doc-en/par-section-en.tex

   1 \section{Sectioning a solid with a plane}
   2
   3 \subsection{Drawing the \Index{intersection} between a plane and a solid}
   4
   5 \subsubsection{The parameters}
   6
   7 The option \texttt{intersectionplan=\{[a b c d]\}} allows the user
   8 to draw the intersection between a plane and a solid. The numbers
   9 between the braces are the coefficients of the affine plane with
  10 equation: $ax+by+cz+d=0$. It is possible to draw the intersection
  11 between a solid and more than one plane by placing the appropriate
  12 parameters in order, as in the following example.
  13
  14 The drawing is activated with \texttt{\texttt{\Lkeyword{intersectiontype}=0}} or any
  15 value $\geq0$.
  16
  17 The colour of the intersection line is chosen with the option
  18 \texttt{\Lkeyword{intersectioncolor}=(bleu) (rouge) etc.}. In the same order,
  19 the thickness of the appropriate line
  20 \texttt{\Lkeyword{intersectionlinewidth}=1 2 etc.} (dimensions in picas) is
  21 set up.
  22
  23
  24 The hidden parts, drawn with dashed lines, will be shown with
  25 \Lkeyword{action}=\Lkeyval{draw}.
  26
  27 \begin{LTXexample}[width=7cm]
  28 \begin{pspicture}(-3,-2)(3,7.5)
  29 \psset{viewpoint=50 20 20 rtp2xyz,Decran=50}
  30 \psset{lightsrc=viewpoint} \psSolid[object=cylindre,
  31    ngrid=1 24,
  32    r=2,
  33    fillcolor=yellow!25,
  34    intersectiontype=0,
  35    intersectionplan={
  36       [0 0 1 -1]
  37       [0 0 1 -2]
  38       [0 0 1 -3]
  39       [0.894 0 0.447 -1.8]},
  40    intersectioncolor=(bleu) (rouge) (vert) (rose),
  41    intersectionlinewidth=1 1.5 1.8 2.2]
  42 \axesIIID(2,2,6)(3,3,7)
  43 \end{pspicture}
  44 \end{LTXexample}
  45
  46 \subsection{Slicing a solid}
  47
  48 \subsubsection{Slicing a filled solid}
  49
  50 The object under consideration is a cylinder. The plane that
  51 slices the object is defined by:
  52
  53  \begin{verbatim}
  54   plansepare={[a b c d]}
  55  \end{verbatim}
  56
  57 The two parts are not drawn, but memorised with the name
  58 \texttt{\Lkeyword{name}=partiescylindre}:
  59
  60  \begin{verbatim}
  61 \psset{solidmemory}
  62 \psSolid[object=cylindre,
  63    r=2,h=6
  64    ngrid=6 24,
  65    plansepare={[0.707 0 0.707 0]},
  66    name=partiescylindre,
  67    action=none](0,0,-3)
  68  \end{verbatim}
  69
  70
  71 Then they are displayed separately using their respective index
  72 numbers. The numbering of the two parts is determined by the
  73 direction of the normal to the \Index{slicing} plane: 0 if above the
  74 normal, 1 if below. For both parts, the sliced face carries the
  75 number 0. If there are several sliced faces, as may happen in the
  76 case of a torus, they are numbered 0, 1 etc.
  77
  78
  79  \begin{verbatim}
  80 \psSolid[object=load,
  81    load=partiescylindre1,
  82    fillcolor={[rgb]{0.7 1 0.7 }},
  83    fcol=0 (1 1 0.7 setrgbcolor)]
  84 \psSolid[object=load,
  85    load=partiescylindre0,RotZ=60,
  86    fillcolor={[rgb]{0.7 1 0.7 }},
  87    fcol=0 (1 1 0.7 setrgbcolor)](0,4,0)
  88  \end{verbatim}
  89
  90 \begin{center}
  91 \begin{pspicture}(-4,-5)(7,4)
  92 \psframe(-4,-5)(7,4)
  93 \psset{viewpoint=50 -40 10 rtp2xyz,Decran=50,linecolor=darkgray}
  94 %\lightsource
  95 \psset{lightsrc=viewpoint}
  96 \psSolid[object=grille,action=draw,
  97    base=-3 5 -3 5,
  98    linecolor=red](0,0,-3)
  99 \psset{solidmemory}
 100 \psSolid[object=cylindre,
 101    r=2,h=6,
 102    ngrid=6 24,
 103    plansepare={[0.707 0 0.707 0]},
 104    name=partiescylindre,
 105    action=none](0,0,-3)
 106 \psSolid[object=load,
 107    load=partiescylindre1,
 108    fillcolor={[rgb]{0.7 1 0.7 }},
 109    fcol=0 (1 1 0.7 setrgbcolor)]
 110 \psSolid[object=load,
 111    load=partiescylindre0,RotZ=90,
 112    fillcolor={[rgb]{0.7 1 0.7 }},
 113    fcol=0 (1 1 0.7 setrgbcolor)](0,4,0)
 114 \psSolid[object=plan,action=draw,
 115    definition=equation,
 116    args={[0.707 0 0.707 0] 90},
 117    base=-2 2 -3 3,planmarks,showBase]
 118 \axesIIID(0,0,0)(2.5,2.5,3.5)
 119 \end{pspicture}
 120 \end{center}
 121
 122 \subsubsection{Slicing a \Index{hollow solid}}
 123
 124 The options \verb+rm=0,hollow+ allow us to not only remove a
 125 face \verb+rm=0+ but also to see inside it \Lkeyword{hollow}.
