Your code looks very clean to me. However, I prefer to write _thick_line a little more compact.

sub _thick_line {
  my ( $image, $thickness, $color, $x1, $y1, $x2, $y2 ) = @_;

  my $angle   = _angle( $x1, $y1, $x2, $y2 );
  my $mirror1 = $angle + 90;
  my $mirror2 = $angle - 90;

  my $m1c = $thickness * cos( deg2rad $mirror1 ) / 2;
  my $m1s = $thickness * sin( deg2rad $mirror1 ) / 2;
  my $m2c = $thickness * cos( deg2rad $mirror2 ) / 2;
  my $m2s = $thickness * sin( deg2rad $mirror2 ) / 2;

  $image->polygon(
    aa     => 1,
    color  => $color,
    points => [
                [ $x1 + $m1c, $y1 + $m1s ],    # bottom left
                [ $x2 + $m1c, $y2 + $m1s ],    # bottom right
                [ $x2 + $m2c, $y2 + $m2s ],    # top right
                [ $x1 + $m2c, $y1 + $m2s ],    # top left
    ],
  );
}
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