
=head1 Draw 


Draw 3d scene as 2d image with lighting and shadowing to assist the human
observer in reconstructing the original 3d scene.

PhilipRBrenan@yahoo.com, 2004, Perl License


=head2 Synopsis


Example t/draw.t      

 #!perl -w
 #______________________________________________________________________
 # Test drawing.
 # philiprbrenan@yahoo.com, 2004, Perl License    
 #______________________________________________________________________
 
 use Math::Zap::Draw;
 use Math::Zap::Cube unit=>'cu';
 use Math::Zap::Triangle;
 use Math::Zap::Vector;
 
 use Test::Simple tests=>1;
 
 #_ Draw _______________________________________________________________
 # Draw this set of objects.
 #______________________________________________________________________
 
 $l = 
 draw 
   ->from    (vector( 10,   10,  10))
   ->to      (vector(  0,    0,   0))
   ->horizon (vector(  1,  0.5,   0))
   ->light   (vector( 20,   30, -20))
 
     ->object(triangle(vector( 0,  0,  0), vector( 8,  0,  0), vector( 0,  8,  0)),                         'red')
     ->object(triangle(vector( 0,  0,  0), vector( 0,  0,  8), vector( 0,  8,  0)),                         'green')
     ->object(triangle(vector( 0,  0,  0), vector(12,  0,  0), vector( 0,  0, 12)) - vector(2.5,  0,  2.5), 'blue')
     ->object(triangle(vector( 0,  0,  0), vector( 8,  0,  0), vector( 0, -8,  0)),                         'pink')
     ->object(triangle(vector( 0,  0,  0), vector( 0,  0,  8), vector( 0, -8,  0)),                         'orange')
     ->object(cu()*2+vector(3,5,1), 'lightblue')
 
 ->print;
 
 $L = <<'END'; 
 #!perl -w
 use Math::Zap::Draw;
 use Math::Zap::Triangle;
 use Math::Zap::Vector;
 
 draw
 ->from    (vector(10, 10, 10))
 ->to      (vector(0, 0, 0))
 ->horizon (vector(1, 0.5, 0))
 ->light   (vector(20, 30, -20))
   ->object(triangle(vector(0, 0, 0), vector(8, 0, 0), vector(0, 8, 0)), 'red')
   ->object(triangle(vector(0, 0, 0), vector(0, 0, 8), vector(0, 8, 0)), 'green')
   ->object(triangle(vector(-2.5, 0, -2.5), vector(9.5, 0, -2.5), vector(-2.5, 0, 9.5)), 'blue')
   ->object(triangle(vector(0, 0, 0), vector(8, 0, 0), vector(0, -8, 0)), 'pink')
   ->object(triangle(vector(0, 0, 0), vector(0, 0, 8), vector(0, -8, 0)), 'orange')
   ->object(triangle(vector(3, 5, 1), vector(5, 5, 1), vector(3, 7, 1)), 'lightblue')
   ->object(triangle(vector(5, 7, 1), vector(5, 5, 1), vector(3, 7, 1)), 'lightblue')
   ->object(triangle(vector(3, 5, 3), vector(5, 5, 3), vector(3, 7, 3)), 'lightblue')
   ->object(triangle(vector(5, 7, 3), vector(5, 5, 3), vector(3, 7, 3)), 'lightblue')
   ->object(triangle(vector(3, 5, 1), vector(3, 7, 1), vector(3, 5, 3)), 'lightblue')
   ->object(triangle(vector(3, 7, 3), vector(3, 7, 1), vector(3, 5, 3)), 'lightblue')
   ->object(triangle(vector(5, 5, 1), vector(5, 7, 1), vector(5, 5, 3)), 'lightblue')
   ->object(triangle(vector(5, 7, 3), vector(5, 7, 1), vector(5, 5, 3)), 'lightblue')
   ->object(triangle(vector(3, 5, 1), vector(3, 5, 3), vector(5, 5, 1)), 'lightblue')
   ->object(triangle(vector(5, 5, 3), vector(3, 5, 3), vector(5, 5, 1)), 'lightblue')
   ->object(triangle(vector(3, 7, 1), vector(3, 7, 3), vector(5, 7, 1)), 'lightblue')
   ->object(triangle(vector(5, 7, 3), vector(3, 7, 3), vector(5, 7, 1)), 'lightblue')
 ->done;
 END
 
 ok($l eq $L);



=head2 Description


This package supplies methods to draw a scene, containing three dimensional
objects, as a two dimensional image, using lighting and shadowing to assist the
human observer in reconstructing the original three dimensional scene.