 126
 127 \begin{center}
 128 \begin{pspicture}(-4,-5)(7,4)
 129 \psframe(-4,-5)(7,4)
 130 \psset{viewpoint=50 -40 10 rtp2xyz,Decran=50,linecolor=darkgray}
 131 %\lightsource
 132 \psset{lightsrc=viewpoint}
 133 \psSolid[object=grille,action=draw,
 134    base=-3 5 -3 5,
 135    linecolor=red](0,0,-3)
 136 \psset{solidmemory}
 137 \psSolid[object=cylindre,
 138    r=2,h=6,
 139    ngrid=6 24,
 140    plansepare={[0.707 0 0.707 0.5]},
 141    name=partiescylindre,
 142    action=none](0,0,-3)
 143 \psSolid[object=load,
 144    load=partiescylindre1,
 145    fillcolor={[rgb]{0.7 1 0.7 }},
 146    rm=0,hollow,
 147    incolor={[rgb]{1 1 0.7}}]
 148 \psSolid[object=load,
 149    load=partiescylindre0,RotZ=90,
 150    fillcolor={[rgb]{0.7 1 0.7 }},
 151    rm=0,hollow,
 152    incolor={[rgb]{1 1 0.7}}](0,4,0)
 153 \psSolid[object=plan,action=draw,
 154    definition=equation,
 155    args={[0.707 0 0.707 0.5] 90},
 156    base=-2 2 -3 3,planmarks,showBase]
 157 \composeSolid
 158 \end{pspicture}
 159 \end{center}
 160
 161
 162 \subsection{\Index{Slice} of a \Index{pyramid}}
 163
 164 \subsubsection{Highlighting the \Index{contour lines} and first slice}
 165
 166 This pyramid is generated as \texttt{\Lkeyword{object}=\Lkeyval{new}} by giving a list
 167 of the coordinates of the vertices, and the vertices of each face.
 168
 169 \begin{verbatim}
 170    sommets=
 171       0 -2 0 %% 0
 172       -2 0 0 %% 1
 173       0 4 0  %% 2
 174       4 0 0  %% 3
 175       0 0 5, %% 4
 176    faces={
 177     [3 2 1 0]
 178     [4 0 3]
 179     [4 3 2]
 180     [4 2 1]
 181     [4 1 0]
 182 }
 183 \end{verbatim}
 184
 185 In the first diagram, the slicing lines are highlighted.
 186
 187  \begin{verbatim}
 188    intersectiontype=0,
 189    intersectionplan={[0 0 1 -1] [0 0 1 -2]},
 190    intersectionlinewidth=1 2,
 191    intersectioncolor=(bleu) (rouge)
 192  \end{verbatim}
 193
 194 Then we cut off the upper part, and draw the slicing plane as
 195 well.
 196
 197  \begin{verbatim}
 198 \psSolid[object=new,
 199    sommets=
 200       0 -2 0 %% 0
 201       -2 0 0 %% 1
 202       0 4 0  %% 2
 203       4 0 0  %% 3
 204       0 0 5, %% 4
 205   faces={
 206      [3 2 1 0]
 207      [4 0 3]
 208      [4 3 2]
 209      [4 2 1]
 210      [4 1 0]},
 211    plansepare={[0 0 1 -2]},
 212    name=firstSlice,
 213    action=none]
 214 \psSolid[object=load,action=draw*,
 215         load=firstSlice1]
 216 \psSolid[object=plan,
 217        definition=equation,
 218        args={[0 0 1 -2]},
 219        base=-3 5 -3 5,action=draw]
 220  \end{verbatim}
 221
 222 To avoid having to repeatedly type the vertices and faces of the
 223 pyramid, we save these data to the files:
 224 \begin{itemize}
 225   \item \texttt{Pyramid-couleurs.dat}
 226   \item \texttt{Pyramid-faces.dat}
 227   \item \texttt{Pyramid-sommets.dat}
 228   \item \texttt{Pyramid-io.dat}
 229 \end{itemize}
 230 thanks to the command \Lkeyword{action}=\Lkeyval{writesolid}:
 231
 232  \begin{verbatim}
 233 \psSolid[object=new,
 234    sommets=
 235       0 -2 0 %% 0
 236       -2 0 0 %% 1
 237       0 4 0  %% 2
 238       4 0 0  %% 3
 239       0 0 5, %% 4
 240   faces={
 241      [3 2 1 0]
 242      [4 0 3]
 243      [4 3 2]
 244      [4 2 1]
 245      [4 1 0]
 246 },file=./Pyramid,fillcolor=yellow!50,
 247   action=writesolid]
 248  \end{verbatim}
 249
 250 All these lines of code could then be removed and, thereafter, we
 251 would recall the data with the command:
 252
 253  \begin{verbatim}
 254 \psSolid[object=datfile,
 255          file=./Pyramid]
 256  \end{verbatim}
 257
