doc-en/par-surfacesparametrees-en.tex

   1 \section{Parameterised surfaces}
   2
   3 \subsection{The method}
   4
   5 The parameterised \Index{surfaces} are setup as $[x(u,v),y(u,v),z(u,v)]$ and administered thanks to the macro \Lcs{psSolid} by the option
   6 \texttt{\Lkeyword{object}=\Lkeyval{surfaceparametree}} and defined either in \textit{Reverse Polish Notation}(\textit{RPN}):
   7
   8
   9 \begin{verbatim}
  10 \defFunction{shell}(u,v){1.2 v exp u Sin dup mul v Cos mul mul}% x(u,v)
  11                         {1.2 v exp u Sin dup mul v Sin mul mul}% y(u,v)
  12                         {1.2 v exp u Sin u Cos mul mul}        % z(u,v)
  13 \end{verbatim}
  14
  15 or in \textit{algebraic notation}:
  16
  17 \begin{verbatim}
  18 \defFunction[algebraic]{shell}(u,v){1.2^v*(sin(u)^2*cos(v))}% x(u,v)
  19                                    {1.2^v*(sin(u)^2*sin(v))}% y(u,v)
  20                                    {1.2^v*(sin(u)*cos(u))}  % z(u,v)
  21 \end{verbatim}
  22
  23 The range for the values of $u$ and $v$ are defined within the option
  24 \texttt{\Lkeyword{range}=$\mathtt{u_{min}}$ $\mathtt{u_{max}}$ $\mathtt{v_{min}}$ %$
  25 $\mathtt{v_{max}}$}.
  26
  27 The drawing of the function is activated with
  28 \texttt{\Lkeyword{function}=name}, this name is implied when the parametric equations are written:
  29 \verb+\defFunction{name}...+
  30
  31 Any other choice of $u$ and $v$ are accepted. Let's remind that the argument of
  32 \texttt{Sin} and \texttt{Cos} must be in radians those of \texttt{sin} and
  33 \texttt{cos} in degrees if \textit{RPN} is  used. Within the algebraic notation, the argument is in radians.
  34
  35
  36 \subsection{Example 1: a \Index{sea shell}}
  37 \newcommand\quadrillage{%
  38 \psset{linecolor={[cmyk]{1,0,1,0.5}}}\green
  39 \multido{\ix=-4+1}{9}{%
  40     \psPoint(\ix\space,4,-3){X1}
  41     \psPoint(\ix\space,4 .2 add,-3){X2}
  42     \psline(X1)(X2)
  43     \uput[-120](X1){\small\ix}}
  44 \multido{\iy=-4+1}{9}{%
  45     \psPoint(-4,\iy\space,-3){Y1}
  46     \psPoint(-4 .2 sub,\iy\space,-3){Y2}
  47     \psline(Y1)(Y2)
  48     \uput[0](Y1){\small\iy}}
  49 \multido{\iz=-3+1}{7}{%
  50     \psPoint(4,4,\iz\space){Z1}
  51     \psPoint(4,4 .2 add,\iz\space){Z2}
  52     \psline(Z1)(Z2)
  53     \uput[l](Z1){\small\iz}}
  54 \psPoint(0,4 0.5 add,-3){X0}
  55 \uput[-120](X0){$x$}
  56     \psPoint(-4 .5 sub,0,-3){Y0}
  57 \uput[0](Y0){$y$}}
  58 \begin{LTXexample}[width=7.8cm]
  59 \psset{unit=0.75}
  60 \begin{pspicture}(-5.5,-6)(4.5,4)
  61 \psframe*(-5.5,-6)(4.5,4)
  62 \psset[pst-solides3d]{viewpoint=20 120 30 rtp2xyz,
  63   Decran=15,lightsrc=-10 15 10}
  64 % Parametric Surfaces
  65 \psSolid[object=grille,base=-4 4 -4 4,
  66   action=draw*,linecolor={[cmyk]{1,0,1,0.5}}]
  67   (0,0,-3)
  68 \defFunction{shell}(u,v)
  69   {1.2 v exp u Sin dup mul v Cos mul mul}
  70   {1.2 v exp u Sin dup mul v Sin mul mul}
  71   {1.2 v exp u Sin u Cos mul mul}
  72 \psSolid[object=surfaceparametree,
  73   linecolor={[cmyk]{1,0,1,0.5}},
  74   base=0 pi pi 4 div neg 5 pi mul 2 div,
  75   fillcolor=yellow!50,incolor=green!50,
  76   function=shell,linewidth=0.5\pslinewidth,ngrid=25]%
  77 \psSolid[object=parallelepiped,a=8,b=8,c=6,
  78   action=draw,linecolor={[cmyk]{1,0,1,0.5}}]%
  79 \quadrillage
  80 \end{pspicture}
  81 \end{LTXexample}
  82
  83 \begin{LTXexample}[width=7.8cm]
  84 \psset{unit=0.75}
  85 \begin{pspicture}(-5,-4)(5,6)
  86 \psframe*(-5,-4)(5,6)
  87 \psset[pst-solides3d]{viewpoint=20 20 -10 rtp2xyz,
  88   Decran=15,lightsrc=5 10 2}
  89 % Parametric Surfaces
  90 \psSolid[object=grille,base=-4 4 -4 4,
  91   action=draw*,linecolor=red](0,0,-3)
  92 \defFunction[algebraic]{shell}(u,v)
  93   {1.21^v*(sin(u)*cos(u))}
  94   {1.21^v*(sin(u)^2*sin(v))}
  95   {1.21^v*(sin(u)^2*cos(v))}
  96 %% \defFunction{shell}(u,v)
  97 %%    {1.2 v exp u Sin u Cos mul mul}
  98 %%    {1.2 v exp u Sin dup mul v Sin mul mul}
  99 %%    {1.2 v exp u Sin dup mul v Cos mul mul}
 100 \psSolid[object=surfaceparametree,
 101    linecolor={[cmyk]{1,0,1,0.5}},
 102    base=0 pi pi 4 div neg 5 pi mul 2 div,
 103    fillcolor=green!50,incolor=yellow!50,
 104    function=shell,linewidth=0.5\pslinewidth,
 105    ngrid=25]%
 106 \white%
 107 \gridIIID[Zmin=-3,Zmax=4,linecolor=white,
 108   QZ=0.5](-4,4)(-4,4)
 109 \end{pspicture}
 110 \end{LTXexample}
 111
 112
 113
 114 \subsection{Example 2: a \Index{helix}}
 115 \begin{LTXexample}[width=5.5cm]
 116 \psset{unit=0.75}
 117 \begin{pspicture}(-3,-4)(3,6)
 118 \psset[pst-solides3d]{viewpoint=20 10 2,Decran=20,
 119   lightsrc=20 10 10}
 120 % Parametric Surfaces
 121 \defFunction{helix}(u,v)
 122   {1 .4 v Cos mul sub u Cos mul 2 mul}
 123   {1 .4 v Cos mul sub u Sin mul 2 mul}
 124   {.4 v Sin mul u .3 mul add}
 125 \psSolid[object=surfaceparametree,linewidth=0.5\pslinewidth,
 126   base=-10 10 0 6.28,fillcolor=yellow!50,incolor=green!50,
 127   function=helix,
 128   ngrid=60 0.4]%
 129 \gridIIID[Zmin=-3,Zmax=3](-2,2)(-2,2)
 130 \end{pspicture}
 131 \end{LTXexample}
 132
 133
 134 \subsection{Example 3: a \Index{cone}}
 135 \begin{LTXexample}[width=10cm]
 136 \psset{unit=0.5}
 137 \begin{pspicture}(-9,-7)(10,12)
 138 \psframe*(-9,-7)(10,12)
 139 \psset[pst-solides3d]{
 140   viewpoint=20 5 10,
 141   Decran=50,lightsrc=20 10 5}
 142 \psSolid[
 143   object=grille,base=-2 2 -2 2,
 144   linecolor=white](0,0,-2)
 145 % Parametric Surfaces
 146 \defFunction{cone}(u,v)
 147   {u v Cos mul}{u v Sin mul}{u}
 148 \psSolid[object=surfaceparametree,
 149    base=-2 2 0 2 pi mul,
 150    fillcolor=yellow!50,
 151    incolor=green!50,function=cone,
 152    linewidth=0.5\pslinewidth,
 153    ngrid=25 40]%
 154 \psset{linecolor=white}\white
 155 \gridIIID[Zmin=-2,Zmax=2]
 156   (-2,2)(-2,2)
 157 \end{pspicture}
 158 \end{LTXexample}
 159
 160
 161 \subsection{An advised website}
 162 You will find on the website:
 163
 164 \centerline{\url{http://k3dsurf.sourceforge.net/}}
 165
 166 an excellent software to represent surfaces with numerous examples of parameterised surfaces and others.
 167
 168 \endinput
 169