>1$ then \verb!\Fct! takes +the initial value of \verb!\Fa! to begin a loop: diminishing \verb!\Fa! by $1$ and updating \verb!\Fct! by multiplying its old value with the new value of \verb!\Fa!. +The loop is defined by the control sequence \verb!\Factor! and is ended when \verb!\Fa! is diminished to $1$; then, the factorial function obtains its value from +the one given back after calling \verb!\FACTORIAL! with one parameter. +\begin{verbatim} +\newcount\Fa\newcount\Fct +\newcount\tempA +\def\Factor{\ifnum\Fa=1\relax\else\advance\Fa by -1\multiply\Fct by \Fa\Factor\fi} +\def\FACTORIAL#1{\Fa=#1 \ifnum\Fa=0 \Fct=1\relax\else\Fct=\Fa \Factor\fi% +\global\tempA=\Fct} +\end{verbatim} +Since \TeX\ limits the largest integer number to $2^{30}-1$, we can evaluate only to +$$\textmtt{\symbol{92}FACTORIAL\{$12$\}}=12!=479001600.$$ +Besides, the last value of \verb!\Fct!, which is right the one obtained from calling the function, is assigned to a global integer variable \verb!\tempA! for use +in other procedures. Finally, make the calling sequence +$$\textmtt{\symbol{92}FACTORIAL\{k\}\symbol{92}the\symbol{92}Fct}$$ +to obtain and show up the value of $k!$. From these values we derive binomial coefficients by the formula +$$\binom{m}{n}=\frac{m!}{n!(m-n)!}.$$ +A simple procedure that makes division of $m!$ by $n!$, then of the result by $(m-n)!$ can be performed by the control sequence +\verb!\BINOMIAL! with two parameters as follows +\begin{verbatim} +\newcount\BINOM +\newcount\temp\newcount\tmp +\def\BINOMIAL#1#2{% +\temp=#1\advance\temp by -#2 +\FACTORIAL{#1} +\tmp=\tempA +\FACTORIAL{\temp} +\temp=\tempA +\divide\tmp by \temp +\FACTORIAL{#2} +\temp=\tempA +\divide\tmp by \temp\global\BINOM=\tmp} +\end{verbatim} +Again, the last value of \verb!\tmp! is assigned to a global integer variable \verb!\BINOM! for later use. Because the vector function of a B\'{e}zier +curve of degree $6$ is only needed, so are the binomial coefficients \verb!\BINOMIAL{!$6$\verb!}{!$k$\verb!}!, $k=0,1,\ldots,6$. +\subsubsection{Bernstein polynomials} +The construction of Bernstein functions is based on an iterative multiplication of the same value $t$ by a given number $m$ of times, +then the process is repeated with the value $1-t$ by a given number $n$ of times. The last result is multiplied by a variable that holds +the value of $\binom{m+n}{m}$. We will use an integer variable \verb!\kc! to count times of multiplication for the same factor $t$ or $1-t$. +The procedure of iterative multiplication \verb!\xmult! with one parameter $m$ or $n$ is a recursive one that is performed +iteratively until \verb!\kc! reaches the value $m$ or $n$. +\begin{verbatim} +\newdimen\Xa\newdimen\Yb\newcount\kc +\def\xmult#1{\ifnum\kc<#1\advance\kc by 1\Yb=\decimal\Xa\Yb\xmult{#1} + \else\relax\fi} +\end{verbatim} +Finally, the value of $t^m(1-t)^n$ that is hold by the variable \verb!\Yb! is multiplied by \verb!