 258 \begin{center}
 259 \psset{unit=0.75}
 260 \begin{pspicture}(-5,-2)(5,7)
 261 %\psframe(-5,-2)(5,7)
 262 \psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
 263 %\lightsource
 264 \psset{lightsrc=viewpoint}
 265 \psSolid[object=grille,
 266    base=-3 5 -3 5,
 267    linecolor=gray]
 268 %% create the pyramid with base area in xy-plane
 269 \psSolid[object=new,
 270    sommets=
 271       0 -2 0 %% 0
 272       -2 0 0 %% 1
 273       0 4 0  %% 2
 274       4 0 0  %% 3
 275       0 0 5, %% 4
 276 faces={
 277    [3 2 1 0]
 278    [4 0 3]
 279    [4 3 2]
 280    [4 2 1]
 281    [4 1 0]
 282 }, action=draw*,
 283    intersectiontype=0,
 284    intersectionplan={[0 0 1 -1]
 285                      [0 0 1 -2]},
 286    intersectionlinewidth=1 2,
 287    intersectioncolor=(bleu) (rouge)]
 288 \axesIIID[linecolor=blue](4,4,5)(5,5,6)
 289 \end{pspicture}
 290 \hfill
 291 \begin{pspicture}(-5,-2)(5,7)
 292 %\psframe(-5,-2)(5,7)
 293 \psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
 294 %\lightsource
 295 \psset{lightsrc=viewpoint}
 296 \psSolid[object=grille,
 297    base=-3 5 -3 5,
 298    linecolor=gray]
 299 \psset{solidmemory}
 300 %% create the pyramid with base area in xy-plane
 301 \psSolid[object=new,
 302    sommets=
 303       0 -2 0 %% 0
 304       -2 0 0 %% 1
 305       0 4 0  %% 2
 306       4 0 0  %% 3
 307       0 0 5, %% 4
 308 faces={
 309    [3 2 1 0]
 310    [4 0 3]
 311    [4 3 2]
 312    [4 2 1]
 313    [4 1 0]
 314 },
 315    plansepare={[0 0 1 -2]},
 316    name=firstSlice,
 317    action=none]
 318 \psSolid[object=load,action=draw*,
 319         load=firstSlice1]
 320 \psSolid[object=plan,
 321        definition=equation,
 322        args={[0 0 1 -2]},
 323        base=-3 5 -3 5,action=draw]
 324 \axesIIID[linecolor=blue](4,4,2)(5,5,6)
 325 \end{pspicture}
 326 \end{center}
 327
 328 \subsubsection{The second \Index{slice} and its insertion within the \Index{pyramid}}
 329
 330 Having removed the upper part \texttt{firstSlice0} (which no
 331 longer appears), we slice the frustum of the pyramid
 332 \texttt{firstSlice1}, and keep the upper part of this as
 333 \texttt{secondSlice0}, then we record it and insert it into a wire
 334 frame model of the pyramid:
 335
 336
 337  \begin{verbatim}
 338 \psset{solidmemory}
 339 \psSolid[object=datfile,
 340          file=./Pyramid,
 341    plansepare={[0 0 1 -2]},
 342    name=firstSlice,
 343    action=none]
 344 \psSolid[object=load,
 345         load=firstSlice1,
 346         action=none,
 347         plansepare={[0 0 1 -1]},
 348         name=secondSlice]
 349 \psSolid[object=load,action=draw*,
 350         load=secondSlice0]
 351 \psSolid[object=load,
 352        load=secondSlice0,
 353        file=./slicePyramid,
 354       action=writesolid]
 355 \psSolid[object=datfile,fillcolor=yellow!50,
 356          file=./slicePyramid]
 357  \end{verbatim}
 358
 359
 360 \begin{center}
 361 \psset{unit=0.75}
 362 \begin{pspicture}(-5,-2)(4,7)
 363 %\psframe(-4,-2)(4,7)
 364 \psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
 365 %\lightsource
 366 \psset{lightsrc=viewpoint}
 367 \psSolid[object=grille,
 368    base=-3 5 -3 5,
 369    linecolor=gray]
 370 \psset{solidmemory}
 371 %% create the pyramid with base area in xy-plane
 372 %\psSolid[object=new,
 373 %   sommets=
 374 %      0 -2 0 %% 0
 375 %      -2 0 0 %% 1
 376 %      0 4 0  %% 2
 377 %      4 0 0  %% 3
 378 %      0 0 5, %% 4
 379 %  faces={
 380 %     [3 2 1 0]
 381 %     [4 0 3]
 382 %     [4 3 2]
 383 %     [4 2 1]
 384 %     [4 1 0]
 385 %},file=./Pyramid,fillcolor=yellow!50,
