579 lines
17 KiB
Perl
579 lines
17 KiB
Perl
package Slic3r::Geometry;
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use strict;
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use warnings;
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require Exporter;
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our @ISA = qw(Exporter);
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our @EXPORT_OK = qw(
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PI epsilon slope line_atan lines_parallel three_points_aligned
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line_point_belongs_to_segment points_coincide distance_between_points
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line_length midpoint point_in_polygon point_in_segment segment_in_segment
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point_is_on_left_of_segment polyline_lines polygon_lines nearest_point
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point_along_segment polygon_segment_having_point polygon_has_subsegment
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polygon_has_vertex polyline_length can_connect_points deg2rad rad2deg
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rotate_points move_points remove_coinciding_points clip_segment_polygon
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sum_vectors multiply_vector subtract_vectors dot perp polygon_points_visibility
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line_intersection bounding_box bounding_box_intersect
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clip_segment_complex_polygon longest_segment angle3points
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);
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use Slic3r::Geometry::DouglasPeucker ();
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use XXX;
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use constant PI => 4 * atan2(1, 1);
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use constant A => 0;
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use constant B => 1;
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use constant X => 0;
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use constant Y => 1;
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our $parallel_degrees_limit = abs(deg2rad(3));
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our $epsilon = 1E-4;
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sub epsilon () { $epsilon }
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sub slope {
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my ($line) = @_;
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return undef if abs($line->[B][X] - $line->[A][X]) < epsilon; # line is vertical
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return ($line->[B][Y] - $line->[A][Y]) / ($line->[B][X] - $line->[A][X]);
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}
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sub line_atan {
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my ($line) = @_;
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return atan2($line->[B][Y] - $line->[A][Y], $line->[B][X] - $line->[A][X]);
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}
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sub lines_parallel {
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my ($line1, $line2) = @_;
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return abs(line_atan($line1) - line_atan($line2)) < $parallel_degrees_limit;
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}
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sub three_points_aligned {
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my ($p1, $p2, $p3) = @_;
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return lines_parallel([$p1, $p2], [$p2, $p3]);
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}
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# this subroutine checks whether a given point may belong to a given
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# segment given the hypothesis that it belongs to the line containing
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# the segment
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sub line_point_belongs_to_segment {
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my ($point, $segment) = @_;
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#printf " checking whether %f,%f may belong to segment %f,%f - %f,%f\n",
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# @$point, map @$_, @$segment;
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my @segment_extents = (
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[ sort { $a <=> $b } map $_->[X], @$segment ],
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[ sort { $a <=> $b } map $_->[Y], @$segment ],
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);
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return 0 if $point->[X] < ($segment_extents[X][0] - epsilon) || $point->[X] > ($segment_extents[X][1] + epsilon);
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return 0 if $point->[Y] < ($segment_extents[Y][0] - epsilon) || $point->[Y] > ($segment_extents[Y][1] + epsilon);
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return 1;
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}
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sub points_coincide {
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my ($p1, $p2) = @_;
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return 1 if abs($p2->[X] - $p1->[X]) < epsilon && abs($p2->[Y] - $p1->[Y]) < epsilon;
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return 0;
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}
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sub distance_between_points {
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my ($p1, $p2) = @_;
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return sqrt((($p1->[X] - $p2->[X])**2) + ($p1->[Y] - $p2->[Y])**2);
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}
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sub line_length {
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my ($line) = @_;
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return distance_between_points(@$line[A, B]);
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}
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sub longest_segment {
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my (@lines) = @_;
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my ($longest, $maxlength);
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foreach my $line (@lines) {
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my $line_length = line_length($line);
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if (!defined $longest || $line_length > $maxlength) {
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$longest = $line;
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$maxlength = $line_length;
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}
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}
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return $longest;
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}
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sub midpoint {
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my ($line) = @_;
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return [ ($line->[B][X] + $line->[A][X]) / 2, ($line->[B][Y] + $line->[A][Y]) / 2 ];
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}
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sub point_in_polygon {
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my ($point, $polygon) = @_;
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my ($x, $y) = @$point;
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my @xy = map @$_, @$polygon;
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# Derived from the comp.graphics.algorithms FAQ,
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# courtesy of Wm. Randolph Franklin
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my $n = @xy / 2; # Number of points in polygon
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my @i = map { 2*$_ } 0..(@xy/2); # The even indices of @xy
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my @x = map { $xy[$_] } @i; # Even indices: x-coordinates
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my @y = map { $xy[$_ + 1] } @i; # Odd indices: y-coordinates
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my ($i, $j);
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my $side = 0; # 0 = outside; 1 = inside
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for ($i = 0, $j = $n - 1; $i < $n; $j = $i++) {
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if (
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# If the y is between the (y-) borders...
