pstricks、psSolid 和 pstODEsolve II

Question

同样，正如相关问题，我们可以将二次微分方程应用于z(x)到是后处理步骤中表面顶点的坐标。

只在表面上绘制了一条曲线。添加更多经过后处理并修改初始条件的初级 ODE 解的曲线作为练习。只需阅读问题的评论并重复部分中的相应代码块即可“third pass“ 和 ”first pass”。您必须将重复代码中的“ intermediateResult”和“ processedTable”替换为唯一名称，例如“ intermediateResult1”和“ processedTable1”。

现在，为了得到最终的图片，我们需要三遍编译。第二遍处理中间表面顶点坐标，这在 TeX 内存方面相当耗费。我们在这里使用 DVI 输出模式来避免内存问题*.tex。*.pdflualatex

这是下面代码（example.tex）的三次编译过程，以获得所示的图片：

latex example
dvips example
ps2pdf -DNOSAFER example.ps

lualatex --output-format=dvi example
dvips example
ps2pdf -DNOSAFER example.ps

latex example
dvips example
ps2pdf -DNOSAFER example.ps

在此处输入图片描述

文件example.tex：

\documentclass[border=10pt]{standalone}
\usepackage{pstricks}
\usepackage{pst-plot}%, pst-math}
\usepackage{pst-ode}
\usepackage{pst-solides3d}
\psset{algebraic,unit=2}

\makeatletter
\define@key[psset]{}{valuerange}[1-]{\expandafter\pst@getrange#1\@nil}
\def\pst@getrange#1-#2\@nil{%
  \ifx\relax#1\relax \def\pst@startvalue{1 }\else\def\pst@startvalue{#1 }\fi%
  \ifx\relax#2\relax \def\pst@endvalue{1e32 }\else\def\pst@endvalue{#2 }\fi%
}
\psset{valuerange=1-}

\def\listplotIIID{\def\pst@par{}\pst@object{listplotIIID}}
\def\listplotIIID@i#1{%
\@nameuse{beginplot@\psplotstyle}%
\addto@pscode{%
   /viewpointXYZ {\pst@solides@viewpoint} def
   /Decran \pst@solides@Decran\space def % distance de l'ecran
    viewpointXYZ /ZpointVue ED /YpointVue ED /XpointVue ED
  /THETA {YpointVue XpointVue atan} bind def
  /PHI   {ZpointVue XpointVue dup mul YpointVue dup mul add sqrt atan} bind def
  /Dobs  {XpointVue dup mul YpointVue dup mul add ZpointVue dup mul add sqrt} bind def
  XpointVue YpointVue ZpointVue /viewpoint defpoint3d
    /XYZ [#1] def
  /@tabXYZ [
    0 3 XYZ length 3 sub {
     /i exch def
     XYZ i get
     XYZ i 1 add get
     XYZ i 2 add get
     i 3 div dup /ii ED
       \pst@startvalue ge {
         ii \pst@endvalue le {
         3dto2d
         \pst@number\psunit mul exch
         \pst@number\psunit mul exch
       } { pop pop pop } ifelse
     }{ pop pop pop } ifelse
    }  for
    ] bind def
 [ @tabXYZ aload pop
    }%
\@nameuse{endplot@\psplotstyle}}%
\makeatother

\def\r(#1){sin(#1)}
\def\rs(#1){cos(#1)}
\def\rss(#1){-sin(#1)}
\def\rssrs(#1){-sin(#1)*cos(#1)}
\def\zsq(#1){sin(#1)*sin(#1)*(1+cos(#1)+sin(2*#1)*sin(2*#1)/16)}
\def\zsszs(#1){(1/2)*sin(2*#1)*(1+cos(#1)+sin(2*#1)*sin(2*#1)/16)+(1/2)*sin(#1)*sin(#1)*(-sin(#1)+sin(4*#1)/8)}

%for parsing lines of xyz.dat files (files with list of vertices)
\def\getFirstOfThree#1 #2 #3;{#1}
\def\getSecondOfThree#1 #2 #3;{#2}
\def\getThirdOfThree#1 #2 #3;{#3}