 386 %  action=writesolid]
 387 \psSolid[object=datfile,
 388          file=./Pyramid,
 389    plansepare={[0 0 1 -2]},
 390    name=firstSlice,
 391    action=none]
 392 \psSolid[object=plan,
 393    definition=equation,
 394    args={[0 0 1 -1]},action=draw,
 395    base=-3 5 -3 5]
 396 \psSolid[object=load,
 397         load=firstSlice1,
 398         action=none,
 399         plansepare={[0 0 1 -1]},
 400         name=secondSlice]
 401 %\psSolid[object=load,action=draw*,
 402 %        load=secondSlice0]
 403 %\psSolid[object=load,
 404 %       load=secondSlice0,
 405 %       file=./slicePyramid,
 406 %      action=writesolid]
 407 \psSolid[object=datfile,fillcolor=yellow!50,
 408          file=./slicePyramid]
 409 \psSolid[object=plan,
 410        definition=equation,
 411        args={[0 0 1 -2]},
 412        base=-3 5 -3 5,action=draw]
 413 \axesIIID[linecolor=blue](0,0,2)(5,5,6)
 414 \end{pspicture}
 415 \hfill
 416 \begin{pspicture}(-4,-2)(6,7)
 417 \psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
 418 %\lightsource
 419 \psset{lightsrc=viewpoint}
 420 \psSolid[object=grille,
 421    base=-3 5 -3 5,
 422    linecolor=gray]
 423 \psSolid[object=datfile,fillcolor=yellow!50,
 424          file=./slicePyramid]
 425 \psSolid[object=datfile,
 426          file=./Pyramid,action=draw]
 427 \axesIIID[linecolor=blue](4,4,2)(5,5,6)
 428 \end{pspicture}
 429 \end{center}
 430
 431 \subsection{Slicing an \Index{octahedron} with a plane parallel to one of its faces}
 432
 433 \subsubsection{The view inside}
 434
 435 Recall that there are options \verb+rm=0,hollow+ that allow us,
 436 on the one hand, to remove a face \verb+rm=0+ and, on the other,
 437 to look inside \Lkeyword{hollow}.
 438
 439 In the following example, we shall start by generating the
 440 required objects without drawing them (\texttt{\Lkeyword{action}=\Lkeyval{none}}).
 441
 442 We construct the octahedron, giving the center of the face with
 443 index $1$ the name $G$, then define the point $H$ which satisfies
 444 $\overrightarrow{OH} = 0.8\,\overrightarrow{OG}$. After that we
 445 define $P$ to be the plane through $H$ parallel to the face of the
 446 octahedron with index $1$. Finally, we slice the octahedron using
 447 the plane $P$.
 448
 449 \begin{LTXexample}[width=7.5cm]
 450 \begin{pspicture}(-3.5,-3)(4.5,5)
 451 \psset{viewpoint=100 5 10 rtp2xyz,Decran=80,
 452    lightsrc=viewpoint,solidmemory,action=none}
 453 \psSolid[object=octahedron,
 454    a=4,name=my_octahedron,]
 455 \psSolid[object=point,
 456    definition=solidcentreface,
 457    args=my_octahedron 1,
 458    name=G,]
 459 \psSolid[object=point,
 460    definition=mulv3d,
 461    args=G .8,
 462    name=H,]
 463 \psSolid[object=plan,
 464    definition=solidface,
 465    args=my_octahedron 1,
 466    base=-4 4 -4 4,
 467    name=P,](H,,)
 468 \psSolid[object=load,
 469    load=my_octahedron,
 470    plansepare=P,
 471    name=part]
 472 \psSolid[object=load,load=part1,
 473          rm=0,hollow,action=draw**,
 474          fillcolor={[rgb]{0.7 1 0.7}},
 475          incolor={[rgb]{1 1 0.7}},]
 476 \psSolid[object=plan,args=P,
 477        action=draw,showBase]
 478 \psSolid[object=line,
 479    args=0 0 0 H,
 480    linestyle=dashed,]
 481 \psProjection[object=point,plan=P,args=0 0,
 482     fontsize=20,pos=cl,text=H,phi=90,]
 483 \axesIIID[linecolor=blue,linewidth=0.4pt](0,0,0)(4,4,4)
 484 \end{pspicture}
 485 \end{LTXexample}
 486
 487 \subsubsection{Regarding the solid as filled}
 488
 489 The option \verb+fcol=0 (YellowOrange)+ allows us to colour the
 490 face with index 0.
 491
 492 \begin{LTXexample}[width=7.5cm]
 493 \begin{pspicture}(-3.5,-3)(4.5,5)
 494 \psset{viewpoint=100 5 10 rtp2xyz,Decran=80,
 495    lightsrc=viewpoint,solidmemory,action=none}
 496 \psSolid[object=octahedron,
 497    a=4,name=my_octahedron,]
 498 \psSolid[object=point,
 499    definition=solidcentreface,
 500    args=my_octahedron 1,
 501    name=G,]
 502 \psSolid[object=point,
 503    definition=mulv3d,
 504    args=G .8,
 505    name=H,]
 506 \psSolid[object=plan,
 507    definition=solidface,
 508    args=my_octahedron 1,
 509    base=-4 4 -4 4,
 510    name=P,](H,,)
 511 \psSolid[object=load,
 512    load=my_octahedron,
 513    plansepare=P,
 514    name=part]
 515 \psSolid[object=load,
 516    load=part1,
 517    fcol=0 (YellowOrange),
 518    action=draw**,
 519    fillcolor={[rgb]{0.7 1 0.7}},]
 520 \psSolid[object=plan,args=P,
 521        action=draw,showBase]
 522 \psSolid[object=line,
 523    args=0 0 0 H,
 524    linestyle=dashed,]
 525 \psProjection[object=point,plan=P,args=0 0,
 526     fontsize=20,pos=cl,text=H,phi=90,]
 527 \axesIIID[linecolor=blue,linewidth=0.4pt](0,0,0)(4,4,4)
 528 \end{pspicture}
 529 \end{LTXexample}
 530
 531 \subsubsection{The two parts of a sliced solid}
 532
 533 You will recall that the direction of the normal of the slicing
 534 plane determines the numbering of the two parts: 0 if above the
 535 normal, 1 if below. For both parts, the sliced face carries the
 536 number 0. If there are several sliced faces, as in the case of the
 537 torus, they are numbered 0, 1 etc.
 538
 539 Using two steps, we memorise both parts of the sliced solid:
 540
 541  \begin{verbatim}
 542 \psSolid[object=load,
 543    load=my_octahedron,
 544    plansepare=P,
 545    name=part]
 546  \end{verbatim}
 547
 548 Then we position and render each part:
 549
 550  \begin{verbatim}
 551 \psSolid[object=load,
 552          fcol=0 (YellowOrange),
 553          fillcolor={[rgb]{0.7 1 0.7}},
 554         load=part1]
 555 \psSolid[object=load,
 556          fillcolor={[rgb]{0.7 1 0.7}},
 557         load=part0](H 2 mulv3d,,)
 558 \composeSolid
 559  \end{verbatim}
 560
 561
 562 \begin{LTXexample}[width=7.5cm]
 563 \begin{pspicture}(-3.5,-3)(4.5,5)
 564 \psset{viewpoint=100 5 20 rtp2xyz,Decran=150,
 565    lightsrc=viewpoint,solidmemory,action=none}
 566 \psSolid[object=octahedron,
 567    a=2,name=my_octahedron,]
 568 \psSolid[object=point,
 569    definition=solidcentreface,
 570    args=my_octahedron 1,
 571    name=G,]
 572 \psSolid[object=point,
 573    definition=mulv3d,
 574    args=G .7,
 575    name=H,]
 576 \psSolid[object=plan,
 577    definition=solidface,
 578    args=my_octahedron 1,
 579    base=-4 4 -4 4,
 580    name=P,](H,,)
 581 \psSolid[object=load,
 582    load=my_octahedron,
 583    plansepare=P,
 584    name=part]
 585 \psset{action=draw**}
 586 \psSolid[object=load,
 587    load=part1,
 588    fcol=0 (YellowOrange),
 589    fillcolor={[rgb]{0.7 1 0.7}},]
 590 \psSolid[object=load,
 591          fillcolor={[rgb]{0.7 1 0.7}},
 592         load=part0](H 2 mulv3d,,)
 593 \composeSolid
 594 \end{pspicture}
 595 \end{LTXexample}
 596
 597 \subsection{Slices of a cube}
 598
 599 \subsubsection{Highlighting the edges of the cut}
 600
 601 \begin{LTXexample}[width=8cm]
 602 \psset{viewpoint=100 30 20 rtp2xyz,Decran=150}
 603 \begin{pspicture}(-4,-3)(4,5)
 604 \psset{solidmemory}
 605 \psSolid[object=plan,definition=normalpoint,
 606    args={1 1 1 [1 1 1]},action=none,name=P]
 607 \psSolid[object=cube,a=2,action=draw,
