Polyhèdres en 3D avec le canvas

 

GS - Le code ci-dessous affiche des polyhèdres en 3d avec le modèle de Lambert (ombrage plat) ou en mode filaire. Il utilise uniquement le canvas. Le modèle d'illumination de Lambert est uniforme [1]. L'intensité de la couleur d'une face est proportionelle à l'angle entre sa normale et la direction d'un rayon lumineux.

- Une version startkit avec plus d'exemples est disponible ici [2]

- Une version tclet est visible ici [3] (sources [4])

L'algorithme de gestion des faces cachées fonctionne bien avec des objets convexes mais est très limité avec les autres types. Voir par exemple le tore ou la navette spatiale comme mauvais exemples.

 # polyhedra.tcl
 # Author:      Gerard Sookahet
 # Date:        30 Mai 2005
 # Description: Rotating polyhedra using a 'standard' tk canvas.
 #              Flat shading and wireframe mode.

 package require Tk 8.4

 bind all <Escape> {exit}

 proc Barycenter {lcoords} {
  set X 0
  set Y 0
  set n [llength $lcoords]

  foreach vtx $lcoords {
     foreach {x y} $vtx {
        set X [expr {$X + $x}]
        set Y [expr {$Y + $y}]
     }
  }
  return [list [expr {$X/$n}] [expr {$Y/$n}]]
 }

 proc CrossProduct {x1 y1 z1 x2 y2 z2} {
  return [list [expr {$y1*$z2 - $y2*$z1}] \
               [expr {$z1*$x2 - $z2*$x1}] \
               [expr {$x1*$y2 - $x2*$y1}]]
 }

 proc DotProduct {x1 y1 z1 x2 y2 z2} {
  return [expr {$x1*$x2 + $y1*$y2 + $z1*$z2}]
 }

 proc MatrixVectorProduct {M V} {
  set x [lindex $V 0]
  set y [lindex $V 1]
  set z [lindex $V 2]
  return [list [expr {[lindex $M 0 0]*$x+[lindex $M 1 0]*$y+[lindex $M 2 0]*$z}] \
               [expr {[lindex $M 0 1]*$x+[lindex $M 1 1]*$y+[lindex $M 2 1]*$z}] \
               [expr {[lindex $M 0 2]*$x+[lindex $M 1 2]*$y+[lindex $M 2 2]*$z}]]
 }

 proc MatrixProduct {M1 M2} {
  set M {{0 0 0 0} {0 0 0 0} {0 0 0 0} {0 0 0 0}}

  for {set i 0} {$i<4} {incr i} {
     for {set j 0} {$j<4} {incr j} {
        lset M $i $j 0
        for {set k 0} {$k<4} {incr k} {
           lset M $i $j [expr {[lindex $M $i $j]+[lindex $M1 $i $k]*[lindex $M2 $k $j]}]
        }
     }
  }
  return $M
 }

 proc MatrixRotation { ax ay az } {
  set sax [expr {sin($ax)}]
  set cax [expr {cos($ax)}]
  set say [expr {sin($ay)}]
  set cay [expr {cos($ay)}]
  set saz [expr {sin($az)}]
  set caz [expr {cos($az)}]
  set Mx {{1 0 0 0} {0 0 0 0} {0 0 0 0} {0 0 0 1}}
  set My {{0 0 0 0} {0 1 0 0} {0 0 0 0} {0 0 0 1}}
  set Mz {{0 0 0 0} {0 0 0 0} {0 0 1 0} {0 0 0 1}}

  # Rotation matrix around X axis with angle ax
  lset Mx 1 1 $cax
  lset Mx 1 2 $sax
  lset Mx 2 1 [expr {-1*$sax}]
  lset Mx 2 2 $cax
  # Rotation matrix around Y axis with angle ay
  lset My 0 0 $cay
  lset My 0 2 [expr {-1*$say}]
  lset My 2 0 $say
  lset My 2 2 $cay
  # Rotation matrix around Z axis with angle az
  lset Mz 0 0 $caz
  lset Mz 0 1 $saz
  lset Mz 1 0 [expr {-1*$saz}]
  lset Mz 1 1 $caz
  return [MatrixProduct [MatrixProduct $Mx $My] $Mz]
 }