\BINOM!, the value $\binom{6}{m}$. +Thus, Bernstein polynomials are given by the following macro \verb!\BERNSTEIN! with three parameters that hold values of +$m$, $n$ and $t$, respectively. +\begin{verbatim} +\newdimen\BSTemp +\def\BERNSTEIN#1#2#3{\Xa=#3pt\kc=0\Yb=1pt\xmult{#1}\kc=0\Xa=-\Xa +\advance\Xa by 1pt\xmult{#2}\BINOMIAL{6}{#1}\global\BSTemp=\BINOM\Yb} +\end{verbatim} +Also, values of \verb!\BERNSTEIN! are assigned to a global variable. +\subsubsection{Vector functions of a plane B\'{e}zier curve of degree $6$} +We will construct here the vector function of a plane and closed B\'{e}zier curve with the control points +$M_0(x_0,y_0)$, $M_1(x_1,y_1)$, \dots, $M_6(x_6,y_6)$ that is chosen in such a way that the curve is tangent to +a line $(\Delta)$ at its initial point $M_0$. According to the mentioned properties of a B\'{e}zier curve, the requirements +for the curve are satisfied when $M_6=M_0$ and $M_0$, $M_1$, $M_5$ are all on the line $(\Delta)$. +\begin{figure}[htbp] +\centering\begin{pspicture}(0,0)(5,4.3) +\psset{unit=0.6cm} +\newcount\pos\def\drawpts#1#2{\newdimen\xrp\xrp=2pt\newdimen\yrp\yrp=2pt\advance\xrp by #1pt\advance\yrp by #2pt\psdot(\decimal\xrp,\decimal\yrp) +\rput[rt](\decimal\xrp,\decimal\yrp){$M_{\the\pos}$}\advance\pos by 1} +\drawpts{0}{0}\drawpts{-1.5}{2}\drawpts{2}{5}\drawpts{6}{3}\drawpts{5}{-1}\drawpts{1.5}{-2} +\parametricplot[algebraic,linewidth=0.5pt]{0}{0.5}{\XBsix{2}{1}|\YBsix{2}{1}} +\parametricplot[algebraic,linewidth=0.5pt]{0.5}{1}{\XBsix{2}{1}|\YBsix{2}{1}} +\psline[linewidth=0.2pt](0.5,4)(3.5,0) +\end{pspicture} +\caption{A B\'{e}zier curve of degree $6$ with the required control points.}\label{Fig13} +\end{figure} +In case $n=6$, from (\ref{eqn9}) we derive the vector function $\mathbf{r}(t)$ with the two components +\begin{equation}\label{eqn10} +X(t)=\sum_{i=0}^6\binom{6}{i}t^i(1-t)^{6-i}x_i,\quad Y(t)=\sum_{i=0}^6\binom{6}{i}t^i(1-t)^{6-i}y_i. +\end{equation} +We first evaluate values of $X(t)$ by the procedures that just have been constructed, with a choice of $x_i$, $i=1,\ldots,6$, from a given $x_0$ +and the requirements of the B\'{e}zier curve as in Figure \ref{Fig13}. For example, we may take +$$x_1=x_0-1.5,\quad x_2=x_0+2,\quad x_3=x_0+6,\quad x_4=x_0+5,\quad x_5=x_0+1.5,\quad x_6=x_0.$$ +Moreover, we may change the shape of the curve by taking a factor $\alpha$ for $X(t)$. In short, we will make a procedure to evaluate the sum of terms +$\binom{6}{i}t^i(1-t)^{6-i}(\alpha x_i)$, $i=0,1,\ldots,6$. In fact, it is a procedure to add up values into a global variable \verb!\XBST! that is called +by a macro \verb!\XBC! with three declaring parameters for $x_i$, $t$ and $\alpha$. The value of $\alpha$ is used last when being multiplied by a +variable that holds $X(t)$. Such a procedure is suggested to be the following macro +\begin{verbatim} +\newdimen\Xrf\newdimen\Yrf +\newdimen\XBST\newdimen\YBST +\def\XBC#1#2#3{% +\BERNSTEIN{0}{6}{#2} +\Xrf=#1pt +\XBST=\decimal\BSTemp\Xrf +\BERNSTEIN{1}{5}{#2} +\Xrf=#1pt\advance\Xrf by -1.5pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{2}{4}{#2} +\Xrf=#1pt\advance\Xrf by 2pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{3}{3}{#2} +\Xrf=#1pt\advance\Xrf by 6pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{4}{2}{#2} +\Xrf=#1pt\advance\Xrf by 5pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{5}{1}{#2} +\Xrf=#1pt\advance\Xrf by 1.5pt +\advance\XBST by \decimal\BSTemp\Xrf +\BERNSTEIN{6}{0}{#2} +\Xrf=#1pt +\advance\XBST by \decimal\BSTemp\Xrf +\global\XBST=#3\XBST} +\end{verbatim} +In the macro, a control sequence of the form \verb!\decimal\Xdim! is for obtaining the decimal value of a variable \verb!\Xdim! without units +(\verb!pt!, by default). This can be defined by +\begin{verbatim} +\def\xch{\catcode`\p=12 \catcode`\t=12}\def\ych{\catcode`\p=11 \catcode`\t=11} +\xch \def\dec#1pt{#1}\ych \gdef\decimal#1{\expandafter\dec \the#1} +\end{verbatim} +Besides, the last line in the definition of \verb!\XBC! (\verb!\global\XBST=#3\XBST!) signifies \verb!\XBST! is adjusted for multiplication by +$\alpha$ when holding the current value of the sum $X(t)$. In Table \ref{bc-Table1}, some values +of \verb!\XBC! are compared with those of a simpler procedure in Maple\footnote{\footnotesize Maple +is a computer algebra system. It was first developed in 1980 by the Symbolic Computation Group at the University of +Waterloo in Waterloo, Ontario, Canada. Maple supports numeric and symbolic computations +and can be used as a programming language, which resembles Pascal.}. +\begin{table}[htbp] +\centering\begin{tabular}{|c|c|c|c|c|c|}\hline +&$(-2.4,0.23,1.5)$&$(4.2,0.45,1.1)$&$(-3.4,0.7,1.24)$&$(-0.89,0.4,1.5)$&$(5.21,0.5,2.15)$\\ \hline +\texttt{\symbol{92}XBC}&$\XBC{-2.4}{0.23}{1.5}\decimal\XBST$&$\XBC{4.2}{0.45}{1.1}\decimal\XBST$&$\XBC{-3.4}{0.7}{1.24}\decimal\XBST$&$\XBC{-0.89}{0.4}{1.5}\decimal\XBST$&$\XBC{5.21}{0.5}{2.15}\decimal\XBST$\\ \hline +Maple&$-2.41044$&$8.13235$&$-0.13696$&$2.78628$&$18.76008$\\ \hline +\end{tabular} +\caption{A comparison between some values of \textmtt{\symbol{92}XBC} and those of a procedure in Maple.}\label{bc-Table1} +\end{table} + +Likewise, from a given $y_0$, we may have a choice of $y_i$, $i=1,\ldots,6$, as follows +$$y_1=y_0+2,\quad y_2=y_0+5,\quad y_3=y_0+3,\quad y_4=y_0-1,\quad y_5=y_0-2,\quad y_6=y_0.$$ +The macro \verb!\YBC! is constructed in a quite analogous way as for \verb!\XBC! to evaluate values of $Y(t)$ in (\ref{eqn10}), +and the value $\beta Y(t)$ ($\beta$ is again an adjustment factor) is hold by a global variable \verb!\YBST!. + +\subsubsection{Partition of a simply connected plane domain by a rectangular grid} + +We consider here a simply connected plane domain $D$ surrounded by a B\'{e}zier curve of degree $6$ that will be depicted by +the procedure \verb!\NetDraw!. This is called together with its five parameters whose values are assigned to the five local variables +with their following meaning: +\begin{align*} +\texttt{\symbol{92}CellNum:}&\text{ The number of horizontal and vertical cells of the grid.}\\ +\texttt{\symbol{92}Xref:}&\text{ The value of $x_0$.}\\ +\texttt{\symbol{92}Yref:}&\text{ The value of $y_0$.}\\ +\texttt{\symbol{92}Xfact:}&\text{ The value of $\alpha$.}\\ +\texttt{\symbol{92}Yfact:}&\text{ The value of $\beta$.} +\end{align*} +We denote by $R$ a rectangular domain containing $D$ and having edges parallel to the coordinate axes. +A grid of rectangular cells will be depicted on $R$. According to the given values of $x_0$, $y_0$, $\alpha$ and $\beta$, the vertices of $R$ +can be chosen as the points $(\verb!\XMin!,\verb!\YMax!)$, $(\verb!\XMax!,\verb!\YMax!)$, $(\verb!\XMax!,\verb!\YMin!)$, $(\verb!\XMin!,\verb!\YMin!)$, where +\begin{align*} +\textmtt{\symbol{92}XMin}&=\alpha(\textmtt{\symbol{92}Xref}-1.5),&&\textmtt{\symbol{92}XMax}=\alpha(\textmtt{\symbol{92}Xref}+6),\\ +\textmtt{\symbol{92}YMin}&=\beta(\textmtt{\symbol{92}Yref}-2),&&\textmtt{\symbol{92}YMax}=\beta(\texttt{\symbol{92}Yref}+5). +\end{align*} +Then, the horizontal step size \verb!\Xsize! and the vertical step +size \verb!\Ysize! may be taken as +$$\mtt\\\texttt{Xsize}=\frac{\mtt\\\texttt{XMax}-\mtt\\\texttt{XMin}}{\mtt\\\texttt{CellNum}},\quad +\mtt\\\texttt{Ysize}=\frac{\mtt\\\texttt{YMax}-\mtt\\\texttt{YMin}}{\mtt\\\texttt{CellNum}}.$$ + +At first, the grid is made by using the structure \verb!\multido! to draw vertical and horizontal lines whose equations are +$$x=\texttt{\symbol{92}XMin}+i\texttt{\symbol{92}Xsize},\quad +y=\texttt{\symbol{92}YMin}+i\texttt{\symbol{92}Ysize},\quad i=0,\ldots,\texttt{\symbol{92}CellNum}.$$ +Next, the B\'{e}zier curve is depicted by the command +$$\textmtt{\symbol{92}parametricplot[algebraic,fillstyle=solid]\{$0$\}\{$1$\}\{$\alpha X(t)$|$\beta Y(t)$\}}$$ +and it is approximated by the sequence of points $(X_i,Y_i)$, $i=1,\ldots,N$, where $X_i$ and $Y_i$ are given from +the calling sequences +$$\textmtt{\symbol{92}XBC\{\symbol{92}XRef\}\{$t_i$\}\{\symbol{92}Xfact\}},\quad +\textmtt{\symbol{92}YBC\{\symbol{92}YRef\}\{$t_i$\}\{\symbol{92}Yfact\}}.$$ + +Finally, the most important algorithm is to determine cells that have points in common with the B\'{e}zier curve, +and to color them. The chosen ``filter'' method here can be described as follows: for each point $(X_i,Y_i)$ of the approximate sequence, +$i=1,\ldots, N$, we examine cells from left to right in horizontal direction, and from below to above in vertical direction. As soon as +being found, the cell containing $(X_i,Y_i)$ is specifically colored by the command \textmtt{\mtt\\pspolygon[fillstyle=solid,$\ldots$]} passing its four +vertices; then, we examine the next point $(X_{i+1},Y_{i+1})$, and so on. Actually, a structure of three nested \verb!