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($y[$i] <= $y && $y < $y[$j]) || ($y[$j] <= $y && $y < $y[$i])
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and
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# ...the (x,y) to infinity line crosses the edge
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# from the ith point to the jth point...
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($x < ($x[$j] - $x[$i]) * ($y - $y[$i]) / ($y[$j] - $y[$i]) + $x[$i])
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) {
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$side = not $side; # Jump the fence
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}
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}
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# if point is not in polygon, let's check whether it belongs to the contour
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if (!$side && 0) {
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return 1 if polygon_segment_having_point($polygon, $point);
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}
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return $side;
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}
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sub point_in_segment {
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my ($point, $line) = @_;
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my ($x, $y) = @$point;
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my @line_x = sort { $a <=> $b } $line->[A][X], $line->[B][X];
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my @line_y = sort { $a <=> $b } $line->[A][Y], $line->[B][Y];
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# check whether the point is in the segment bounding box
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return 0 unless $x >= ($line_x[0] - epsilon) && $x <= ($line_x[1] + epsilon)
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&& $y >= ($line_y[0] - epsilon) && $y <= ($line_y[1] + epsilon);
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# if line is vertical, check whether point's X is the same as the line
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if ($line->[A][X] == $line->[B][X]) {
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return 1 if abs($x - $line->[A][X]) < epsilon;
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}
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# calculate the Y in line at X of the point
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my $y3 = $line->[A][Y] + ($line->[B][Y] - $line->[A][Y])
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* ($x - $line->[A][X]) / ($line->[B][X] - $line->[A][X]);
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return abs($y3 - $y) < epsilon ? 1 : 0;
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}
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sub segment_in_segment {
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my ($needle, $haystack) = @_;
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# a segment is contained in another segment if its endpoints are contained
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return point_in_segment($needle->[A], $haystack) && point_in_segment($needle->[B], $haystack);
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}
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sub point_is_on_left_of_segment {
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my ($point, $line) = @_;
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return (($line->[B][X] - $line->[A][X])*($point->[Y] - $line->[A][Y])
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- ($line->[B][Y] - $line->[A][Y])*($point->[X] - $line->[A][X])) > 0;
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}
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sub polyline_lines {
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my ($polygon) = @_;
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my @lines = ();
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my $last_point;
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foreach my $point (@$polygon) {
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push @lines, [ $last_point, $point ] if $last_point;
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$last_point = $point;
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}
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return @lines;
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}
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sub polygon_lines {
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my ($polygon) = @_;
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return polyline_lines([ @$polygon, $polygon->[0] ]);
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}
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sub nearest_point {
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my ($point, $points) = @_;
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my ($nearest_point, $distance);
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foreach my $p (@$points) {
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my $d = distance_between_points($point, $p);
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if (!defined $distance || $d < $distance) {
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$nearest_point = $p;
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$distance = $d;
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return $p if $distance < epsilon;
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}
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}
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return $nearest_point;
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}
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# given a segment $p1-$p2, get the point at $distance from $p1 along segment
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sub point_along_segment {
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my ($p1, $p2, $distance) = @_;
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my $point = [ @$p1 ];
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my $line_length = sqrt( (($p2->[X] - $p1->[X])**2) + (($p2->[Y] - $p1->[Y])**2) );
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for (X, Y) {
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if ($p1->[$_] != $p2->[$_]) {
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$point->[$_] = $p1->[$_] + ($p2->[$_] - $p1->[$_]) * $distance / $line_length;
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}
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}
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return $point;
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}
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# given a $polygon, return the (first) segment having $point
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sub polygon_segment_having_point {
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my ($polygon, $point) = @_;
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foreach my $line (polygon_lines($polygon)) {
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return $line if point_in_segment($point, $line);
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}
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return undef;
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}
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# return true if the given segment is contained in any edge of the polygon
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sub polygon_has_subsegment {
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my ($polygon, $segment) = @_;
