doc-en/par-tube-en.tex

   1 \section{Tubes}
   2
   3 This section is about to substitute a curve in two or three dimensions (2D or 3D),
   4 that are setup parameterised, by a \Index{tube}, where the initial curve is the axes and
   5 we can choose the radius and grid.  We find some mathematical elements concerning
   6 these objects on the following websites:
   7
   8 \centerline{\url{http://fr.wikipedia.org/wiki/Tube_(math\%C3\%A9matiques)}}
   9
  10 \centerline{\url{http://www.mathcurve.com/surfaces/tube/tube.shtml}}
  11
  12 As usual, the \texttt{pst-solides3d} package offers two possibilities to draw the tubes:
  13 \begin{itemize}
  14   \item via PSTricks and the argument \Lkeyword{object} of \Lcs{psSolid}
  15   \item directly with \verb+\codejps+
  16 \end{itemize}
  17
  18 \encadre{It is often advisable to calculate in advance, by hand or with a preferred software, the first derivatives of the parametric functions which define the coordinates.}
  19
  20 However, if this derivative isn't defined explicitly by the user, the package makes some approximate calculations, but the result then is not always sufficient.
  21
  22
  23 \subsection{Usage with PSTricks}
  24
  25 \subsubsection{Give your curves a relief}
  26 ``\textit{Donnez du relief \`{a} vos courbes}'', this is the title of the article
  27 from Robert \textsc{Ferr\'{e}ol}, available on:
  28
  29 \url{http://mapage.noos.fr/r.ferreol/atelecharger/textes/relief/courbes\%20en\%20relief.html}
  30
  31 from who  the following functions were borrowed and which are analogous to a
  32 Lissajous figure enrolled around a cylinder.
  33
  34
  35 \begin{LTXexample}[pos=t]
  36 \begin{pspicture}(-3.5,-4)(4,4)
  37 \psset{lightsrc=80 30 30,viewpoint=100 45 30 rtp2xyz,Decran=110,linewidth=0.2pt}
  38 \defFunction[algebraic]{Func}(t){2.5*cos(t)}{2.5*sin(t)}{2*cos(5*t)}
  39 \defFunction[algebraic]{Func'}(t){-2.5*sin(t)}{2.5*cos(t)}{-10*sin(5*t)}
  40 \psSolid[object=courbe,range=0 6.28,hue=0 1 0.7 1,
  41    ngrid=360 8,function=Func,r=0.15]
  42 \end{pspicture}
  43 \end{LTXexample}
  44
  45
  46 The argument \texttt{\Lkeyword{object}=\Lkeyval{courbe}} with the parameters
  47 \Lkeyword{r}, \Lkeyword{function} and \Lkeyword{range} is used to specify
  48 the radius of the tube, the name of the function to be used and the range.
  49
  50 We can also refine the grid with the optional argument
  51 \texttt{\Lkeyword{ngrid}=$n_1$ $n_2$} where $n_1$ represents the number of
  52 vertices of a section of a tube (if $n_1 = 6$, this gives a tube with a
  53 hexagonal section) and $n_2$ represents the number of divisions along it.
  54
  55
  56 \subsubsection{A \Index{hairline curve} is produced  with the radius \texttt{r=0}}
  57
  58 And thus, no fear to specify the derived function.
  59
  60 \begin{LTXexample}[pos=t]
  61 \begin{pspicture}(-3.5,-4)(4,4)
  62 \psset{lightsrc=80 30 30,viewpoint=100 45 30 rtp2xyz,Decran=110}
  63 \defFunction[algebraic]{FI}(t){2.5*cos(t)}{2.5*sin(t)}{2*cos(5*t)}
  64 \psSolid[object=courbe,range=0 6.28,linewidth=2pt,linecolor=blue,function=FI,r=0]
  65 \end{pspicture}
  66 \end{LTXexample}
  67
  68
  69 \subsection{Usage with \texttt{\textbackslash{}codejps}}
  70
  71 The syntax is \texttt{\textbackslash{}codejps\{t\_min t\_max (name\_function)
  72 radius\_tube [n1 n2] newtube\}}.