 608    intersectiontype=0,
 609    intersectionplan=P,
 610    intersectionlinewidth=2,
 611    intersectioncolor=(rouge),
 612 ](1,1,1)
 613 \psProjection[object=point,
 614    args=0 0,fontsize=10,pos=dc,
 615    text=H,phi=-30,plan=P,
 616 ]
 617 \psSolid[object=line,
 618    linestyle=dashed,
 619    args=0 0 0 1 1 1]
 620 \psSolid[object=vecteur,
 621    linecolor=red,
 622    args=1 1 1 .7 mulv3d](1,1,1)
 623 \axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
 624 \end{pspicture}
 625 \end{LTXexample}
 626
 627 \subsubsection{Showing the sliced cube with its hexagonal cut face}
 628
 629 \begin{LTXexample}[width=8cm]
 630 \psset{viewpoint=100 30 20 rtp2xyz,Decran=150}
 631 \begin{pspicture}(-4,-3)(4,5)
 632 \psset{solidmemory}
 633 \psSolid[object=plan,action=none,definition=normalpoint,
 634    args={1 1 1 [1 1 1]},name=P]
 635 \psSolid[object=cube,a=2,
 636    plansepare=P,
 637    action=none,
 638    name=parts_cube,
 639 ](1,1,1)
 640 \psSolid[object=load,
 641          load=parts_cube1,
 642          fcol=0 (Dandelion),
 643          fillcolor={[rgb]{0.7 1 0.7}},
 644 ]
 645 \psProjection[object=point,
 646    args=0 0,fontsize=10,pos=dc,
 647    text=H,phi=-30,plan=P,
 648 ]
 649 \psSolid[object=vecteur,
 650    linecolor=red,
 651    args=1 1 1 .7 mulv3d](1,1,1)
 652 \axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
 653 \end{pspicture}
 654 \end{LTXexample}
 655
 656 \subsubsection{The sliced cube in various positions}
 657
 658 Where we use the option that allows us to memorise a solid, in
 659 order to put the truncated cube, after undergoing various
 660 transformations, down on its cut face.
 661
 662  \begin{verbatim}
 663 \psset{solidmemory}
 664 \psSolid[object=datfile,
 665          fcol=0 (Dandelion),
 666          fillcolor={[rgb]{0.7 1 0.7}},
 667          name=C1,
 668          action=none,
 669          file=./cubeHexagone]
 670  \end{verbatim}
 671
 672
 673 \begin{center}
 674 \begin{pspicture}(-3,-3)(3,3)
 675 \psframe(-3,-2)(3,3)
 676 \psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
 677 %\lightsource
 678 \psset{lightsrc=viewpoint}
 679 \psSolid[object=grille,
 680    base=-2 3 -2 3,
 681    linecolor=gray]
 682 \psSolid[object=datfile,
 683          fcol=0 (Dandelion),
 684          fillcolor={[rgb]{0.7 1 0.7}},
 685          file=./cubeHexagone]
 686 \axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
 687 \end{pspicture}
 688
 689 \hfil
 690
 691 \begin{pspicture}(-2,-3)(4,3)
 692 \psframe(-2,-2)(4,3)
 693 \psset{viewpoint=100 -30 20 rtp2xyz,Decran=100}
 694 %\lightsource
 695 \psset{lightsrc=viewpoint}
 696 \psSolid[object=grille,
 697    base=-2 3 -2 3,
 698    linecolor=gray]
 699 \psSolid[object=datfile,
 700          fcol=0 (Dandelion),
 701          fillcolor={[rgb]{0.7 1 0.7}},
 702          file=./cubeHexagone]
 703 \axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
 704 \end{pspicture}
 705
 706 \begin{pspicture}(-3,-2)(3,3)
 707 \psframe(-3,-2)(3,3)
 708 \psset{viewpoint=100 225 20 rtp2xyz,Decran=100}
 709 %\lightsource
 710 \psset{lightsrc=viewpoint}
 711 \psSolid[object=grille,
 712    base=-2 3 -2 3,
 713    linecolor=gray]
 714 \psSolid[object=datfile,
 715          fcol=0 (Dandelion),
 716          fillcolor={[rgb]{0.7 1 0.7}},
 717          file=./cubeHexagone]
 718 \axesIIID[linecolor=blue](0,0,0)(2.5,2.5,2.5)
 719 \end{pspicture}
 720 \hfil
 721 \begin{pspicture}(-3,-2)(3,3)
 722 \psframe(-3,-2)(3,3)
 723 \psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
 724 %\lightsource
 725 \psset{lightsrc=viewpoint}
 726 \psSolid[object=grille,
 727    base=-2 3 -2 3,
 728    linecolor=gray]
 729 \psset{solidmemory}
 730 \psSolid[object=datfile,
 731          fcol=0 (Dandelion),
 732          fillcolor={[rgb]{0.7 1 0.7}},
 733          name=C1,
 734          action=none,
 735          file=./cubeHexagone]
 736 \codejps{C1 {-1.5 -1.5 0 translatepoint3d} solidtransform
 737 drawsolid**
 738 }
 739 \axesIIID[linecolor=blue](0,0,0)(2.5,2.5,2.5)
 740 \end{pspicture}
 741
 742 \begin{pspicture}(-3,-2)(3,4)
 743 \psframe(-3,-2)(3,3)
 744 \psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
 745 %\lightsource
 746 \psset{lightsrc=viewpoint}
 747 \psSolid[object=grille,
 748    base=-2 3 -2 3,
 749    linecolor=gray]
 750 \psset{solidmemory}
 751 \psSolid[object=datfile,
 752          fcol=0 (Dandelion),
 753          fillcolor={[rgb]{0.7 1 0.7}},
 754          name=C1,
 755          action=none,
 756          file=./cubeHexagone]
 757 \codejps{C1 {-1.5 -1.5 0 translatepoint3d} solidtransform
 758             {0 0 45 rotateOpoint3d} solidtransform
 759 drawsolid**
 760 }
 761 %\composeSolid
 762 \axesIIID[linecolor=blue](0,0,0)(2.5,2.5,2.5)
 763 \end{pspicture}
 764 \hfil
 765 \begin{pspicture}(-3,-2)(3,4)
 766 \psframe(-3,-2)(3,3)
 767 \psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
 768 %\lightsource
 769 \psset{lightsrc=viewpoint}
 770 \psSolid[object=grille,
 771    base=-2 3 -2 3,
 772    linecolor=gray]
 773 \psset{solidmemory}
 774 \psSolid[object=datfile,
 775          fcol=0 (Dandelion),
 776          fillcolor={[rgb]{0.7 1 0.7}},
 777          name=C1,
 778          action=none,
 779          file=./cubeHexagone]
 780 \codejps{C1 {-1.5 -1.5 0 translatepoint3d} solidtransform
 781             {0 0 45 rotateOpoint3d} solidtransform
 782             {-35.2644 -90 add 0 0 rotateOpoint3d} solidtransform
 783 drawsolid*
 784 }
 785 \axesIIID[linecolor=blue](1,2.5,0.5)(2.5,3,2.5)
 786 \end{pspicture}
 787 \end{center}
 788
 789
 790 \subsection{Multiple sections}
 791
 792 \subsubsection{Slicing a sphere with PStricks}
 793
 794 \begin{LTXexample}[width=8cm]
 795 \begin{pspicture}(-4,-4)(4,4)
 796 \psset{viewpoint=100 20 20 rtp2xyz,Decran=75}
 797 \psset{solidmemory,lightsrc=viewpoint}
 798 \codejps{
 799    /coeff 0.75 def /rO 4 def /OH coeff rO mul neg def}%
 800 \psSolid[object=sphere,
 801    r=rO,ngrid=9 18,
 802    plansepare={[1 0 0 OH]},
 803    name=part,
 804    action=none]
 805 \psSolid[object=load,
 806    load=part1,plansepare={[-1 0 0 OH]},action=none,name=part]
 807 \psSolid[object=load,
 808    load=part1,plansepare={[0 1 0 OH]},action=none,name=part]
 809 \psSolid[object=load,
 810    load=part1,plansepare={[0 -1 0 OH]},action=none,name=part]
 811 \psSolid[object=load,
 812    load=part1,plansepare={[0 0 1 OH]},action=none,name=part]
 813 \psSolid[object=load,
 814    load=part1,plansepare={[0 0 -1 OH]},action=none,name=part]
 815 \psSolid[object=load,hue=.1 .8 0.5 1,
 816    load=part1](0,0,0)
 817 \composeSolid
 818 \end{pspicture}
 819 \end{LTXexample}
 820
 821 \subsubsection{Multiple sections of a \Index{parallelepiped}}
 822
 823 Multiple sections are better carried out inside a PostScript loop,
 824 within \verb+\codejps+; it's easier and quicker!
 825
 826 In this example, the original solid is a parallelepiped.
 827 Truncations of the vertices and chamfering of the edges are
 828 effected by means of slicing planes, starting off with the
 829 vertices and finishing with the edges.