There are many existing packages to perform this important task: this
package Math::Zap::Is the only one to make the attempt in Pure Perl. Pending the
$VERSION=1.03;
power of Petaflop Parallel Perl (when we will be set free from C), this
approach is slow. However, it is not so slow as to be completely useless
for simple scenes as might be encountered inside, say for instance, beam
lines used in high energy particle physics, the owners of which often
have large Perl computers.

The key advantage of this package is that is open: you can manipulate
both the objects to be drawn and the drawing itself all in Pure Perl.

 
 package Math::Zap::Draw;
 $VERSION=1.03;
 use Math::Zap::Vector check=>'vectorCheck';
 use Math::Zap::Vector2;
 use Math::Zap::Triangle2  newnnc=>'triangle2Newnnc';
 use Math::Zap::Triangle;
 use Math::Zap::Color;
 use Tk;
 use Carp;
 
 use constant debug=>0;
 


=head2 Constructors



=head3 draw


Constructor

 
 sub draw() {bless {}}
 


=head2 Methods



=head3 from 


Set view point 

 
 sub from($$)
  {my ($d) =         check(@_[0..0]); # Drawing 
   my ($v) = vectorCheck(@_[1..1]); # Vector
 
   $d->{from} = $v;
   $d;
  }
 


=head3 to


Viewing this point

 
 sub to($$)
  {my ($d) =         check(@_[0..0]); # Drawing 
   my ($v) = vectorCheck(@_[1..1]); # Vector
 
   $d->{to} = $v;
   $d;
  }
 


=head3 Horizon


Sets the direction of the horizon.

 
 sub horizon($$)
  {my ($d) =         check(@_[0..0]); # Drawing 
   my ($v) = vectorCheck(@_[1..1]); # Vector
 
   $d->{horizon} = $v;
   $d;
  }
 


=head3 light


Light source position

 
 sub light($$)
  {my ($d) =         check(@_[0..0]); # Drawing 
   my ($v) = vectorCheck(@_[1..1]); # Vector
 
   $d->{light} = $v;
   $d;
  }
 


=head3 withControls


Display a window allowing the user to set to,from,horizon,light

 
 sub withControls($)
  {my ($d) =         check(@_[0..0]); # Drawing 
 
   $d->{withControls} = 1;
   $d;
  }


=head3 object


Draw this object

 
 sub object($$$)
  {my ($d) = check(@_[0..0]); # Drawing 
   my ($o) =       @_[1..1];  # Object to be drawn
   my ($c) =       @_[2..2];  # Color of object's surfaces
 
   if ($o->can('triangulate'))
    {push @{$d->{triangles}}, $o->triangulate($c);
    }
   else
    {die "Cannot draw $o";
    }
   $d;
  }
 


=head3 done 


Draw the complete object list

 
 sub done($)
  {my ($d) = check(@_[0..0]); # Drawing 
   &fission($d);
   &new($d);
  }
 


=head2 Methods



=head3 print


Print the complete object list as a triangles in a reusable manner.

 
 sub print($)
  {my ($d) = check(@_[0..0]); # Drawing
   my $l = << 'END';
 #!perl -w
 use Math::Zap::Draw;
 use Math::Zap::Triangle;
 use Math::Zap::Vector;
 
 draw
 END
   $l .= '->from    ('. $d->{from}   ->print .")\n";
   $l .= '->to      ('. $d->{to}     ->print .")\n";
   $l .= '->horizon ('. $d->{horizon}->print .")\n";
   $l .= '->light   ('. $d->{light}  ->print .")\n";
 
   for my $p(@{$d->{triangles}}) # Triangulation
    {$l .= '  ->object('. $p->{triangle}->print .', \''. $p->{color}. "\')\n";
    }
   $l .= "->done;\n";
  }
 


=head3 check


Check its a drawing 

 
 sub check(@)
  {if (debug)
    {for my $t(@_)
      {confess "$t is not a drawing" unless ref($t) eq __PACKAGE__;
      }
    } 
   return (@_)
  }
 


