man Math::Clipper (3): Polygon clipping in 2D

SYNOPSIS

use Math::Clipper ':all';
my $clipper = Math::Clipper->new;
$clipper->add_subject_polygon( [ [-100, 100], [ 0, -200], [100, 100] ] );
$clipper->add_clip_polygon( [ [-100, -100], [100, -100], [ 0, 200] ] );
my $result = $clipper->execute(CT_DIFFERENCE);
# $result is now a reference to an array of three triangles
$clipper->clear();
# all data from previous operation cleared
# object ready for reuse
# Example with floating point coordinates:
# Clipper requires integer input.
# These polygons won't work.
my $poly_1 = [
[-0.001, 0.001],
[0, -0.002],
[0.001, 0.001]
];
my $poly_2 = [
[-0.001, -0.001],
[0.001, -0.001],
[0, 0.002]
];
# But we can have them automatically scaled up (in place) to a safe integer range
my $scale = integerize_coordinate_sets( $poly_1 , $poly_2 );
$clipper->add_subject_polygon( $poly_1 );
$clipper->add_clip_polygon( $poly_2 );
my $result = $clipper->execute(CT_DIFFERENCE);
# to convert the results (in place) back to the original scale:
unscale_coordinate_sets( $scale, $result );
# Example using 32 bit integer math instead of the default 53 or 64
# (less precision, a bit faster)
my $clipper32 = Math::Clipper->new;
my $scale32 = integerize_coordinate_sets( { bits=>32 } , $poly_1 , $poly_2 );
$clipper32->add_subject_polygon( $poly_1 );
$clipper32->add_clip_polygon( $poly_2 );
my $result32 = $clipper->execute(CT_DIFFERENCE);
unscale_coordinate_sets( $scale32, $result32 );

DESCRIPTION

"Clipper" is a C++ (and Delphi) library that implements polygon clipping.

Exports

The module optionally exports a few constants to your namespace. Standard Exporter semantics apply (including the ":all" tag).

The list of exportable constants is comprised of the clip operation types (which should be self-explanatory):

    CT_INTERSECTION
    CT_UNION
    CT_DIFFERENCE
    CT_XOR

Additionally, there are constants that set the polygon fill type during the clipping operation:

    PFT_EVENODD
    PFT_NONZERO
    PFT_POSITIVE
    PFT_NEGATIVE

CONVENTIONS

INTEGERS: Clipper 4.x works with polygons with integer coordinates. Data in floating point format will need to be scaled appropriately to be converted to the available integer range before polygons are added to a clipper object. (Scaling utilities are provided here.)

A Polygon is represented by a reference to an array of 2D points. A Point is, in turn, represented by a reference to an array containing two numbers: The X and Y coordinates. A 1x1 square polygon example:

  [ [0, 0],
    [1, 0],
    [1, 1],
    [0, 1] ]

Sets of polygons, as returned by the "execute" method, are represented by an array reference containing 0 or more polygons.

Clipper also has a polygon type that explicitly associates an outer polygon with any additional polygons that describe ``holes'' in the filled region of the outer polygon. This is called an ExPolygon. The data structure for an ExPolygon is as follows,:

  { outer => [ <polygon> ],
    holes => [ 
               [ <polygon> ],
               [ <polygon> ],
               ...
             ]
  
  }

Clipper additionally offers an export type named PolyTree which represents several nested polygons by assigning each one to its parent. The PolyTree structure is an arrayref looking like this one:

  [
      { outer => [ ..points.. ], children => [] },
      {
         outer => [ ..points.. ],
         children => [
            { hole => [ ..points.. ], children => [] },
            { hole => [ ..points.. ], children => [] },
         ],
      }
  ]

Each item is a hashref which may contain either the contour or the hole key, containing the polygon points. It also contains a children key containing an arrayref of hashrefs itself, and so on. The Clipper documentation reports that it's more computationally expensive to process (roughly 5-10% slower), it should only be used when parent-child polygon relationships are needed and not just polygon coordinates.