 # Compute normal vector and norm for each face
 # -------------------------------------------------------------------
 proc NormalVector {lvtx lcnx} {
  set lnv {}
  set lmv {}

  foreach face $lcnx {
     foreach {nx ny nz} [CrossProduct \
     [expr {[lindex $lvtx [lindex $face 1] 0] - [lindex $lvtx [lindex $face 0] 0]}] \
     [expr {[lindex $lvtx [lindex $face 1] 1] - [lindex $lvtx [lindex $face 0] 1]}] \
     [expr {[lindex $lvtx [lindex $face 1] 2] - [lindex $lvtx [lindex $face 0] 2]}] \
     [expr {[lindex $lvtx [lindex $face 2] 0] - [lindex $lvtx [lindex $face 1] 0]}] \
     [expr {[lindex $lvtx [lindex $face 2] 1] - [lindex $lvtx [lindex $face 1] 1]}] \
     [expr {[lindex $lvtx [lindex $face 2] 2] - [lindex $lvtx [lindex $face 1] 2]}]] {}
     lappend lnv [list $nx $ny $nz]
     lappend lmv [DotProduct $nx $ny $nz $nx $ny $nz]
  }
  return [list $lnv $lmv]
 }

 # 2D projection
 # -------------------------------------------------------------------
 proc Projection {x y z M} {
     global scx scy vdist

   set nx [expr  {[lindex $M 0 0]*$x+[lindex $M 1 0]*$y+[lindex $M 2 0]*$z}]
   set ny [expr  {[lindex $M 0 1]*$x+[lindex $M 1 1]*$y+[lindex $M 2 1]*$z}]
   set nz [expr {([lindex $M 0 2]*$x+[lindex $M 1 2]*$y+[lindex $M 2 2]*$z+10)/$vdist}]
   return [list [expr {$nx/$nz+$scx/2.0}] [expr {$ny/$nz+$scy/2.0}]]
 }

 # Apply transformations to vertex coordinates
 # -------------------------------------------------------------------
 proc Transformations {lvtx lnv} {
     global ax ay az

  update
  set lnew {}
  set lvn {}
  # Compute matrix rotation
  set M [MatrixRotation $ax $ay $az]
  set i 0
  # Apply projection
  foreach vtx $lvtx {
     lappend lnew [Projection [lindex $vtx 0] [lindex $vtx 1] [lindex $vtx 2] $M]
     incr i
  }
  # Normal vector rotation
  foreach v $lnv {lappend lvn [MatrixVectorProduct $M $v]}

  return [list $M $lnew $lvn]
 }

 # Compute color entensity for each face
 # -------------------------------------------------------------------
 proc Intensity {lnv lmv lvv} {
  set lclr {}
  set v [DotProduct [lindex $lvv 0] [lindex $lvv 1] [lindex $lvv 2] \
                    [lindex $lvv 0] [lindex $lvv 1] [lindex $lvv 2]]
  set i 0
  foreach nv $lnv {
     set clr 31
     set a [DotProduct [lindex $nv 0]  [lindex $nv 1]  [lindex $nv 2] \
                       [lindex $lvv 0] [lindex $lvv 1] [lindex $lvv 2]]
     set b [expr {sqrt([lindex $lmv $i]*$v)}]
     set clr [expr {round(31*($a/$b))}]
     lappend lclr [expr {$clr < 0 ? 31 : [expr {32 - $clr}]}]
     incr i
  }
  return $lclr
 }

 # Start the display and rotation loop
 # -------------------------------------------------------------------
 proc DisplayModel {w s} {
     global stop
     global display
     global ax ay az tx ty tz

  $w.c delete all
  set stop 0
  set ax 0.2
  set ay 0.1
  set az 0.3
  set tx 0
  set ty 0
  set tz 0

  set d $display
  foreach {t lvtx lcnx lclr} [ReadData $s] {}
  $w.c create text 10 10 -anchor w -fill white -text $t
  foreach {lnv lmv} [NormalVector $lvtx $lcnx] {}

  set lpoly [DisplayInit $w $d $lcnx $lclr]

  if {$d == "shaded"} then {
   for {set i 1} {$i<=820} {incr i} {
      if $stop break
      set ax [expr {$ax-0.02}]
      set az [expr {$az+0.02}]
      set ay [expr {$ay+0.025}]
      after 40
      DisplayShaded $w $lpoly $lvtx $lcnx $lnv $lmv
   }
  } else {
   for {set i 1} {$i<=820} {incr i} {
      if $stop break
      set ax [expr {$ax-0.02}]
      set az [expr {$az+0.02}]
      set ay [expr {$ay+0.025}]
      after 40
      Display $w $lpoly $lvtx $lcnx $lnv $lmv
   }
  }
 }

 # Data structure for models with vertices and connectivity
 # -------------------------------------------------------------------
 proc ReadData { n } {

     set lvtx {}
     set lcnx {}
     set lclr {}
     set txt ""

  switch $n {
  tetrahedron {
    set txt "tetrahedron: 4 faces 4 vertices 5 edges"
    set a [expr {1.0/sqrt(3.0)}]
    set lvtx [list [list $a $a $a] [list $a -$a -$a] \
                   [list -$a $a -$a] [list -$a -$a $a]]
    set lcnx {{0 3 1} {2 0 1} {3 0 2} {1 3 2}}
  }

  cube {
    set txt "cube: 6 faces 8 vertices 12 edges"
    set lvtx {{0.7 0.7 0.7} {-0.7 0.7 0.7} {-0.7 -0.7 0.7} {0.7 -0.7 0.7}
              {0.7 0.7 -0.7} {-0.7 0.7 -0.7} {-0.7 -0.7 -0.7} {0.7 -0.7 -0.7}}
    set lcnx {{4 7 6 5} {0 1 2 3} {3 2 6 7} {4 5 1 0} {0 3 7 4} {5 6 2 1}}
  }

  octahedron {
    set txt "octahedron 8 faces  6 vertices 16 edges"
    set lvtx {{1 0 0} {0 1 0} {-1 0 0} {0 -1 0} {0 0 1} {0 0 -1}}
    set lcnx {{0 1 4} {1 2 4} {2 3 4} {3 0 4}
              {1 0 5} {2 1 5} {3 2 5} {0 3 5}}
  }

  dodecahedron {
    set txt "dodecahedron 12 faces 20 vertices 30 edges"
    set s3 [expr sqrt(3)]
    set s5 [expr sqrt(5)]
    set alpha [expr {sqrt(2.0/(3 + $s5))/$s3}]
    set beta [expr {(1.0 + sqrt(6.0/(3 + $s5) - 2 + 2*sqrt(2.0/(3.0 + $s5))))/$s3}]
    set gamma [expr {1.0/$s3}]
    set lvtx [list \
 	[list -$alpha 0 $beta] \
 	[list $alpha 0 $beta] \
 	[list -$gamma -$gamma -$gamma] \
 	[list -$gamma -$gamma $gamma] \
 	[list -$gamma $gamma -$gamma] \
 	[list -$gamma $gamma $gamma] \
 	[list $gamma -$gamma -$gamma] \
 	[list $gamma -$gamma $gamma] \
 	[list $gamma $gamma -$gamma] \
 	[list $gamma $gamma $gamma] \
 	[list $beta $alpha 0] \
 	[list $beta -$alpha 0] \
 	[list -$beta $alpha  0] \
 	[list -$beta -$alpha 0] \
 	[list -$alpha 0 -$beta] \
 	[list $alpha 0 -$beta] \
 	[list 0 $beta $alpha] \
 	[list 0 $beta -$alpha] \
 	[list 0 -$beta $alpha] \
 	[list 0 -$beta -$alpha]]
     set lcnx {{0 1 9 16 5} {1 0 3 18 7} {1 7 11 10 9} {11 7 18 19 6}
               {8 17 16 9 10} {2 14 15 6 19} {2 13 12 4 14} {2 19 18 3 13}
               {3 0 5 12 13} {6 15 8 10 11} {4 17 8 15 14} {4 12 5 16 17}}
  }

  icosahedron {
    set txt "icosahedron: 20 faces 12 vertices 30 edges"
    set X 0.525731112119133606
    set Z 0.850650808352039932
    set lvtx [list [list -$X 0.0 $Z] [list $X 0.0 $Z] [list -$X 0.0 -$Z] \
                   [list $X 0.0 -$Z] [list 0.0 $Z $X] [list 0.0 $Z -$X] \
                   [list 0.0 -$Z $X] [list 0.0 -$Z -$X] [list $Z $X 0.0] \
                   [list -$Z $X 0.0] [list $Z -$X 0.0] [list -$Z -$X 0.0]]
    set lcnx {{4 0 1} {9 0 4} {5 9 4} {5 4 8}
              {8 4 1} {10 8 1} {3 8 10} {3 5 8}
              {2 5 3} {7 2 3} {10 7 3} {6 7 10}
              {11 7 6} {0 11 6} {1 0 6} {1 6 10}
              {0 9 11} {11 9 2} {2 9 5} {2 7 11}}
  }
  }

  for {set i 0} {$i <= [llength $lcnx]} {incr i} {
     lappend lclr "0000[format %2.2x 255]"
  }
  return [list $txt $lvtx $lcnx $lclr]
 }

 # Initialization of canvas with polygonal objects filled or not
 # -------------------------------------------------------------------
 proc DisplayInit {w d lcnx lclr} {
  set lpoly {}
  set i 0
  if {$d == "shaded"} then {
    foreach cnx $lcnx {
       lappend lpoly [$w.c create polygon \
                     [string repeat " 0" [expr {2*[llength $cnx]}]] \
                     -fill "#[lindex $lclr $i]"]
       incr i
    }
  } else {
    foreach cnx $lcnx {
       lappend lpoly [$w.c create polygon \
                     [string repeat " 0" [expr {2*[llength $cnx]}]] \
                     -fill black -outline blue]
    }
  }
  return $lpoly
 }

 # Flat shaded display with gradient color
 # -------------------------------------------------------------------
 proc DisplayShaded {w lpoly lvtx lcnx lnv lmv} {
  update

  set lgradB {}
  foreach {M lnew lvn} [Transformations $lvtx $lnv] {}
  # Light vector is set to <1 1 -1>
  foreach i [Intensity $lvn $lmv [list 1 1 -1]] {
     lappend lgradB [format %2.2x [expr {100+154*$i/32}]]
  }
  set i 0
  foreach cnx $lcnx {
     set lcoords {}
     foreach j $cnx {lappend lcoords [lindex $lnew $j]}
     # Backface culing for hidden face. Not removed but only reduced to a point
     if {[lindex $lvn $i 2] < 0} {
       eval $w.c coords [lindex $lpoly $i] [join $lcoords]
       $w.c itemconfigure [lindex $lpoly $i] -fill "#0000[lindex $lgradB $i]"
     } else {
       $w.c coords [lindex $lpoly $i] [string repeat " [join [Barycenter $lcoords]]" [llength $cnx]]
     }
     incr i
  }
 }

 # Wireframe display
 # -------------------------------------------------------------------
 proc Display {w lpoly lvtx lcnx lnv lmv} {
  update

  foreach {M lnew lvn} [Transformations $lvtx $lnv] {}
  set i 0
  foreach cnx $lcnx {
     set lcoords {}
     foreach j $cnx {lappend lcoords [lindex $lnew $j]}
     # Backface culing for hidden face. Not removed but only reduced to a point
     if {[lindex $lvn $i 2] < 0} {
       eval $w.c coords [lindex $lpoly $i] [join $lcoords]
     } else {
       $w.c coords [lindex $lpoly $i] [string repeat " [join [Barycenter $lcoords]]" [llength $cnx]]
     }
     incr i
  }
 }

 # -------------------------------------------------------------------
 proc Main {} {
     global stop
     global display
     global scx scy vdist

  set w .tdc
  catch {destroy $w}
  toplevel $w
  wm withdraw .
  wm title $w "Rotating polyhedra in Tk canvas "

  set display shaded
  set scx 420
  set scy 420
  set vdist 1200

  pack [canvas $w.c -width $scx -height $scy -bg white -bg black -bd 0]
  $w.c delete all

  set f1 [frame $w.f1 -relief sunken -borderwidth 2]
  pack $f1 -fill x
  button $f1.brun -text Stop -command {set stop 1}
  button $f1.bq -text Quit -command exit
  label $f1.l1 -text "  "
  radiobutton $f1.rbs -text "Shaded" -variable display -value shaded
  radiobutton $f1.rbw -text "Wireframe" -variable display -value wireframe
  eval pack [winfo children $f1] -side left
  set f2 [frame $w.f2 -relief sunken -borderwidth 2]
  pack $f2 -fill x
  foreach i {tetrahedron cube octahedron dodecahedron icosahedron} {
     button $f2.b$i -text $i -command "DisplayModel $w $i"
  }
  eval pack [winfo children $f2] -side left
  set f3 [frame $w.f3 -relief sunken -borderwidth 2]
  pack $f3 -fill x
  label $f3.l1 -text "View distance "
  scale $f3.sca -from 300 -to 1600 -length 300 \
                -orient horiz -bd 1 -showvalue true -variable vdist
  eval pack [winfo children $f3] -side left
 }

 Main

dc Impressionant ! Je ne pensais pas qu'en si peu de lignes, on pouvait faire cela, voire même qu'on pouvait faire cela avec Tk seul. Encore une fois : Bravo !

J'ai néanmoins parfois des problèmes de faces cachées avec le tore notamment.

GS C'est normal car il s'agit d'objets non-convexes. Il faudrait dès lors utiliser des algorithmes plus subtils (Z-buffer) mais aussi plus gourmands car ils travaillent au niveau du pixel. Merci pour les compliments. La Force est avec Tcl-Tk ;-)


ulis du très beau travail !


dc dans la page http://gersoo.frJL