doc-en/par-definirplanquelconque-en.tex

   1 \section{Defining a \Index{projection plane}}
   2
   3 The \Index{plane} of projection is defined with the option
   4 \texttt{\Lkeyword{plan}=plantype} which expects an argument \textit{type of
   5 plane}. The creation of such an argument invariably happens
   6 through the command \verb+\psSolid[object=plan]+ (see the relevant
   7 paragraph of chapter 4 and the example below in sub-paragraph
   8 \textit{Labels\/} of the paragraph \textit{Points}).
   9 \endinput
  10
  11 \section{Specifying a general projection plane}
  12
  13 To define a plane of projection, we assume that the drawing to be
  14 projected resides on the plane $Oxy$, and the user has to specify
  15 the images of the origin $O$ and the basis $\overrightarrow{\imath}$,
  16 $\overrightarrow{\jmath}$, and $\overrightarrow{k}$.
  17 If they wish to abbreviate the syntax, users are required at most
  18 to specify the image of $O$ and the image of $\overrightarrow{k}$,
  19 in other words the image of the origin and the components of the
  20 normal to the plane of projection.
  21
  22 The package then suggests an orientation of the plane of
  23 projection. If users are not happy with this orientation, they can
  24 specify it themselves.
  25
  26 The following paragraphs detail the proposed syntax.
  27
  28
  29 \subsection{The origin}
  30
  31 \texttt{(x,y,z)} are the projected coordinates of the origin of
  32 the plane, which are either numerical values or expressions that
  33 PostScript can handle.
  34
  35
  36 \texttt{\textbackslash psProjection[\ldots](1,2,3)} positions the
  37 origin of the plane at the coordinates $(1,2,3)$.
  38
  39
  40 \texttt{\textbackslash psProjection[\ldots](0.5 0.5 add,2 sqrt,1.5 2
  41 exp)} positions the origin of the plane at the coordinates
  42 $(1,\sqrt{2},1.5^2)$.
  43
  44
  45 If no coordinates are chosen (by the end of the command), it is
  46 interpreted as $(0,0,0)$, placing the origin at $O$.
  47
  48
  49 \subsection{Defining the normal to a plane}
  50
  51 There are four ways to define a normal to a plane, two of which
  52 have an option to rotate the coordinate system of the plane around
  53 that normal: \Lkeyword{normal}
  54
  55 \subsubsection{Method 1: giving the components of the normal vector}
  56
  57
  58 In this case \texttt{\Lkeyword{normal}=nx ny nz}, the argument consists of
  59 3 values: the components of the normal vector. For example
  60 \texttt{\Lkeyword{normal}=0 0 1} for the plane $Oxy$.
  61
  62 \newpage
  63
  64 \begin{LTXexample}[width=6.5cm]
  65 \psset{unit=0.5}
  66 \begin{pspicture}(-9,-6.5)(6,6)
  67 \psframe*[linecolor=blue!50](-6,-6)(6,6)
  68 \psset{lightsrc=10 -20 50,viewpoint=50 20 30 rtp2xyz,Decran=50}
  69 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
  70 \psSolid[object=grille,base=-4 4 -4 4]}
  71 \defFunction[algebraic]{f1}(x){3*cos(x)}{3*sin(x)}{}
  72 \psProjection[object=courbeR2,
  73    range=-3.14 3.14,
  74    linecolor=blue,
  75    normal=0 0 1,
  76    function=f1]
  77 \axesIIID(0,0,0)(4,4,4)
  78 \psProjection[object=chemin,
  79   linewidth=.1,
  80   linecolor=red,
  81   normal=1 0 0,
  82   path=newpath
  83     0 0 smoveto
  84     -1 0 slineto]
  85 \rput(0,-6.75){%
  86   \psframebox[linecolor=blue!50]
  87    {\texttt{$\backslash${}defFunction[algebraic]%
  88     \{f\}(x)\{3*cos(x)\}\{3*sin(x)\}\{\}}}}
  89 \end{pspicture}
  90 \end{LTXexample}
  91
  92 \newpage
  93 If we call
  94 $\big(\overrightarrow{i}(1,0,0),\overrightarrow{j}(0,1,0),\overrightarrow{k}(0,0,1)\big)$
  95 the basis of the referencing coordinate system and if
  96 $\big(\overrightarrow{I},\overrightarrow{J},\overrightarrow{K}\big)$
  97 is the basis of the coordinate system of the plane to be defined,