\multido! loops +is used to perform this filter method, and the loop for counting points $(X_i,Y_i)$ is the most outer one. The algorithm to examine if +a cell contains $(X_i,Y_i)$ may have the form of +\begin{verbatim} + \ifdim\YBST<y\relax\else\advance y by \Ysize + \ifdim\YBST>y\relax\else + \ifdim\XBST<x\relax\else\advance x by \Xsize + \ifdim\XBST>x\relax\else + \pspolygon[fillstyle=solid,fillcolor=red] + (x,y)(x+\Xsize,y)(x+\Xsize,y+\Ysize)(x,y+\Ysize)(x,y)\relax + \fi\fi\fi\fi +\end{verbatim} +The single characters ``\verb!x!'', ``\verb!y!'' and the opertation ``\verb!+!'' will be replaced appropriately in the procedure \verb!\NetDraw! below. +For increasingly chosen values of $N$, we may know if the set of cells having points in common with the curve can cover the curve itself? +On the other hand, to increase the number of approximate points is corresponding to reduce the increment in the most outer loop \verb!\multido!. +For instance, if we take $N=100$, then the variable \verb!\nz! in the loop should be declared as $\texttt{\mtt\\nz}=0.00+0.01$ because +$t_i\in[0,1]$. + +In summary, to illustrate a partition of a simply connected plane domain, we can apply the following procedure \verb!\NetDraw!. As mentioned above, +step by step, \verb!\NetDraw! can: draw the boundary of a plane domain $D$ by a B\'{e}zier curve of degree $6$, draw a partition of a rectangle $R$ containing $D$ +by a grid of rectangular cells, and color cells having points in common with the boundary of $D$. +\bigskip +%\vskip2ex +\hrule +\begin{verbatim} +\def\NetDraw#1#2#3#4#5{% +\newcount\CellNum +\newdimen\XRef +\newdimen\YRef +\newdimen\Xfact +\newdimen\Yfact +\newdimen\XMin +\newdimen\XMax +\newdimen\YMin +\newdimen\YMax +\newdimen\Xsize +\newdimen\Ysize +\newdimen\tempx +\newdimen\tempy +\CellNum=#1 +\XRef=#2pt +\YRef=#3pt +\Xfact=#4pt +\Yfact=#5pt +\XMin=\decimal\Xfact\XRef \advance\XMin by -1.50\Xfact +\XMax=\decimal\Xfact\XRef \advance\XMax by 6.00\Xfact +\YMin=\decimal\Yfact\YRef \advance\YMin by -2.00\Yfact +\YMax=\decimal\Yfact\YRef \advance\YMax by 5.00\Yfact +\Xsize=\XMax \advance\Xsize by -\XMin \divide\Xsize by \CellNum +\Ysize=\YMax \advance\Ysize by -\YMin \divide\Ysize by \CellNum +\parametricplot[algebraic,fillstyle=solid,fillcolor=yellow!80,plotpoints=200, +linewidth=0.5pt]{0}{1}{\XBsix{\decimal\XRef}{\decimal\Xfact}| +\YBsix{\decimal\YRef}{\decimal\Yfact}} +\multido{\nz=0.00+0.005}{200}{\XBC{\decimal\XRef}{\nz}{\decimal\Xfact} +\YBC{\decimal\YRef}{\nz}{\decimal\Yfact} +\multido{\nx=\decimal\XMin+\decimal\Xsize}{\the\CellNum} +{\tempx=\nx pt\multido{\ny=\decimal\YMin+\decimal\Ysize}{\the\CellNum} +{\tempy=\ny pt\ifdim\YBST<\tempy\relax\else\advance\tempy by \Ysize% + \ifdim\YBST>\tempy\relax\else + \ifdim\XBST<\tempx\relax\else\advance\tempx by \Xsize + \ifdim\XBST>\tempx\relax\else + \pspolygon[fillstyle=solid,fillcolor=blue!90,linecolor=black, + linewidth=0.2pt] + (\nx,\ny)(\decimal\tempx,\ny)(\decimal\tempx,\decimal\tempy) + (\nx,\decimal\tempy)(\nx,\ny)\relax\fi\fi\fi\fi} +}} +\advance\CellNum by 1 +\multido{\nx=\decimal\XMin+\decimal\Xsize}{\the\CellNum} +{\psline[linewidth=0.2pt](\nx,\decimal\YMax)(\nx,\decimal\YMin)} +\multido{\ny=\decimal\YMin+\decimal\Ysize}{\the\CellNum} +{\psline[linewidth=0.2pt](\decimal\XMin,\ny)(\decimal\XMax,\ny)} +\parametricplot[algebraic,linecolor=white,plotpoints=200,linewidth=0.5pt]{0}{1} +{\XBsix{\decimal\XRef}{\decimal\Xfact}|\YBsix{\decimal\YRef}{\decimal\Yfact}}} +\end{verbatim}\hrule\bigskip + +In the definition of \verb!\NetDraw!, the expressions $\alpha X(t)$, $\beta Y(t)$ are declared in the \verb!algebraic! form by the macros \verb!\XBsix!, +\verb!\YBsix! with two parameters that hold values of $x_0,\alpha$ and $y_0,\beta$, respectively. These macros are given by +the following definitions: +\begin{verbatim} +\def\XBsix#1#2{% +(1-t)^6*(#1)*(#2)+6*t*(1-t)^5*(#1-1.5)*(#2)+15*t^2*(1-t)^4*(#1+2)*(#2)+ +20*t^3*(1-t)^3*(#1+6)*(#2)+15*t^4*(1-t)^2*(#1+5)*(#2)+ +6*t^5*(1-t)*(#1+1.5)*(#2)+t^6*(#1)*(#2)} +\def\YBsix#1#2{% +(1-t)^6*(#1)*(#2)+6*t*(1-t)^5*(#1+2)*(#2)+15*t^2*(1-t)^4*(#1+5)*(#2)+ +20*t^3*(1-t)^3*(#1+3)*(#2)+15*t^4*(1-t)^2*(#1-1)*(#2)+ +6*t^5*(1-t)*(#1-2)*(#2)+t^6*(#1)*(#2)} +\end{verbatim} +\subsection{Examples} +Firstly, let us see the effect of change for the shape of a domain when adjusting its boundary by taking different values of $\alpha$ and $\beta$. The +calling sequence for this purpose may have the form of +\begin{align*} +&\textmtt{\mtt\\parametricplot[algebraic,plotpoints=200,linewidth=0.5pt]\{$0$\}\{$1$\}}\\ +&\textmtt{\mtt\{XBsix\{$x_0$\}\{$\alpha$\}|\mtt\\YBsix\{$y_0$\}\{$\beta$\}\mtt\}} +\end{align*} +and its result is given in Figure \ref{Fig14}. +\begin{figure}[htbp] +\centering\begin{pspicture}(0,0.5)(3.5,3) +\psset{unit=12pt} +\parametricplot[algebraic,plotpoints=200,linewidth=0.5pt]{0}{1}{\XBsix{2}{1.5}|\YBsix{2}{1.5}} +\end{pspicture} +\begin{pspicture}(0,0.5)(5,3.5) +\psset{unit=12pt} +\parametricplot[algebraic,plotpoints=200,linewidth=0.5pt]{0}{1}{\XBsix{2}{2}|\YBsix{2}{1.5}} +\end{pspicture} +\begin{pspicture}(0,1)(4,3.5) +\psset{unit=12pt} +\parametricplot[algebraic,plotpoints=200,linewidth=0.5pt]{0}{1}{\XBsix{2}{1.5}|\YBsix{2}{2}} +\end{pspicture} +\caption{From left to right, corresponding to the couple of values: +$\alpha=2,\,\beta=2$; $\alpha=2.5,\,\beta=2$; $\alpha=2,\,\beta=2.5$.}\label{Fig14} +\end{figure} + +The Table \ref{bc-table2} below provides three partitions of a domain $D$ with cells that decrease in size. In each case, \verb!\NetDraw! can +determine the cells that have points in common with the boundary curve and color them. The calling sequence here +takes a simple form +$$\textmtt{\mtt\\NetDraw\{$c$\}\{$x_0$\}\{$y_0$\}\{$\alpha$\}\{$\beta$\}}$$ +The numerical argument $c$ is assigned to the local variable \verb!