 830
 831 \begin{LTXexample}[width=7cm]
 832 \begin{pspicture}(-3.5,-4)(3.5,4)
 833 \psset{viewpoint=100 -20 10 rtp2xyz,Decran=100}
 834 %\lightsource
 835 \psset{lightsrc=viewpoint}
 836 \codejps{
 837 4 4 6 newparallelepiped
 838 45 90 360 {
 839 /iAngle exch def
 840   /n_x iAngle cos 35.2644 cos mul def
 841   /n_y iAngle sin 35.2644 cos mul def
 842   /n_z 35.2644 sin def
 843   /distance 2 3 add 3 sqrt div neg def
 844 [ n_x n_y n_z distance]
 845 solidplansepare
 846 } for
 847 45 90 360 {
 848 /iAngle exch def
 849   /n_x iAngle cos 35.2644 cos mul def
 850   /n_y iAngle sin 35.2644 cos mul def
 851   /n_z 35.2644 sin neg def
 852   /distance 2 3 add 3 sqrt div neg def
 853 [ n_x n_y n_z distance]
 854 solidplansepare
 855 } for
 856 45 90 360 {
 857 /iAngle exch def
 858 % plan : ax+by+cz-d=0
 859 [ iAngle cos % a
 860   iAngle sin % b
 861   0 % c
 862   -2.5 % -d
 863 ] solidplansepare
 864 } for
 865 dup [.5 .2] solidputhuecolors
 866 solidlightOn
 867 drawsolid*}
 868 \end{pspicture}
 869 \end{LTXexample}
 870 \subsection{Sections of a torus}
 871 %\begin{pspicture}(-6,-4)(6,4)
 872 %\psSolid[r1=3,r0=1.5,
 873 %         object=tore,
 874 %%         ngrid=18 60,
 875 %         file=./tore1860,action=writesolid]
 876 %\end{pspicture}
 877
 878 \begin{center}
 879 \begin{pspicture}(-6,-4)(6,4)
 880 \pstVerb{/Ampl 3 2 sqrt mul def}%
 881 \psset{viewpoint=50 -10 30 rtp2xyz,Decran=50}
 882 %\lightsource
 883 \psset{lightsrc=viewpoint}
 884 \psframe[fillstyle=solid,fillcolor=gray!50](-6,-4)(6,4)
 885 %\psset{solidmemory}
 886 %\psSolid[object=datfile,file=./tore1860,
 887 %         plansepare={[1 0 0 -1.5]},
 888 %         name=lemniscate,
 889 %         action=none](0,0,0)
 890 %\psSolid[object=load,
 891 %        load=lemniscate1,
 892 %       file=./tore1860lemniscate,action=writesolid](0,0,0)
 893 \defFunction[algebraic]{lemniscate}(t){1.5}{Ampl*sin(t)/(1+cos(t)*cos(t))}{Ampl*sin(t)*cos(t)/(1+cos(t)*cos(t))}
 894 \psSolid[object=datfile,file=./tore1860lemniscate,
 895         fcol=0 (0.5 0.72 0.5 setrgbcolor)
 896              1 (0.5 0.72 0.5 setrgbcolor),
 897         fillcolor=green!30,
 898         intersectiontype=0,
 899         intersectionplan={
 900         [1 0 0 -1.5]},
 901         intersectioncolor=(rouge),
 902         intersectionlinewidth=2.2]
 903 \psSolid[object=courbe,r=0,linewidth=2pt,
 904         range=0 6.28,
 905         linecolor=red,
 906         function=lemniscate]%
 907 \psSolid[object=plan,action=draw,
 908    definition=equation,
 909    args={[1 0 0 -1.5] 90},
 910    base=-5 5 -2 2,planmarks,showBase]
 911 %\composeSolid
 912 \end{pspicture}
 913 \end{center}
 914 \begin{center}
 915 \begin{pspicture}(-6,-4)(6,4)
 916 \pstVerb{/Ampl 3 2 sqrt mul def}%
 917 \psset{viewpoint=50 -10 30 rtp2xyz,Decran=50}
 918 %\lightsource
 919 \psset{lightsrc=viewpoint}
 920 \psframe[fillstyle=solid,fillcolor=gray!50](-6,-4)(6,4)
 921 \defFunction[algebraic]{lemniscate}(t){1.5}{Ampl*sin(t)/(1+cos(t)*cos(t))}{Ampl*sin(t)*cos(t)/(1+cos(t)*cos(t))}
 922 \psSolid[object=datfile,file=./tore1860lemniscate,
 923         hollow,
 924         rm=0 1,
 925         fillcolor=green!30,incolor=yellow!50]
 926 \psSolid[object=courbe,r=0,linewidth=2pt,
 927         range=0 6.28,
 928         linecolor=red,
 929         function=lemniscate]%
 930 \psSolid[object=plan,action=draw,
 931    definition=equation,
 932    args={[1 0 0 -1.5] 90},
 933    base=-5 5 -2 2,planmarks,showBase]
 934 \end{pspicture}
 935 \end{center}
 936 \subsection{Some more examples}
 937 \begin{enumerate}
 938 \item
 939 You will find a \textit{jps} coded version of this document
 940 within the \verb+\codejps+ command in the following document:
 941
 942 \centerline{\url{http://melusine.eu.org/syracuse/mluque/solides3d2007/sections}}
 943 \item A lesson about conic sections on:
 944
 945 \centerline{\url{http://melusine.eu.org/syracuse/mluque/solides3d2007/sections/sections-cone}}
 946 \item A lesson about cylindrical sections on:
 947
 948 \centerline{\url{http://melusine.eu.org/syracuse/mluque/solides3d2007/sections/section-cylindre}}
 949 \item A lesson about sections of a torus on:
 950
 951 \centerline{\url{http://melusine.eu.org/syracuse/mluque/solides3d2007/sections/section-tore}}
 952 \end{enumerate}
 953
 954
 955 \endinput