  98 with $\overrightarrow{K}$ being the chosen normal vector, then the
  99 following relations are verified and should be kept in mind:
 100 \begin{enumerate}
 101   \item $\overrightarrow{J}=\overrightarrow{K}\wedge \overrightarrow{i}$
 102   \item $\overrightarrow{I}=\overrightarrow{J}\wedge \overrightarrow{K}$
 103 \end{enumerate}
 104 If $\overrightarrow{K}=\overrightarrow{i}$ then
 105 $\overrightarrow{J}=\overrightarrow{j}$
 106
 107 \encadre{With the convention: $\overrightarrow{K}$ is drawn in
 108  red, $\overrightarrow{J}$ in blue and $\overrightarrow{I}$ in green.} %$
 109
 110 \vfill
 111 \begin{minipage}{0.27\linewidth}
 112 \psset{unit=0.4}
 113 \centerline{\texttt{[normal=0 0 1]}}
 114
 115 \begin{pspicture}(-6,-6)(4,7)%
 116 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 117 \psset{lightsrc=10 -20 50,viewpoint=50 20 30 rtp2xyz,Decran=60}
 118 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 119 \psSolid[object=grille,base=-4 4 -4 4]}
 120 \axesIIID(0,0,0)(4,4,4)
 121 \psProjection[object=chemin,
 122       linewidth=.2,
 123       linecolor=red,
 124       normal=1 0 0,
 125       path=newpath
 126             0 0 smoveto
 127             -1 0 slineto]
 128 \psProjection[object=chemin,
 129       linewidth=.2,
 130       linecolor=blue,
 131       normal=0 0 1,
 132       path=newpath
 133             0 0 smoveto
 134             0 1 slineto]
 135 \psProjection[object=chemin,
 136       linewidth=.2,
 137       linecolor=green,
 138       normal=0 0 1,
 139       path=newpath
 140             0 0 smoveto
 141             1 0 slineto]
 142 \end{pspicture}
 143 \end{minipage}
 144 \hfill
 145 \begin{minipage}{0.27\linewidth}
 146 \psset{unit=0.4}
 147 \centerline{\texttt{[normal=1 0 0]}}
 148
 149 \begin{pspicture}(-6,-6)(4,7)
 150 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 151 \psset{lightsrc=50 20 30,viewpoint=50 20 30,Decran=70}
 152 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 153 \psSolid[object=grille,base=-4 4 -4 4,RotY=90]}
 154 \axesIIID(0,0,0)(4,4,4)
 155 \psProjection[object=chemin,
 156       linewidth=.2,
 157       linecolor=green,
 158       normal=1 0 0,
 159       path=newpath
 160             0 0 smoveto
 161             1 0 slineto]
 162 \psProjection[object=chemin,
 163       linewidth=.2,
 164       linecolor=blue,
 165       normal=1 0 0,
 166       path=newpath
 167             0 0 smoveto
 168             0 1 slineto]
 169 \psProjection[object=chemin,
 170       linewidth=.2,
 171       linecolor=red,
 172       normal=0 0 1,
 173       path=newpath
 174             0 0 smoveto
 175             1 0 slineto]
 176 \end{pspicture}
 177 \end{minipage}
 178 \hfill
 179 \begin{minipage}{0.27\linewidth}
 180 \psset{unit=0.4}
 181
 182 \centerline{\texttt{[normal=0 1 0]}}
 183
 184 \begin{pspicture}(-6,-6)(4,7)
 185 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 186 \psset{lightsrc=50 60 30,viewpoint=50 60 30,Decran=70}
 187 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 188 \psSolid[object=grille,base=-4 4 -4 4,RotX=-90]}
 189 \axesIIID(0,0,0)(4,4,4)
 190 \psProjection[object=chemin,
 191       linewidth=.2,
 192       linecolor=green,
 193       normal=0 1 0,
 194       path=
 195          newpath
 196             0 0 smoveto
 197             1 0 slineto]
 198 \psProjection[object=chemin,
 199       linewidth=.2,
 200       linecolor=blue,
 201       normal=0 1 0,
 202       path=
 203          newpath
 204             0 0 smoveto
 205             0 1 slineto]
 206 \psProjection[object=chemin,
 207       linewidth=.2,
 208       linecolor=red,
 209       normal=0 0 1,
 210       path=
 211          newpath
 212             0 0 smoveto
 213             0 1 slineto]
 214 \end{pspicture}
 215 \end{minipage}
 216
 217 \vfill
 218
 219 \begin{minipage}{0.27\linewidth}
 220 \psset{unit=0.4}