\CellNum!. The number of approximate points $(X_i,Y_i)$ for the boundary curve +is chosen to be $200$, by default, corresponding to the increment $\Delta\mtt{z}=0.005$. The number $N$ can be made larger and the filter procedure +for determining required cells may be more exact. In Table \ref{bc-table2}, the B\'{e}zier curves are depicted with the choice +$x_0=y_0=\alpha=\beta=2$. +\begin{table}[htbp] +\centering\begin{tabular}{cc} +\includegraphics[width=6cm]{b6of1}&\includegraphics[width=6cm]{b6of2}\\ +\includegraphics[width=6cm]{b6of3}&\includegraphics[width=6cm]{b6of4} +\end{tabular} +\caption{The partitions of $D$ with $c=18$, $c=36$ and $c=63$.}\label{bc-table2} +\end{table} + +\acknw +I am very grateful to +\begin{itemize} +\item Timothy Van Zandt, Herbert Vo\ss, Dominique Rodriguez and Michael Sharpe for helping me with +their great works on \texttt{PSTricks}. +\item H\`an Th\hantt\rlap\accentcircflx\ Th\`anh for helping me with his pdf\hskip.03em\LaTeX\ program. +\item Robert Fuster for his very useful package \texttt{calculator}. +\end{itemize} +\begin{thebibliography}{10} +\bibitem{one} Dominique Rodriguez, Michael Sharpe \&\ Herbert Vo\ss. \textsl{\texttt{pstricks-add}: Additional Macros for PSTricks\/}. +Version 3.60, +\url{http://ctan.org/tex-archive/graphics/pstricks/contrib}, 2013 +\bibitem{two} Timothy Van Zandt, Michael Sharpe \&\ Herbert Vo\ss. \textsl{\texttt{pst-node}: Nodes and node connections}. +Version 1.29, +\url{http://ctan.org/tex-archive/graphics/pstricks/contrib}, 2013 +\bibitem{three} Helmut Kopka \&\ Patrick W. Daly. \textsl{Guide to \LaTeX \/}. +Addison-Wesley, Fourth Edition, 2004, ISBN 0321173856 +\bibitem{four} Timothy Van Zandt. \textsl{User's Guide\/}. Version 1.5,\\ +\url{http://ctan.org/tex-archive/graphics/pstricks/base}, 2007 +\bibitem{five}Eitan M. Gurari. \textsl{Writing With \TeX \/}, McGraw-Hill, Inc., 1994, ISBN 0-07-025207-6 +\bibitem{six} Robert Fuster. \textsl{\texttt{calculator-calculus}: Scientific Calculations With \LaTeX \/}. Version 1.0a, +\url{http://ctan.org/tex-archive/macros/latex/contrib/calculator}, 2012 +\end{thebibliography} + +\end{document} +\begin{thebibliography}{10} +\bibitem{one} Dominique Rodriguez, Michael Sharpe \&\ Herbert Vo\ss. \textsl{\texttt{pstricks-add}: Additional Macros for PSTricks\/}. +Version 3.60, +\url{http://ctan.org/tex-archive/graphics/pstricks/contrib}, 2013 +\bibitem{two} Timothy Van Zandt, Michael Sharpe \&\ Herbert Vo\ss. \textsl{\texttt{pst-node}: Nodes and node connections}. +Version 1.29, +\url{http://ctan.org/tex-archive/graphics/pstricks/contrib}, 2013 +\bibitem{three} Helmut Kopka \&\ Patrick W. Daly. \textsl{Guide to \LaTeX \/}. +Addison-Wesley, Fourth Edition, 2004, ISBN 0321173856 +\bibitem{four} Timothy Van Zandt. \textsl{User's Guide\/}. Version 1.5,\\ +\url{http://ctan.org/tex-archive/graphics/pstricks/base}, 2007 +\bibitem{five}Eitan M. Gurari. \textsl{Writing With \TeX \/}, McGraw-Hill, Inc., 1994, ISBN 0-07-025207-6 +\bibitem{six} Robert Fuster. \textsl{\texttt{calculator-calculus}: Scientific Calculations With \LaTeX \/}. Version 1.0a, +\url{http://ctan.org/tex-archive/macros/latex/contrib/calculator}, 2012 +\end{thebibliography} |