=head3 is


Test its a drawing 

 
 sub is(@)
  {for my $t(@_)
    {return 0 unless ref($t) eq __PACKAGE__;
    }
   'draw';
  }
 


=head3 showFissionFragments


Show fission fragments: the objects to be drawn are triangulated
where-ever they may intersect.  It is useful to see these sub triangles
when debugging.  See also L</fission>.

 
 sub showFissionFragments($)
  {my ($d) =         check(@_[0..0]); # Drawing 
 
   $d->{showFissionFragments} = 1;
   $d;
  }
 


=head3 Fission


Fission the triangles that intersect.  See L</showFissionFragments>

 
 sub fission($)
  {my ($d) = check(@_[0..0]);    # Drawing 
   my @P   = @{$d->{triangles}}; # Triangles to be fissoned
   my $tested;                   # Source triangles already tested
 
 #_ Draw ________________________________________________________________
 # Check each pair of triangles
 #_______________________________________________________________________
 
   L: for(;;)
    {for   (my $i = 0; $i < scalar(@P); ++$i)
      {my $p = $P[$i];
       next unless defined($p);
 
 #_ Draw ________________________________________________________________
 # Check against triangle 
 #_______________________________________________________________________
 
       for (my $j = $i+1; $j < scalar(@P); ++$j)
        {my $q = $P[$j];
         next unless defined($q);
         my ($t, @t, @T);
 
 #_ Draw ________________________________________________________________
 # Already tested
 #_______________________________________________________________________
 
         next if $tested->{$p->{plane}}{$q->{plane}};
         $tested->{$p->{plane}}{$q->{plane}} = 1;
         $tested->{$q->{plane}}{$p->{plane}} = 1;
         next if $p->{triangle}->parallel($q->{triangle});
 
 #_ Draw ________________________________________________________________
 # Divide intersecting triangles
 #_______________________________________________________________________
 
         @t = $p->{triangle}->divide($q->{triangle});
         @T = $q->{triangle}->divide($p->{triangle});
 
 #_ Draw ________________________________________________________________
 # Add divisions to list of triangles
 #_______________________________________________________________________
 
         next unless @t > 1 or @T > 1;
         delete $P[$i];
         delete $P[$j];
 
         push @P, {triangle=>$_, color=>$q->{color}, plane=>$q->{plane}} for(@t);
         push @P, {triangle=>$_, color=>$p->{color}, plane=>$p->{plane}} for(@T);
         next L;
        }
      }
     last;
    }
 
 #_ Draw ________________________________________________________________
 # Update list of triangles to be drawn
 #_______________________________________________________________________
 
   my @p;
   for my $p(@P)
    {push @p, $p if defined($p);
    } 
   $d->{triangles} = [@p];
  }
 


=head3 new 


New drawing - not a constructor

 
 sub new($)
  {my ($d) = check(@_[0..0]); # Drawing 
   &newCanvas ($d);
   &newControl($d);
   &drawing   ($d, 1);
   MainLoop;
  } 
 


=head3 newCanvas


Canvas for drawing

 
 sub newCanvas($)
  {my ($d) = check(@_[0..0]); # Drawing 
   my $m = $d->{MainWindowCanvas}

        = new MainWindow;
  my $c = $d->{canvas}
        = $m->Canvas(-background=>'yellow')->pack(-expand=>1, -fill=>'both');

  $d->{canvas}{width}  = $c->cget(-width=>);
  $d->{canvas}{height} = $c->cget(-height=>);

  $c->CanvasBind('<Configure>' => [$d=>'configure', Ev('w'), Ev('h')]);
 }
 

=head3 newControl


Controls for drawing

 
 sub newControl()
  {my ($d) = check(@_[0..0]);                   # Drawing 
   my  $m  = $d->{MainWindowControls} = new MainWindow;
 
   my $a11 = $d->{a11} = $m->Label(-text=>'View point');
   my $a12 = $d->{a12} = $m->Entry(-textvariable=>\$d->{from}->{x});
   my $a13 = $d->{a13} = $m->Entry(-textvariable=>\$d->{from}->{y});
   my $a14 = $d->{a14} = $m->Entry(-textvariable=>\$d->{from}->{z});
   my $a21 = $d->{a21} = $m->Label(-text=>'Looking to');
   my $a22 = $d->{a22} = $m->Entry(-textvariable=>\$d->{to}->{x});
   my $a23 = $d->{a23} = $m->Entry(-textvariable=>\$d->{to}->{y});
   my $a24 = $d->{a24} = $m->Entry(-textvariable=>\$d->{to}->{z});
   my $a31 = $d->{a31} = $m->Label(-text=>'Horizontal');
   my $a32 = $d->{a32} = $m->Entry(-textvariable=>\$d->{horizon}->{x});
   my $a33 = $d->{a33} = $m->Entry(-textvariable=>\$d->{horizon}->{y});
   my $a34 = $d->{a34} = $m->Entry(-textvariable=>\$d->{horizon}->{z});
   my $a41 = $d->{a41} = $m->Label(-text=>'Lit from');
   my $a42 = $d->{a42} = $m->Entry(-textvariable=>\$d->{light}->{x});
   my $a43 = $d->{a43} = $m->Entry(-textvariable=>\$d->{light}->{y});
   my $a44 = $d->{a44} = $m->Entry(-textvariable=>\$d->{light}->{z});
   my $a51 = $d->{a51} = $m->Button(-text=>'Redraw', -command=>sub{&drawing($d, 1)});
   my $a52 = $d->{a52} = $m->Button(-text=>'In');
   my $a53 = $d->{a53} = $m->Button(-text=>'Out');
   my $a54 = $d->{a54} = $m->Button(-text=>'Quit',   -command=>sub{exit(0)});
 
   $a11->grid($a12, $a13, $a14);
   $a21->grid($a22, $a23, $a24);
   $a31->grid($a32, $a33, $a34);
   $a41->grid($a42, $a43, $a44);
   $a51->grid($a52, $a53, $a54);
  }
 


=head3 Configure


Configuration of canvas has been changed

 
 sub configure
  {my ($d)    = check(@_[0..0]);                # Drawing 
   my $c      = $d->{canvas};
   $d->{canvas}{width}  = $_[1];
   $d->{canvas}{height} = $_[2];
   &drawing($d, 0);
  } 
 


=head3 drawing


New drawing of objects

 
 sub drawing($$)
  {my ($d)    = check(@_[0..0]);                # Drawing 
   my $zorder = shift;                          # Re-sort of zorder required?
 
 #_ Draw ________________________________________________________________
 # Locate background
 #_______________________________________________________________________
 
   my $from   = $d->{from};                     # View point
   my $lt     = $d->{light};                    # Light     
   my $to     = $d->{to};                       # View towards
   my $hz     = $d->{horizon};                  # Horizon  
   
   my $v = (($from-$to) x $hz)->norm;           # Vertical   in background plane
   my $h = ($v  x ($from-$to))->norm;           # Horizontal in background plane
   my $B = triangle($to, $to+$h, $to+$v);       # Background plane
   $d->{background} = $B;
 
   &zorder($d) if $zorder;                      # Partially order triangles from view point 
   $d->{canvas}->delete('all');                 # Clear canvas
 
 #_ Draw ________________________________________________________________
 # Dimensions of projected image
 #_______________________________________________________________________
 
   my ($mx, $Mx, $my, $My);
   for my $D(@{$d->{triangles}})
    {my $t = $B->project($D->{triangle}, $from); # Project onto background
     $D->{project} = $t;                         # Optimization - record for reuse
 
     my ($ax, $ay) = ($t->a->x, $t->a->y);
     my ($bx, $by) = ($t->b->x, $t->b->y);
     my ($cx, $cy) = ($t->c->x, $t->c->y);
 
     $mx = $ax if !defined($mx) or $mx > $ax;   
     $mx = $bx if !defined($mx) or $mx > $bx;   
     $mx = $cx if !defined($mx) or $mx > $cx;   
     $Mx = $ax if !defined($Mx) or $Mx < $ax;   
     $Mx = $bx if !defined($Mx) or $Mx < $bx;   
     $Mx = $cx if !defined($Mx) or $Mx < $cx;   
 
     $my = $ay if !defined($my) or $my > $ay;   
     $my = $by if !defined($my) or $my > $by;   
     $my = $cy if !defined($my) or $my > $cy;   
     $My = $ay if !defined($My) or $My < $ay;   
     $My = $by if !defined($My) or $My < $by;   
     $My = $cy if !defined($My) or $My < $cy;
    }
 
   my $cw = $d->{canvas}{width};
   my $ch = $d->{canvas}{height};
 
   my $sx = int($d->{canvas}{width} /($Mx-$mx));  
   my $sy = int($d->{canvas}{height}/($My-$my));
   my $s  = $d->{canvas}{scale} = ($sx < $sy ? $sx : $sy);
 
   my $dx = $d->{canvas}{dx} = -$mx * $s + ($cw - $s * ($Mx-$mx)) / 2;
   my $dy = $d->{canvas}{dy} =  $My * $s + ($ch - $s * ($My-$my)) / 2;
 
 #_ Draw ________________________________________________________________
 # Draw each triangle
 #_______________________________________________________________________
 
   for my $D(@{$d->{triangles}})
    {my $T     = $D->{triangle};
     my $color = $D->{color};
     my $p     = $D->{plane};
     my $t     = $D->{project};
 
 # Coordinates of triangle to be drawn
     my @a = ($dx+$t->a->x*$s, $dy-$t->a->y*$s, 
              $dx+$t->b->x*$s, $dy-$t->b->y*$s,
              $dx+$t->c->x*$s, $dy-$t->c->y*$s,
              );
     push @a,  -outline=>'black' if defined($d->{showFissionFragments});
 
 #_ Draw ________________________________________________________________
 # Side towards/away from the light
 #_______________________________________________________________________
 
     my $fb = $T->frontInBehindZ($from, $lt);   
 
     if (!defined($fb) or $fb < 0)              # Towards light              
      {push @a, -fill=>$color;
       $d->{canvas}->createPolygon(@a);
       &shadows($d, $D);
      }
     else                                       # Away from light
      {$d->{canvas}->createPolygon(@a, -fill=>color($color)->dark);
      }                        
    }
  }
 


=head3 shadows


Shadows from a point of illumination

 
 sub shadows($$)
  {my ($d)   = check(@_[0..0]);                           # Drawing 
   my ($p)   =      (@_[1..1]);                           # Current triangle to be drawn
   my $from  = $d->{from};                                # View point       
   my $to    = $d->{to};                                  # Look towards     
   my $light = $d->{light};                               # Position of light
   my $back  = $d->{background};                          # Background
   my $c     = $d->{canvas};                              # Canvas
   my $dx    = $d->{canvas}{dx};                          # Canvas center x
   my $dy    = $d->{canvas}{dy};                          # Canvas center y
   my $s     = $d->{canvas}{scale};                       # Scale factor
 
 #_ Draw ________________________________________________________________
 # Shadow each triangle
 #_______________________________________________________________________
 
   my @s;
   for my $q(@{$d->{triangles}})                                
    {next if $p == $q;                                    # Do not shadow self
     next if $p->{plane} == $q->{plane};                  # Do not shadow stuff in same plane
     my $t = $p->{triangle};                              # Shadowed  triangle
     my $T = $q->{triangle};                              # Shadowing triangle
 #   next if $t->frontInBehindZ($from, $light) > 0;       # Check that plane view point and light
 
     my $b = $t->project($T, $light);                     # Project Shadowing triangle onto shadowed triangle
     my $d = triangle2Newnnc                            # Shadow in shadowed plane coordinates
      (vector2($b->a->x, $b->a->y),
       vector2($b->b->x, $b->b->y),
       vector2($b->c->x, $b->c->y)
      );
     my $D = triangle2Newnnc                            # Shadowed plane 
      (vector2(0,0),
       vector2(1,0),
       vector2(0,1)
      );
     return if $d->narrow();                              # Projected shadow  too narrow?
     return if $D->narrow();                              # Shadowed triangle too narrow?
 
     my @r = $d->ring($D);                                # Ring of common points 
     if (scalar(@r) > 2)                                  # Less than two - small intersection
      {my @a;
       for my $r(@r)                                      # Points of intersection current/shadowing triangle
        {my $sr = $t->convertPlaneToSpace($r);            # Convert intersection to space coords
         last if $T->frontInBehind($light, $sr) == 1;     # $t gives back of shadowing plane
         my $sb = $back->intersectionInPlane($from, $sr); # Project from view point onto background
         push @a, $dx+$sb->x*$s, $dy-$sb->y*$s;           # Save coordinates
        }
 
 #_ Draw ________________________________________________________________
 # Draw shadow
 #_______________________________________________________________________
 
       push @a, -outline=>color($p->{color})->dark, -fill=>color($p->{color})->dark;
       $c->createPolygon(@a);
      }
    } 
  }
 


=head4 zorder


Z-order: order the fission triangles from the back ground to the point
of view:

Compare each triangle with every other, recording for each triangle
which triangles are behind it.

Place all triangles with no triangles behind them with at the start of
the order.

Reprocess the remainder until none left (success) or a cycle is detected
(bad algorithm).

The two triangles to be compared are projected on to the background: if
their projections have no points in common they are unordered, otherwise
use the distance to each triangle from the view point towards the common
point as a measure of which is first.

fission() guarantees that no two triangles intersect, this algorithm
should correctly order each pair of triangles.

 
 sub zorder($)
  {my ($d) = check(@_[0..0]);       # Drawing
 
   my $from = $d->{from};           # View point
   my $back = $d->{background};     # Background
   my @P    = @{$d->{triangles}};   # Triangles to be drawn 
 
 #_ Draw ________________________________________________________________
 # Filter for useful triangles
 #_______________________________________________________________________
 
   my @o;
   for(my $ip = 0; $ip < @P; ++$ip)
    {my $t = $P[$ip]{triangle};
 #   next unless $t->area > .1;     # Ignore small triangles
 #   next if $t->narrow(0);
 
     $o{$ip} = {};
     push @o, $ip;
    }
 
 #_ Draw ________________________________________________________________
 # Relationship
 #_______________________________________________________________________
 
   for my $ip(@o)
    {my $t = $P[$ip]{triangle};
 
     for my $jp(@o)
      {next unless $ip < $jp;
       my $T = $P[$jp]{triangle};
       my $i = $back->project($t, $from);
       my $I = $back->project($T, $from);
 
       my $i2 = triangle2Newnnc(vector2($i->a->x, $i->a->y), vector2($i->b->x, $i->b->y), vector2($i->c->x, $i->c->y));
       my $I2 = triangle2Newnnc(vector2($I->a->x, $I->a->y), vector2($I->b->x, $I->b->y), vector2($I->c->x, $I->c->y));
 #      next if $i2->narrow(0);
 #      next if $I2->narrow(0);
 
       my @c = $i2->pointsInCommon($I2);
       next unless scalar(@c);
 
       for my $c(@c)
        {my $C = $back->convertPlaneToSpace($c);
         my $d = $t->distanceToPlaneAlongLine($from, $C);
         my $D = $T->distanceToPlaneAlongLine($from, $C);
         next if abs($d-$D) < 0.1;  # Points to close in space to disambiguate
 
         $o{$ip}{$jp} = 1 if $d < $D;  # Assumes order does not matter for coplanar triangles
         $o{$jp}{$ip} = 1 if $d > $D;  # Assumes order does not matter for coplanar triangles
         last;
        }
      } 
    }
 
 #_ Draw ________________________________________________________________
 # Order by relationship
 #_______________________________________________________________________
 
   my @p;
   for(;;)
    {my $n = 0;
     for my $i(sort(keys(%o)))
      {unless (keys(%{$o{$i}}))
        {push @p, $P[$i];
         delete $o{$i};
         ++$n; 
         for my $j(keys(%o))
          {delete $o{$j}{$i};
          }
        }
      }
     last unless $n;
    }
   keys(%o) == 0 or warn "Cycle present??";
   $d->{triangles} = [@p];
  }
 


=head2 Exports


Export L</draw>

 
 use Math::Zap::Exports qw(
   draw ()
  );
 
 #_ Draw ________________________________________________________________
 # Package loaded successfully
 #_______________________________________________________________________
 
 1;
 
 
 


=head2 Credits




=head3 Author



philiprbrenan@yahoo.com


=head3 Copyright



philiprbrenan@yahoo.com, 2004


=head3 License



Perl License.