The ``fill type'' of a polygon refers to the strategy used to determine which side of a polygon is the inside, and whether a polygon represents a filled region, or a hole. You may optionally specify the fill type of your subject and clip polygons when you call the "execute" method.

When you specify the NONZERO fill type, the winding order of polygon points determines whether a polygon is filled, or represents a hole. Clipper uses the convention that counter clockwise wound polygons are filled, while clockwise wound polygons represent holes. This strategy is more explicit, but requires that you manage winding order of all polygons.

The EVENODD fill type strategy uses a test segment, with it's start point inside a polygon, and it's end point out beyond the bounding box of all polygons in question. All intersections between the segment and all polygons are calculated. If the intersection count is odd, the inner-most (if nested) polygon containing the segment's start point is considered to be filled. When the intersection count is even, that polygon is considered to be a hole.

For an example case in which NONZERO and EVENODD produce different results see ``NONZERO vs. EVENODD'' section below.

METHODS

new

Constructor that takes no arguments returns a new "Math::Clipper" object.

add_subject_polygon

Adds a(nother) polygon to the set of polygons that will be clipped.

add_clip_polygon

Adds a(nother) polygon to the set of polygons that define the clipping operation.

add_subject_polygons

Works the same as "add_subject_polygon" but adds a whole set of polygons.

add_clip_polygons

Works the same as "add_clip_polygon" but adds a whole set of polygons.

execute

Performs the actual clipping operation. Returns the result as a reference to an array of polygons.

    my $result = $clipper->execute( CT_UNION );

Parameters: the type of the clipping operation defined by one of the constants ("CT_*").

Additionally, you may define the polygon fill types ("PFT_*") of the subject and clipping polygons as second and third parameters respectively. By default, even-odd filling ("PFT_EVENODD") will be used.

    my $result = $clipper->execute( CT_UNION, PFT_EVENODD, PFT_EVENODD );

ex_execute

Like "execute", performs the actual clipping operation, but returns a reference to an array of ExPolygons. (see ``CONVENTIONS'')

pt_execute

Like "execute", performs the actual clipping operation, but returns a PolyTree structure. (see ``CONVENTIONS'')

clear

For reuse of a "Math::Clipper" object, you can call the "clear" method to remove all polygons and internal data from previous clipping operations.

UTILITY FUNCTIONS

integerize_coordinate_sets

Takes an array of polygons and scales all point coordinates so that the values will fit in the integer range available. Returns an array reference containing the scaling factors used for each coordinate column. The polygon data will be scaled in-place. The scaling vector is returned so you can ``unscale'' the data when you're done, using "unscale_coordinate_sets".

    my $scale_vector = integerize_coordinate_sets( $poly1 , $poly2 , $poly3 );

The main purpose of this function is to convert floating point coordinate data to integers. As of Clipper version 4, only integer coordinate data is allowed. This helps make the intersection algorithm robust, but it's a bit inconvenient if your data is in floating point format.

This utility function is meant to make it easy to convert your data to Clipper-friendly integers, while retaining as much precision as possible. When you're done with your clipping operations, you can use the "unscale_coordinate_sets" function to scale results back to your original scale.

Convert all your polygons at once, with one call to "integerize_coordinate_sets", before loading the polygons into your clipper object. The scaling factors need to be calculated so that all polygons involved fit in the available integer space.

By default, the scaling is uniform between coordinate columns (e.g., the X values are scaled by the same factor as the Y values) making all the scaling factors returned the same. In other words, by default, the aspect ratio between X and Y is constrained.

Options may be passed in an anonymous hash, as the first argument, to override defaults. If the first argument is not a hash reference, it is taken instead as the first polygon to be scaled.

    my $scale_vector = integerize_coordinate_sets( {
                                                    constrain => 0, # don't do uniform scaling
                                                    bits => 32     # use the +/-1,073,741,822 integer range
                                                    },
                                                    $poly1 , $poly2 , $poly3
                                                 );

The "bits" option can be 32, 53, or 64. The default will be 53 or 64, depending on whether your Perl uses 64 bit integers AND long doubles by default. (The scaling involves math with native doubles, so it's not enough to just have 64 bit integers.)

Setting the "bits" option to 32 may provide a modest speed boost, by allowing Clipper to avoid calculations with large integer types.

The "constrain" option is a boolean. Default is true. When set to false, each column of coordinates (X, Y) will be scaled independently. This may be useful when the domain of the X values is very much larger or smaller than the domain of the Y values, to get better resolution for the smaller domain. The different scaling factors will be available in the returned scaling vector (array reference).

This utility will also operate on coordinates with three or more dimensions. Though the context here is 2D, be aware of this if you happen to feed it 3D data. Large domains in the higher dimensions could squeeze the 2D data to nothing if scaling is uniform.

unscale_coordinate_sets

This undoes the scaling done by "integerize_coordinate_sets". Use this on the polygons returned by the "execute" method. Pass the scaling vector returned by "integerize_coordinate_sets", and the polygons to ``unscale''. The polygon coordinates will be updated in place.

    unscale_coordinate_sets($scale,$clipper_result);

offset

    my $offset_polygons = offset($polygons, $distance);
    my $offset_polygons = offset($polygons, $distance, $scale, $jointype, $miterlimit);

Takes a reference to an array of polygons ($polygons), a positive or negative offset dimension ($distance), and, optionally, a scaling factor ($scale), a join type ($jointype) and a numeric angle limit for the "JT_MITER" join type.

The polygons will use the NONZERO fill strategy, so filled areas and holes can be specified by polygon winding order.

A positive offset dimension makes filled polygons grow outward, and their holes shrink. A negative offset makes polygons shrink and their holes grow.

Coordinates will be multiplied by the scaling factor before the offset operation and the results divided by the scaling factor. The default scaling factor is 100. Setting the scaling factor higher will result in more points and smoother contours in the offset results.

Returns a new set of polygons, offset by the given dimension.

    my $offset_polygons = offset($polygons, 5.5); # offset by 5.5
        or
    my $offset_polygons = offset($polygons, 5.5, 1000); # smoother results, proliferation of points

WARNING: As you increase the scaling factor, the number of points grows quickly, and will happily consume all of your RAM. Large offset dimensions also contribute to a proliferation of points.

Floating point data in the input is acceptable - in that case, the scaling factor also determines how many decimal digits you'll get in the results. It is not necessary, and generally not desirable to use "integerize_coordinate_sets" to prepare data for this function.

When doing negative offsets, you may find the winding order of the results to be the opposite of what you expect, although this seems to be fixed in recent Clipper versions. Check the order and change it if it is important in your application.

Join type can be one of "JT_MITER", "JT_ROUND" or "JT_SQUARE".

int_offset

    my $offset_polygons = int_offset($polygons, $distance, $scale, $jointype, $miterlimit);

This function is a faster replacement for offset() when input coordinates are integers. If floats are supplied to it, their decimal digits will be truncated so the offset might work on invalid geometry (truncation can lead to self-intersecting polygons). Be sure to only use this one if your input polygons only have integer coordinates.

int_offset2

    my $offset_polygons = int_offset($polygons, $distance1, $distance2, $scale, $jointype, $miterlimit);

This function works like int_offset() but it does two consecutive offsets with the given distances. The purpose of the *offset2 functions is to avoid overhead when two consecutive offsets are needed (scaling/unscaling only happens once, and no conversion to Perl variables happens in between).

ex_int_offset

    my $offset_expolygons = ex_int_offset($polygons, $distance, $scale, $jointype, $miterlimit);

This function works like int_offset() but it does a UNION operation on the resulting polygons and returns an arrayref of ExPolygons.

ex_int_offset2

    my $offset_expolygons = ex_int_offset2($polygons, $distance1, $distance2, $scale, $jointype, $miterlimit);

This function works like ex_int_offset() but it does two consecutive offsets with the given distances before performing the UNION operation.

area

Returns the signed area of a single polygon. A counter clockwise wound polygon area will be positive. A clockwise wound polygon area will be negative. