While you did not ask for any code, I found the problem interesting, so threw together a quick "solution".

Please note: instead of drawing a line to a point I knew to be outside the polygon, I took a horizontal line (Y = point{y}) and examined all the X coordinates where this line intersected the polygon. By ordering these X coordinates you end up with an outside/inside/outside/inside pattern. The only "sticky" spots I found were vertices which lie on the horizontal line, and points which lie on one of the polygon's horizontal lines.

Anyway, here is the code. (If anyone has suggestion on better coding practices, I am always up for that as well! *Smiles*)

use strict;

#--
#-- Script:    within.pl
#-- Purpose:   Check if a point lies within a polygon
#--
#-- Author:    Rhose
#-- Date:      August 31, 2001
#--


#-- Define variables
my @mPolygon;


#-- Define subroutines
sub CheckPoint
{

  #-- Get parameters
  my @pPoint = @_;

  #-- Define constants
  use constant TRUE     => 1;
  use constant FALSE    => 0;

  use constant LOCATION => {
                             0  => 'Outside',
                             1  => 'Inside',
                             2  => 'On'
                           };

  use constant X1       => 0;
  use constant Y1       => 1;
  use constant X2       => 2;
  use constant Y2       => 3;

  #-- Define local variables
  my $mLoc;         # Location of the point (outside, inside, or on)
  my $mOnHoriz;     # Points on a horizontal line cause problems; this
+ catches them
  my $mPtr;         # A pointer used when processing arrays
  my $mSlope;       # The slop of an individual line

  my @mSortY;       # Temporary array to see if a line intersects the 
+Y = $pPoint[Y1] line
  my @mSortX;       # Temporary array to see if the point is on a line
  my @mX;           # Array of all X coords where the Y = $pPoint[Y1] 
+line intersects the polygon

  my %mLine;        # Array of lines which compose the polygon


  #-- Make sure the last point is the same as the first (complete the 
+polygon)
  push @mPolygon, @mPolygon[0,1] if (@mPolygon[0] != @mPolygon[-2] || 
+@mPolygon[1] != @mPolygon[-1]);

  #-- Build the lines array
  for ($mPtr = 0; $mPtr <= $#mPolygon - 3; $mPtr+=2)
  {
    push @{$mLine{'L'.($mPtr/2)}}, @mPolygon[$mPtr..$mPtr+3];
  }

  #-- Check for lines which intersect the Y = $pPoint[Y1] line
  $mOnHoriz = FALSE;
  foreach (keys %mLine)
  {

    #-- Skip lines where the point's Y is not within the line's Ys
    @mSortY = sort $pPoint[Y1],${$mLine{$_}}[Y1],${$mLine{$_}}[Y2];
    next if ($mSortY[1] != $pPoint[Y1]);

    #-- Add the X intersect for the line
    if (${$mLine{$_}}[Y1] == ${$mLine{$_}}[Y2])
    {
      @mSortX = sort $pPoint[X1],${$mLine{$_}}[X1],${$mLine{$_}}[X2];
      if ($mSortX[1] == $pPoint[X1])
      {
        $mOnHoriz = TRUE;
        last;
      }
    }
    else 
    {
      $mSlope = (${$mLine{$_}}[X2] - ${$mLine{$_}}[X1])/(${$mLine{$_}}
+[Y2] - ${$mLine{$_}}[Y1]);
      push @mX, ${$mLine{$_}}[X2] - ($mSlope * (${$mLine{$_}}[Y2] - $p
+Point[Y1]));
    }

  }

  #-- Sort the X coordinates
  @mX = sort @mX;

  #-- Find out where the point is
  if ($mOnHoriz)
  {
    $mLoc = 2;
  }
  else
  {
    $mLoc = 0;
    for ($mPtr = 0; $mPtr <= $#mX; $mPtr++)
    {
 
      #-- Check for on the line
      $mLoc = 2 if ($mX[$mPtr] == $pPoint[X1]);

      #-- Skip vertices
      next if ($mX[$mPtr] == $mX[$mPtr+1]);

      #-- Stop if past the point
      last if ($mX[$mPtr] >= $pPoint[X1]);
 
      #-- Toggle the outside/inside location flag
      $mLoc = 1 - $mLoc;
    }
  }

  #-- Display the results
  print 'Point (',$pPoint[X1],',',$pPoint[Y1],') is ',LOCATION->{$mLoc
+}," the polygon.\n";

}


#-- Check some points
print "mirod's shape:\n";
@mPolygon = (1,0,1,1,2,2,1,3,1,5,2,5,4,4,5,5,6,5,6,0,5,0,4,1,3,0);
CheckPoint(0.5,4);
CheckPoint(1.5,1.5);
CheckPoint(1.5,2);
CheckPoint(2,4);
CheckPoint(3.5,3);
CheckPoint(4,-1);
CheckPoint(4,4);
CheckPoint(4,5);
CheckPoint(5,5);
CheckPoint(5.5,5);
CheckPoint(6,2);
CheckPoint(6,6);
CheckPoint(7,3);

print "\n\n";
print "Triangle:\n";
@mPolygon = (1,0,3,0,3,3);
CheckPoint(1,0);
CheckPoint(3,0);
CheckPoint(3,3);
CheckPoint(2,1.5);
CheckPoint(3,1);
CheckPoint(2,0);
CheckPoint(2.5,1);
CheckPoint(0,2);
CheckPoint(4,2);
CheckPoint(2,-1);


#-- Exit
exit(0);


#--- End of script
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