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foreach my $line (polygon_lines($polygon)) {
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return 1 if segment_in_segment($segment, $line);
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}
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return 0;
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}
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sub polygon_has_vertex {
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my ($polygon, $point) = @_;
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foreach my $p (@$polygon) {
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return 1 if points_coincide($p, $point);
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}
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return 0;
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}
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sub polyline_length {
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my ($polyline) = @_;
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my $length = 0;
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$length += line_length($_) for polygon_lines($polyline);
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return $length;
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}
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sub can_connect_points {
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my ($p1, $p2, $polygons) = @_;
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# check that the two points are visible from each other
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return 0 if grep !polygon_points_visibility($_, $p1, $p2), @$polygons;
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# get segment where $p1 lies
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my $p1_segment;
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for (@$polygons) {
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$p1_segment = polygon_segment_having_point($_, $p1);
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last if $p1_segment;
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}
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# defensive programming, this shouldn't happen
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if (!$p1_segment) {
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die sprintf "Point %f,%f wasn't found in polygon contour or holes!", @$p1;
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}
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# check whether $p2 is internal or external (internal = on the left)
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return point_is_on_left_of_segment($p2, $p1_segment)
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|| point_in_segment($p2, $p1_segment);
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}
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sub deg2rad {
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my ($degrees) = @_;
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return PI() * $degrees / 180;
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}
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sub rad2deg {
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my ($rad) = @_;
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return $rad / PI() * 180;
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}
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sub rotate_points {
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my ($radians, $center, @points) = @_;
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$center ||= [0,0];
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return map {
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[
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$center->[X] + cos($radians) * ($_->[X] - $center->[X]) - sin($radians) * ($_->[Y] - $center->[Y]),
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$center->[Y] + cos($radians) * ($_->[Y] - $center->[Y]) + sin($radians) * ($_->[X] - $center->[X]),
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]
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} @points;
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}
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sub move_points {
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my ($shift, @points) = @_;
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return map [ $shift->[X] + $_->[X], $shift->[Y] + $_->[Y] ], @points;
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}
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# preserves order
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sub remove_coinciding_points {
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my ($points) = @_;
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my %p = map { sprintf('%f,%f', @$_) => "$_" } @$points;
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%p = reverse %p;
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@$points = grep $p{"$_"}, @$points;
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}
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# implementation of Liang-Barsky algorithm
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# polygon must be convex and ccw
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sub clip_segment_polygon {
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my ($line, $polygon) = @_;
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if (@$line == 1) {
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# the segment is a point, check for inclusion
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return point_in_polygon($line, $polygon);
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}
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my @V = (@$polygon, $polygon->[0]);
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my $tE = 0; # the maximum entering segment parameter
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my $tL = 1; # the minimum entering segment parameter
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my $dS = subtract_vectors($line->[B], $line->[A]); # the segment direction vector
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for (my $i = 0; $i < $#V; $i++) { # process polygon edge V[i]V[Vi+1]
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my $e = subtract_vectors($V[$i+1], $V[$i]);
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my $N = perp($e, subtract_vectors($line->[A], $V[$i]));
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my $D = -perp($e, $dS);
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if (abs($D) < epsilon) { # $line is nearly parallel to this edge
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($N < 0) ? return : next; # P0 outside this edge ? $line is outside : $line cannot cross edge, thus ignoring
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}
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my $t = $N / $D;
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if ($D < 0) { # $line is entering across this edge
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if ($t > $tE) { # new max $tE
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$tE = $t;
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return if $tE > $tL; # $line enters after leaving polygon?
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}
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} else { # $line is leaving across this edge
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if ($t < $tL) { # new min $tL
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$tL = $t;
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return if $tL < $tE; # $line leaves before entering polygon?