\IfFileExists{surf-sommets.dat}{%
  \endlinechar=-1% suppress trailing space at input line end
  \newread\xIIfile%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %  third pass
  %
  %  * post-process y coordinates in intermediateResult.dat (curve on surface)
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \pstVerb{% initialise Postscript variable to take the processed table
    true setglobal
    globaldict /processedTable {} put
    false setglobal
  }%
  \openin\xIIfile=intermediateResult.dat%
  %line-wise read table from `intermediateResult.dat', and treat second column: z(2nd column)
  \loop\read\xIIfile to \inputline%
  \ifeof\xIIfile\else%
    \pstODEsolve[algebraic,algebraicT]{processedVal}{1}{0}{\expandafter\getSecondOfThree\inputline;}{2}{0 1}{%
      1 | -sin(x[0])*sqrt(1+cos(x[0])+sin(2*x[0])*sin(2*x[0])/16)
    }%
    %process input line and append result to postscript variable `processedTable' (executable list)
    \pstVerb{
      true setglobal
      globaldict /processedTable [
        processedTable
        \expandafter\getFirstOfThree\inputline;\space
        processedVal exch pop %insert second value of pstODEsolve result and throw the first at t=0 away
        \expandafter\getThirdOfThree\inputline;] cvx put
      false setglobal
    }%
  \repeat%
}{
  \IfFileExists{intermediateSurf-sommets.dat}{%
    \endlinechar=-1% suppress trailing space at input line end
    \newread\xIIfile%
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %  second pass
    %
    %  * post-process vertex y coordinates in intermediateSurf-sommets.dat
    %    write result into surf-sommets.dat
    %  * copy remaining intermediateSurf-*.dat to surf-*.dat files
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    % post-process vertex y coordinates in intermediateSurf-sommets.dat
    \openin\xIIfile=intermediateSurf-sommets.dat%
    \pstVerb{ %ps2pdf will write surf-sommets.dat
      /surfsommets (surf-sommets.dat) (w) file def
      /numberstring 256 string def
      /num2str {numberstring cvs} def
    }
    %line-wise read table from `intermediateSurf-sommets.dat'
    \loop\read\xIIfile to \inputline%
    \ifeof\xIIfile
      \closein\xIIfile
      \pstVerb{surfsommets closefile}
    \else%
      %treat second column (output ODE)
      \pstODEsolve[algebraic,algebraicT]{processedVal}{1}{0}{\expandafter\getSecondOfThree\inputline;}{2}{0 1}{ 1 |
       -sin(x[0])*sqrt(1+cos(x[0])+sin(2*x[0])*sin(2*x[0])/16)}%
       %write resulting table into `surf-sommets.dat'
       \pstVerb{
         surfsommets \expandafter\getFirstOfThree\inputline; num2str writestring
         surfsommets ( ) writestring
         surfsommets processedVal exch pop num2str writestring
         surfsommets ( ) writestring
         surfsommets \expandafter\getThirdOfThree\inputline; num2str writestring
         surfsommets (\string\n) writestring
       }%
    \repeat%
    % copy remaining intermediateSurf-*.dat to surf-*.dat files
    \newwrite\copiedFile
    \openin\xIIfile=intermediateSurf-couleurs.dat%
    \immediate\openout\copiedFile=surf-couleurs.dat%
    \loop\read\xIIfile to \inputline%
    \ifeof\xIIfile
      \closein\xIIfile
      \closeout\copiedFile
    \else%
      \immediate\write\copiedFile{\inputline}
    \repeat%
    \openin\xIIfile=intermediateSurf-faces.dat%
    \immediate\openout\copiedFile=surf-faces.dat%
    \loop\read\xIIfile to \inputline%
    \ifeof\xIIfile
      \closein\xIIfile
      \closeout\copiedFile
    \else%
      \immediate\write\copiedFile{\inputline}
    \repeat%
    \openin\xIIfile=intermediateSurf-io.dat%
    \immediate\openout\copiedFile=surf-io.dat%
    \loop\read\xIIfile to \inputline%
    \ifeof\xIIfile
      \closein\xIIfile
      \closeout\copiedFile
    \else%
      \immediate\write\copiedFile{\inputline}
    \repeat%
  }{
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %  first pass
    %
    %  * solve main ODE (curve on surface) without special treatment for second column
    %    of the solution table
    %  * write solution to file `intermediateResult.dat'
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \pstODEsolve[algebraic,algebraicOutputFormat,saveData]{intermediateResult}{%
      \r(x[2])*sin(x[0]) |
      x[2] | % we process this column in the second pass
      \r(x[2])*cos(x[0])
    }{0}{22}{200}{1.57 1 0.3 0}{
     x[1] | -2*\rs(x[2])*x[1]*x[3]/(\r(x[2]))| x[3] |
     -(\rssrs(x[2])+\zsszs(x[2]))*x[3]*x[3]/(\rs(x[2])*\rs(x[2])+\zsq(x[2]))
     +\rs(x[2])*\r(x[2])*x[1]*x[1]/(\rs(x[2])*\rs(x[2])+\zsq(x[2]))%
    }
  }
}

\begin{document}

\begin{pspicture}(-0.7,-1.1)(3.2,1.15)%
  \psset[pst-solides3d]{viewpoint=20 5 20,Decran=20,lightsrc=viewpoint}%
  \IfFileExists{surf-sommets.dat}{%
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %  third pass
    %
    %  plot readily processed surface from surf-*.dat files
    %  plot readily processed curve in PS variable `processedTable'
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \psSolid[object=datfile, file=surf,
      base=0 3.14 0 6.28,
      inhue=0.01 0.1, hue=0.1 0.01,
      linewidth=0.5pt,
      grid=false,
      opacity=0.9,
      ngrid=200 70,
      intersectiontype=0,intersectionplan={[ 0 0 1 0] [0 1 0 0]},intersectioncolor=(blue) (red),
      intersectionlinewidth=1 2
    ]%
    \listplotIIID[linecolor=black,linewidth=1pt]{processedTable}
  }{%
    \IfFileExists{intermediateSurf-sommets.dat}{%
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      %  second pass
      %
      %  do nothing here
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    }{%
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      %  first pass
      %
      %  write the surface to intermediateSurf-*.dat files for postprocessing in the 2nd pass
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      %the surface definition with the un-processed vertex coordinate y = t
      %(to be post-prcessed in the second pass)
      \defFunction[algebraic]{surf}(t,v){\r(t)*sin(v)}{t}{\r(t)*cos(v)}
      \psSolid[
        file=intermediateSurf,action=writesolid,
        object=surfaceparametree,
        function=surf,
        base=0 3.14 0 6.28,
        inhue=0.01 0.1, hue=0.1 0.01,
        linewidth=0.5pt,
        grid=false,
        opacity=0.9,
        ngrid=200 70,
        intersectiontype=0,intersectionplan={[ 0 0 1 0] [0 1 0 0]},intersectioncolor=(blue) (red),
        intersectionlinewidth=1 2
      ]%
    }%
  }%
  \axesIIID(1,4,1)(2,4.5,2)%
\end{pspicture}

\end{document}

Answer 1