  73
  74
  75 \begin{LTXexample}[pos=t]
  76 \begin{pspicture}(-3.5,-3.5)(4,3.5)
  77 \psset{lightsrc=80 30 30,viewpoint=100 45 90 rtp2xyz,Decran=100,linewidth=0.2pt}
  78 \codejps{
  79 /rpn {tx@AlgToPs begin AlgToPs end cvx exec} def
  80 /xc {((2+1*cos(2.75*t))*cos(t)) rpn } def
  81 /yc {((2+1*cos(2.75*t))*sin(t)) rpn } def
  82 /zc {(1*sin(2.75*t)) rpn } def
  83 /xc' {(-2.75*sin(2.75*t)*cos(t)-(2+cos(2.75*t))*sin(t)) rpn } def
  84 /yc' {(-2.75*sin(2.75*t)*sin(t)+(2+cos(2.75*t))*cos(t)) rpn } def
  85 /zc' {(2.75*cos(2.75*t)) rpn } def
  86 /g { 3 dict begin /t exch def xc yc zc end } def
  87 /g' { % first derivative
  88 3 dict begin /t exch def xc' yc' zc' end } def
  89 /solenoide{
  90 % t_min t_max (name_function) radius_tube [resolution]
  91    0 25.2 (g) 0.1 [360 8] newtube dup [0 1] solidputhuecolors} def
  92 solenoide
  93 drawsolid**
  94 }%
  95 \end{pspicture}
  96 \end{LTXexample}
  97
  98 \subsection{Improving the speed of readout}
  99
 100 The curve with the name ``\textit{horopter}'' is the subject of this website:
 101
 102 \centerline{\url{http://www.mathcurve.com/courbes3d/horoptere/horoptere.shtml}}
 103
 104 \subsubsection{Obtaining the curve directly}
 105
 106 The following lines allow us to calculate the points and draw  the curve.
 107 The \Index{resolution} \texttt{\Lkeyword{ngrid}=72 12} of the curve was increased, so some
 108 more calculation time to produce the result, which some will judge as very long.
 109 \begin{verbatim}
 110 \begin{pspicture}(-7,-2)(7,4)
 111 \psset{lightsrc=80 30 30}
 112 \psset{viewpoint=1000 60 20 rtp2xyz,Decran=1000}
 113 \psframe(-7,-2)(7,4)
 114 \psset{solidmemory}
 115 \codejps{/a 2 def /b 2 def}%
 116 \defFunction[algebraic]{F3}(t)
 117    {a*(1+cos(t))}
 118    {b*tan(t/2)}
 119    {a*sin(t)}
 120 \defFunction[algebraic]{F3'}(t)
 121    {-a*sin(t)}
 122    {b*(1+tan(1/2*t)^2)}
 123    {a*cos(t)}
 124 \psSolid[object=courbe,
 125    range=-2.7468 2.7468,
 126    ngrid=72 12,
 127    function=F3,hue=0 1 0.7 1,
 128    action=none,name=H1,
 129    r=1]%
 130 \psSolid[object=cylindrecreux,
 131          h=20,r=1,RotX=90,
 132          incolor=green!30,action=none,
 133          name=C1,
 134          ngrid=36 36](2,10,0)
 135 \psSolid[object=fusion,
 136    base=H1 C1]
 137 \composeSolid
 138 \end{pspicture}
 139 \end{verbatim}
 140
 141 \subsubsection{Saving the parameters of the curve}
 142
 143 If this curve is used several times, it is advisable to backup all the
 144 characteristics of that curve, like:
 145 coordinates of the vertices, list of colours of the faces with placing
 146 the last command \texttt{\Lkeyword{action}=\Lkeyval{writesolid}}:
 147 \begin{verbatim}
 148 \psSolid[object=fusion,
 149          base=H1 C1,
 150          file=horoptere,
 151          action=writesolid]
 152 \end{verbatim}
 153 The following sequence \Cadre{LaTeX fichier.tex->dvips->GSview
 154   (Windows) or gv (Linux)} will generate 4 files:
 155 \begin{itemize}
 156   \item \texttt{horoptere-couleurs.dat} $\rightarrow$ the colours of the faces;
 157   \item \texttt{horoptere-faces.dat} $\rightarrow$ the list of faces;
 158   \item \texttt{horoptere-sommets.dat} $\rightarrow$ the list of vertices;
 159   \item \texttt{horoptere-io.dat} $\rightarrow$ the number of faces and vertices.
 160 \end{itemize}
 161
 162 then read and execute the files with the command:
 163 \texttt{\textbackslash{}psSolid[object=datfile,file=horoptere]}, the time saved can be quite significant
 164
 165
 166
 167 \encadre{By default, under Windows and Linux, the security of files on the hard drive is activated and doesn't allow to write on the drive.
 168 To deactivate that security option, more or less temporarily, here the two corresponding procedures:
 169
 170 \begin{description}
 171
 172   \item[Linux:] The advice from Jean-Michel \textsc{Sarlat}: the simplest will be to use GhostScript directly, within the console. As there is no image to wait for:
 173
 174 \$$>$ gs  -dNOSAFER  monfichier.ps quit.ps
 175   \item[Windows:] Within the menue \texttt{Options}, the option
 176     \textsf{Security of files} must be turned to unchecked.
 177 \end{description}}
 178
 179 \subsubsection{The plot of the curve}
 180
 181 \begin{LTXexample}[width=9cm]
 182 \begin{pspicture}(-5,-3.5)(4,3)
 183 \psset{lightsrc=80 30 30}
 184 \psset{viewpoint=100 60 20 rtp2xyz,
 185        Decran=75}
 186 \psframe*[linecolor=cyan!30](-4.5,-3)(3.5,3)
 187 \psSolid[object=datfile,file=./horoptere]
 188 \end{pspicture}
 189 \end{LTXexample}
 190
 191
 192 \clearpage
 193 \subsection{Some other examples}
 194
 195
 196 \subsubsection{A straight line}
 197
 198 \begin{LTXexample}[width=7cm]
 199 \begin{pspicture}(-3.5,-2)(3.5,2)
 200 \psset{viewpoint=100 -20 20 rtp2xyz,
 201        Decran=75,unit=0.8}
 202 \psSolid[object=grille,base=-4 4 -4 4]%
 203 \defFunction[algebraic]{FIV}(t){t}{t}{0.5}
 204 \defFunction[algebraic]{FIV'}(t){1}{1}{0}
 205 \psSolid[object=courbe,
 206    range=-4 4, ngrid=16 16,
 207    function=FIV, r=0.5]
 208 \end{pspicture}
 209 \end{LTXexample}
 210
 211
 212
 213 \subsubsection{A \Index{hypocycloid}}
 214
 215 \begin{LTXexample}[width=7cm]
 216 \begin{pspicture}(-3.5,-3)(3.5,3)
 217 \psset{viewpoint=100 20 45 rtp2xyz,
 218        Decran=75,unit=0.7}
 219 \psSolid[object=grille,base=-5 5 -5 5]%
 220 \defFunction[algebraic]{FII}(t)
 221    {4*cos(t)+cos(4*t)/2}
 222    {4*sin(t)-sin(4*t)/2}
 223    {1}
 224 \defFunction[algebraic]{FII'}(t)
 225    {-4*sin(t)-2*sin(4*t)}
 226    {4*cos(t)-2*cos(4*t)}
 227    {0}
 228 \psSolid[object=courbe,
 229    range=0 6.28,ngrid=90 16,
 230    function=FII,r=1]
 231 \end{pspicture}
 232 \end{LTXexample}
 233
 234 \clearpage
 235
 236 \subsubsection{The spring of Gaston}
 237
 238 \begin{LTXexample}[width=7cm]
 239 \begin{pspicture}(-3.5,-4)(3.5,4.5)
 240 \psset{lightsrc=80 30 30,
 241   viewpoint=100 20 20 rtp2xyz,Decran=50}
 242 \defFunction[algebraic]{FIII}(t)
 243   {(t^2+3)*sin(15*t)}
 244   {(t^2+3)*cos(15*t)}{2*t}
 245 \defFunction[algebraic]{FIII'}(t)
 246   {2*t*sin(15*t)+15*(t^2+3)*cos(15*t)}
 247   {2*t*cos(15*t)-15*(t^2+3)*sin(15*t)}{2}
 248 \psSolid[object=courbe,
 249   range=-2 2,ngrid=360 6,
 250   function=FIII,hue=0.2 0.3,
 251   linewidth=0.1pt,r=0.2]
 252 \end{pspicture}
 253 \end{LTXexample}
 254
 255
 256 \endinput
 257