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}
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}
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}
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# $tE <= $tL implies that there is a valid intersection subsegment
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return [
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sum_vectors($line->[A], multiply_vector($dS, $tE)), # = P(tE) = point where S enters polygon
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sum_vectors($line->[A], multiply_vector($dS, $tL)), # = P(tE) = point where S enters polygon
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];
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}
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sub sum_vectors {
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my ($v1, $v2) = @_;
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return [ $v1->[X] + $v2->[X], $v1->[Y] + $v2->[Y] ];
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}
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sub multiply_vector {
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my ($line, $scalar) = @_;
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return [ $line->[X] * $scalar, $line->[Y] * $scalar ];
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}
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sub subtract_vectors {
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my ($line2, $line1) = @_;
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return [ $line2->[X] - $line1->[X], $line2->[Y] - $line1->[Y] ];
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}
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# 2D dot product
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sub dot {
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my ($u, $v) = @_;
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return $u->[X] * $v->[X] + $u->[Y] * $v->[Y];
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}
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# 2D perp product
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sub perp {
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my ($u, $v) = @_;
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return $u->[X] * $v->[Y] - $u->[Y] * $v->[X];
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}
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sub polygon_points_visibility {
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my ($polygon, $p1, $p2) = @_;
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my $our_line = [ $p1, $p2 ];
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foreach my $line (polygon_lines($polygon)) {
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my $intersection = line_intersection($our_line, $line, 1) or next;
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next if grep points_coincide($intersection, $_), $p1, $p2;
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return 0;
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}
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return 1;
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}
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sub line_intersection {
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my ($line1, $line2, $require_crossing) = @_;
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$require_crossing ||= 0;
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my $intersection = _line_intersection(map @$_, @$line1, @$line2);
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return (ref $intersection && $intersection->[1] == $require_crossing)
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? $intersection->[0]
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: undef;
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}
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sub _line_intersection {
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my ( $x0, $y0, $x1, $y1, $x2, $y2, $x3, $y3 );
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if ( @_ == 8 ) {
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( $x0, $y0, $x1, $y1, $x2, $y2, $x3, $y3 ) = @_;
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# The bounding boxes chop the lines into line segments.
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# bounding_box() is defined later in this chapter.
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my @box_a = bounding_box([ [$x0, $y0], [$x1, $y1] ]);
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my @box_b = bounding_box([ [$x2, $y2], [$x3, $y3] ]);
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# Take this test away and the line segments are
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# turned into lines going from infinite to another.
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# bounding_box_intersect() defined later in this chapter.
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return "out of bounding box" unless bounding_box_intersect( 2, @box_a, @box_b );
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}
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elsif ( @_ == 4 ) { # The parametric form.
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$x0 = $x2 = 0;
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( $y0, $y2 ) = @_[ 1, 3 ];
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# Need to multiply by 'enough' to get 'far enough'.
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my $abs_y0 = abs $y0;
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my $abs_y2 = abs $y2;
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my $enough = 10 * ( $abs_y0 > $abs_y2 ? $abs_y0 : $abs_y2 );
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$x1 = $x3 = $enough;
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$y1 = $_[0] * $x1 + $y0;
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$y3 = $_[2] * $x2 + $y2;
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}
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my ($x, $y); # The as-yet-undetermined intersection point.
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my $dy10 = $y1 - $y0; # dyPQ, dxPQ are the coordinate differences
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my $dx10 = $x1 - $x0; # between the points P and Q.
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my $dy32 = $y3 - $y2;
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my $dx32 = $x3 - $x2;
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my $dy10z = abs( $dy10 ) < epsilon; # Is the difference $dy10 "zero"?
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my $dx10z = abs( $dx10 ) < epsilon;
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my $dy32z = abs( $dy32 ) < epsilon;
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my $dx32z = abs( $dx32 ) < epsilon;
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my $dyx10; # The slopes.
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my $dyx32;
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$dyx10 = $dy10 / $dx10 unless $dx10z;
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$dyx32 = $dy32 / $dx32 unless $dx32z;
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# Now we know all differences and the slopes;
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# we can detect horizontal/vertical special cases.
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# E.g., slope = 0 means a horizontal line.
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unless ( defined $dyx10 or defined $dyx32 ) {
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return "parallel vertical";
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}
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elsif ( $dy10z and not $dy32z ) { # First line horizontal.
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$y = $y0;
|
|
$x = $x2 + ( $y - $y2 ) * $dx32 / $dy32;
|
|
}
|
|
elsif ( not $dy10z and $dy32z ) { # Second line horizontal.
|
|
$y = $y2;
|
|
$x = $x0 + ( $y - $y0 ) * $dx10 / $dy10;
|
|
}
|
|
elsif ( $dx10z and not $dx32z ) { # First line vertical.
|
|
$x = $x0;
|
|
$y = $y2 + $dyx32 * ( $x - $x2 );
|
|
}
|
|
elsif ( not $dx10z and $dx32z ) { # Second line vertical.