 221 \centerline{\texttt{[normal=1 0 1]}}
 222
 223 \begin{pspicture}(-6,-6)(4,7)%
 224 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 225 \psset{lightsrc=50 20 20,viewpoint=50 20 20,Decran=70}
 226 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 227 \psSolid[object=grille,base=-4 4 -4 4]
 228 \psSolid[object=grille,base=-4 0 -4 4,RotY=90]
 229 \psSolid[object=grille,base=-2 2 -4 4,RotY=45,linecolor=red](1.414,0,1.414)}
 230 \psPoint(2,0,2){O1}%\psdot(O1)
 231 \axesIIID(2.8,3,2.8)(4,4,4)
 232 \psPoint(1.414,0,1.414){O1}\psPoint(2.414,0,2.414){OK}
 233 \psline[linewidth=.2,linecolor=red](O1)(OK)
 234 \psProjection[object=chemin,
 235       linewidth=.2,
 236       linecolor=blue,
 237       normal=1 0 1,
 238       path=
 239          newpath
 240             0 0 smoveto
 241             0 1 slineto](1.414,0,1.414)
 242 \psProjection[object=chemin,
 243       linewidth=.2,
 244       linecolor=green,
 245       normal=1 0 1,
 246       path=
 247          newpath
 248             0 0 smoveto
 249             1 0 slineto](1.414,0,1.414)
 250 \end{pspicture}
 251 \end{minipage}
 252 \hfill
 253 \begin{minipage}{0.27\linewidth}
 254 \psset{unit=0.4}
 255 \centerline{\texttt{[normal=0 1 1]}}
 256
 257 \begin{pspicture}(-6,-6)(4,7)%
 258 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 259 \psset{lightsrc=30 20 20,viewpoint=30 20 20,Decran=45}
 260 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 261 \psSolid[object=grille,base=-4 4 -4 4]
 262 \psSolid[object=grille,base=-4 0 -4 4,RotY=90,RotZ=90]
 263 \psSolid[object=grille,base=-4 4 -2 2,RotX=-45,linecolor=red](0,1.414,1.414)}
 264 \axesIIID(2.8,3,2.8)(4,4,4)
 265 \psPoint(0,1.414,1.414){O1}\psPoint(0,2.414,2.414){OK}
 266 \psline[linewidth=.2,linecolor=red](O1)(OK)
 267 \psProjection[object=chemin,
 268       linewidth=.2,
 269       linecolor=blue,
 270       normal=0 1 1,
 271       path=
 272          newpath
 273             0 0 smoveto
 274             0 1 slineto](0,1.414,1.414)
 275 \psProjection[object=chemin,
 276       linewidth=.2,
 277       linecolor=green,
 278       normal=0 1 1,
 279       path=
 280          newpath
 281             0 0 smoveto
 282             1 0 slineto](0,1.414,1.414)
 283 \end{pspicture}
 284 \end{minipage}
 285 \hfill
 286 \begin{minipage}{0.27\linewidth}
 287 \psset{unit=0.4}
 288 \centerline{\texttt{[normal=1 1 0]}}
 289
 290 \begin{pspicture}(-6,-6)(4,7)%
 291 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 292 \psset{lightsrc=30 20 20,viewpoint=30 20 20,Decran=45}
 293 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 294 \psSolid[object=grille,base=0 2 -4 4,RotY=90,RotZ=45,linecolor=red]
 295         (1.414,1.414,0)
 296 \psSolid[object=grille,base=-4 4 -4 4]
 297 \psSolid[object=grille,base= -2 0 -4 4,RotY=90,RotZ=45,linecolor=red]
 298         (1.414,1.414,0)}
 299 \axesIIID(2.8,2.8,1)(4,4,4)
 300 \psPoint(1.414,1.414,0){O1}\psPoint(2.414,2.414,0){OK}
 301 \psline[linewidth=.2,linecolor=red](O1)(OK)
 302 \psProjection[object=chemin,
 303       linewidth=.2,
 304       linecolor=blue,
 305       normal=1 1 0,
 306       path=
 307          newpath
 308             0 0 smoveto
 309             0 1 slineto](1.414,1.414,0)
 310 \psProjection[object=chemin,
 311       linewidth=.2,
 312       linecolor=green,
 313       normal=1 1 0,
 314       path=
 315          newpath
 316             0 0 smoveto
 317             1 0 slineto](1.414,1.414,0)
 318 \end{pspicture}
 319 \end{minipage}
 320
 321 \vfill
 322
 323 \newpage
 324
 325 \subsubsection{Method 2: giving the components of the normal vector and an angle
 326 of rotation}
 327
 328 In this case \texttt{\Lkeyword{normal}=nx ny nz A}, the argument takes four
 329 values: the components of the normal vector and the angle of
 330 rotation (in degrees) around that axis.
 331
 332 \begin{center}
 333 \begin{minipage}{0.34\linewidth}
 334 \psset{unit=0.5}
 335 \centerline{\texttt{[normal=1 0 1]}}
 336
 337 \begin{pspicture}(-6,-6)(6,7)%
 338 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 339 \psset{lightsrc=50 20 20,viewpoint=50 20 20,Decran=70}
 340 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 341 \psSolid[object=grille,base=-4 4 -4 4]
 342 \psSolid[object=grille,base=-4 0 -4 4,RotY=90]
 343 \psSolid[object=grille,base=-2 2 -4 4,RotY=45,linecolor=red](1.414,0,1.414)}
 344 \psPoint(2,0,2){O1}%\psdot(O1)
 345 \axesIIID(2.8,3,2.8)(4,4,4)
 346 \psPoint(1.414,0,1.414){O1}\psPoint(2.414,0,2.414){OK}
 347 \psline[linewidth=.2,linecolor=red](O1)(OK)
 348 \psProjection[object=chemin,
 349       linewidth=.2,
 350       linecolor=blue,
 351       normal=1 0 1,
 352       path=
 353          newpath
 354             0 0 smoveto
 355             0 1 slineto](1.414,0,1.414)
 356 \psProjection[object=chemin,
 357       linewidth=.2,
 358       linecolor=green,
 359       normal=1 0 1,
 360       path=
 361          newpath
 362             0 0 smoveto
 363             1 0 slineto](1.414,0,1.414)
 364 \defFunction[algebraic]{fonction}(x){cos(x)}{x}{}
 365 \psProjection[object=courbeR2,
 366    range=-4 4,
 367    normal=1 0 1,
 368    function=fonction](1.414,0,1.414)
 369 \end{pspicture}
 370 \end{minipage}
 371 \hfil
 372 \begin{minipage}{0.34\linewidth}
 373 \psset{unit=0.5}
 374 \centerline{\texttt{[normal=1 0 1 45]}}
 375
 376 \begin{pspicture}(-6,-6)(6,7)%
 377 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 378 \psset{lightsrc=50 20 20,viewpoint=50 30 20,Decran=70}
 379 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 380 \psSolid[object=grille,base=-4 4 -4 4,action=draw]
 381 \psSolid[object=grille,base=-4 4 -4 4,RotY=90,action=draw]}
 382 \axesIIID(0,0,0)(4,4,4)
 383 \psPoint(1.414,0,1.414){O1}\psPoint(2.414,0,2.414){OK}
 384 \psline[linewidth=.2,linecolor=red](O1)(OK)
 385 \psProjection[object=chemin,
 386       linewidth=.02,
 387       linecolor=red,
 388       normal=1 0 1 45,
 389       path=newpath
 390           -2 1 2
 391           {-4 smoveto
 392            0 8 srlineto} for
 393            -4 1 4
 394           {-2 exch smoveto
 395            4 0 srlineto} for
 396             ](1.414,0,1.414)
 397 \psProjection[object=chemin,
 398       linewidth=.2,
 399       linecolor=blue,
 400       normal=1 0 1 45,
 401       path=
 402          newpath
 403             0 0 smoveto
 404             0 1 slineto](1.414,0,1.414)
 405 \psProjection[object=chemin,
 406       linewidth=.2,
 407       linecolor=green,
 408       normal=1 0 1 45,
 409       path=
 410          newpath
 411             0 0 smoveto
 412             1 0 slineto](1.414,0,1.414)
 413 \defFunction[algebraic]{fonction}(x){cos(x)}{x}{}
 414 \psProjection[object=courbeR2,
 415    range=-4 4,
 416    normal=1 0 1 45 ,
 417    function=fonction](1.414,0,1.414)
 418 \end{pspicture}
 419 \end{minipage}
 420 \end{center}
 421
 422 %\newpage
 423 In the second figure, the normal (represented in red) is identical
 424 to the one in the first figure, but the plane is rotated 45
 425 degrees around that normal.
 426
 427
 428 \subsubsection{Method 3: defining the normal by the images of \textit{i}
 429  and \textit{k}}
 430
 431 In this case \texttt{\Lkeyword{normal}=ix iy iz kx ky kz}, the argument
 432 takes six values: the components of the images of
 433 $\overrightarrow{i}$ and $\overrightarrow{k}$, with:
 434 $\overrightarrow{J}=\overrightarrow{K}\wedge \overrightarrow{I}$.
 435
 436 \begin{center}
 437 \begin{minipage}{0.34\linewidth}
 438 \psset{unit=0.5}
 439 \centerline{\texttt{[normal=0 1 0 1 0 0]}}
 440
 441 \begin{pspicture}(-6,-6)(6,7)%
 442 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 443 \psset{lightsrc=50 20 20,viewpoint=50 30 20,Decran=70}
 444 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 445 \psSolid[object=grille,base=-4 4 -4 4]%
 446 \psSolid[object=grille,base=-4 4 -4 4,action=draw,RotX=90,RotZ=90]}%
 447 \defFunction[algebraic]{fonction}(x){x}{x*sin(x)}{}
 448 \axesIIID(0,0,0)(4,4,4)
 449 \psProjection[object=chemin,
 450       linewidth=.1,
 451       linecolor=green,
 452       normal=0 1 0 1 0 0,
 453       path=
 454          newpath
 455             0 0 smoveto
 456             1 0 slineto]
 457 \psProjection[object=chemin,
 458       linewidth=.1,
 459       linecolor=blue,
 460       normal=0 1 0 1 0 0,
 461       path=
 462          newpath
 463             0 0 smoveto
 464             0 1 slineto]
 465 \psProjection[object=chemin,
 466       linewidth=.2,
 467       linecolor=red,
 468       normal=0 0 1,
 469       path=
 470          newpath
 471             0 0 smoveto
 472             1 0 slineto]
 473 \psProjection[object=courbeR2,
 474    range=-4 4,
 475    linecolor=green,
 476    normal=0 1 0 1 0 0,
 477    function=fonction]
 478 \end{pspicture}
 479 \end{minipage}
 480 \hfil
 481 \begin{minipage}{0.34\linewidth}
 482 \psset{unit=0.5}
 483 \centerline{\texttt{[normal=-1 1 0 1 1 2 ]}}
 484
 485 \begin{pspicture}(-6,-6)(6,7)%
 486 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 487 \psset{lightsrc=50 20 20,viewpoint=50 30 20,Decran=70}
 488 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 489 \psSolid[object=grille,base=-4 4 -4 4]}%
 490 \defFunction[algebraic]{fonction}(x){x}{x*sin(x)}{}
 491 \axesIIID(0,0,0)(4,4,4)
 492 \psProjection[object=chemin,
 493       linewidth=.1,
 494       linecolor=green,
 495       normal=-1 1 0 1 1 2 ,
 496       path=
 497          newpath
 498             0 0 smoveto
 499             1 0 slineto]
 500 \psProjection[object=chemin,
 501       linewidth=.1,
 502       linecolor=blue,
 503       normal=-1 1 0 1 1 2 ,
 504       path=
 505          newpath
 506             0 0 smoveto
 507             0 1 slineto]
 508 \psPoint(0,0,0){O}\psPoint(0.4,0.4,0.8){K}
 509 \psline[linewidth=.1,linecolor=red](O)(K)
 510 \psProjection[object=courbeR2,
 511    range=-4 4,
 512    linecolor={[cmyk]{1,0,1,0.5}},
 513    normal=-1 1 0 1 1 2 ,
 514    function=fonction]
 515 \psProjection[object=chemin,
 516       linewidth=.02,
 517       linecolor=red,
 518       normal=-1 1 0 1 1 2,
 519       path=newpath
 520           -4 1 4
 521           {-4 exch smoveto
 522            8 0 srlineto} for
 523            -4 1 4
 524           {-4 smoveto
 525            0 8 srlineto} for]
 526 \end{pspicture}
 527 \end{minipage}
 528 \end{center}
 529
 530
 531 \newpage
 532 \subsubsection{Method 4: defining the normal by the images of
 533   \textit{i}, \textit{k} and an angle of rotation}
 534
 535 In this case \texttt{\Lkeyword{normal}=ix iy iz kx ky kz phi}, the argument
 536 takes seven values: the components of the images of
 537 $\overrightarrow{i}$, $\overrightarrow{k}$ and the angle of
 538 rotation (in degrees) around the normal, with:
 539 $\overrightarrow{J}=\overrightarrow{K}\wedge \overrightarrow{I}$.
 540
 541 \begin{center}
 542 \begin{minipage}{0.34\linewidth}
 543 \psset{unit=0.5}
 544 \centerline{\texttt{[normal=0 1 0 1 0 0 90]}}
 545
 546 \begin{pspicture}(-6,-6)(6,7)%
 547 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 548 \psset{lightsrc=50 20 20,viewpoint=50 30 20,Decran=70}
 549 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 550 \psSolid[object=grille,base=-4 4 -4 4]%
 551 \psSolid[object=grille,base=-4 4 -4 4,action=draw,RotX=90,RotZ=90]}%
 552 \defFunction[algebraic]{fonction}(x){x}{x*sin(x)}{}
 553 \axesIIID(0,0,0)(4,4,4)
 554 \psProjection[object=chemin,
 555       linewidth=.1,
 556       linecolor=green,
 557       normal=0 1 0 1 0 0 90,
 558       path=
 559             0 0 smoveto
 560             1 0 slineto]
 561 \psProjection[object=chemin,
 562       linewidth=.1,
 563       linecolor=blue,
 564       normal=0 1 0 1 0 0 90,
 565       path=
 566             0 0 smoveto
 567             0 1 slineto]
 568 \psProjection[object=chemin,
 569       linewidth=.2,
 570       linecolor=red,
 571       normal=0 0 1,
 572       path=
 573             0 0 smoveto
 574             1 0 slineto]
 575 \psProjection[object=courbeR2,
 576    range=-4 4,
 577    linecolor=green,
 578    normal=0 1 0 1 0 0 90,
 579    function=fonction]
 580 \end{pspicture}
 581 \end{minipage}
 582 \hfil
 583 \begin{minipage}{0.34\linewidth}
 584 \psset{unit=0.5}
 585 \centerline{\texttt{[normal=-1 1 0 1 1 2 90]}}
 586
 587 \begin{pspicture}(-6,-6)(6,7)%
 588 \psframe*[linecolor=blue!50](-6,-6)(6,6)
 589 \psset{lightsrc=50 20 20,viewpoint=50 30 20,Decran=70}
 590 {\psset{linewidth=0.5\pslinewidth,linecolor=gray}
 591 \psSolid[object=grille,base=-4 4 -4 4]}%
 592 \defFunction[algebraic]{fonction}(x){x}{x*sin(x)}{}
 593 \axesIIID(0,0,0)(4,4,4)
 594 \psProjection[object=chemin,
 595   linewidth=.1,
 596   linecolor=green,
 597   normal=-1 1 0 1 1 2 90,
 598   path=newpath
 599     0 0 smoveto
 600     1 0 slineto]
 601 \psProjection[object=chemin,
 602   linewidth=.1,
 603   linecolor=blue,
 604   normal=-1 1 0 1 1 2 90,
 605   path=newpath
 606     0 0 smoveto
 607     0 1 slineto]
 608 \psPoint(0,0,0){O}\psPoint(0.4,0.4,0.8){K}
 609 \psline[linewidth=.1,linecolor=red](O)(K)
 610 \psProjection[object=courbeR2,
 611    range=-4 4,
 612    linecolor={[cmyk]{1,0,1,0.5}},
 613    normal=-1 1 0 1 1 2 90,
 614    function=fonction]
 615 \psProjection[object=chemin,
 616   linewidth=.02,
 617   linecolor=red,
 618   normal=-1 1 0 1 1 2,
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