|
|
$x = $x2;
|
|
$y = $y0 + $dyx10 * ( $x - $x0 );
|
|
}
|
|
elsif ( abs( $dyx10 - $dyx32 ) < epsilon ) {
|
|
# The slopes are suspiciously close to each other.
|
|
# Either we have parallel collinear or just parallel lines.
|
|
|
|
# The bounding box checks have already weeded the cases
|
|
# "parallel horizontal" and "parallel vertical" away.
|
|
|
|
my $ya = $y0 - $dyx10 * $x0;
|
|
my $yb = $y2 - $dyx32 * $x2;
|
|
|
|
return "parallel collinear" if abs( $ya - $yb ) < epsilon;
|
|
return "parallel";
|
|
}
|
|
else {
|
|
# None of the special cases matched.
|
|
# We have a "honest" line intersection.
|
|
|
|
$x = ($y2 - $y0 + $dyx10*$x0 - $dyx32*$x2)/($dyx10 - $dyx32);
|
|
$y = $y0 + $dyx10 * ($x - $x0);
|
|
}
|
|
|
|
my $h10 = $dx10 ? ($x - $x0) / $dx10 : ($dy10 ? ($y - $y0) / $dy10 : 1);
|
|
my $h32 = $dx32 ? ($x - $x2) / $dx32 : ($dy32 ? ($y - $y2) / $dy32 : 1);
|
|
|
|
return [[$x, $y], $h10 >= 0 && $h10 <= 1 && $h32 >= 0 && $h32 <= 1];
|
|
}
|
|
|
|
# 2D
|
|
sub bounding_box {
|
|
my ($points) = @_;
|
|
|
|
my @x = sort { $a <=> $b } map $_->[X], @$points;
|
|
my @y = sort { $a <=> $b } map $_->[Y], @$points;
|
|
|
|
return ($x[0], $y[0], $x[-1], $y[-1]);
|
|
}
|
|
|
|
# bounding_box_intersect($d, @a, @b)
|
|
# Return true if the given bounding boxes @a and @b intersect
|
|
# in $d dimensions. Used by line_intersection().
|
|
sub bounding_box_intersect {
|
|
my ( $d, @bb ) = @_; # Number of dimensions and box coordinates.
|
|
my @aa = splice( @bb, 0, 2 * $d ); # The first box.
|
|
# (@bb is the second one.)
|
|
|
|
# Must intersect in all dimensions.
|
|
for ( my $i_min = 0; $i_min < $d; $i_min++ ) {
|
|
my $i_max = $i_min + $d; # The index for the maximum.
|
|
return 0 if ( $aa[ $i_max ] + epsilon ) < $bb[ $i_min ];
|
|
return 0 if ( $bb[ $i_max ] + epsilon ) < $aa[ $i_min ];
|
|
}
|
|
|
|
return 1;
|
|
}
|
|
|
|
sub clip_segment_complex_polygon {
|
|
my ($line, $polygons) = @_;
|
|
|
|
my @intersections = grep $_, map line_intersection($line, $_, 1),
|
|
map polygon_lines($_), @$polygons or return ();
|
|
|
|
# this is not very elegant, however it works
|
|
@intersections = sort { sprintf("%020f,%020f", @$a) cmp sprintf("%020f,%020f", @$b) } @intersections;
|
|
|
|
shift(@intersections) if !grep(point_in_polygon($intersections[0], $_), @$polygons)
|
|
&& !grep(polygon_segment_having_point($_, $intersections[0]), @$polygons);
|
|
|
|
# defensive programming
|
|
###die "Invalid intersections" if @intersections % 2 != 0;
|
|
|
|
my @lines = ();
|
|
while (@intersections) {
|
|
# skip tangent points
|
|
my @points = map shift @intersections, 1..2;
|
|
next if !$points[1];
|
|
next if points_coincide(@points);
|
|
push @lines, [ @points ];
|
|
}
|
|
return [@lines];
|
|
}
|
|
|
|
sub angle3points {
|
|
my ($p1, $p2, $p3) = @_;
|
|
# p1 is the center
|
|
|
|
my $angle = atan2($p2->[X] - $p1->[X], $p2->[Y] - $p1->[Y])
|
|
- atan2($p3->[X] - $p1->[X], $p3->[Y] - $p1->[Y]);
|
|
|
|
# we only want to return only positive angles
|
|
return $angle <= 0 ? $angle + 2*PI() : $angle;
|
|
}
|
|
|
|
1;
|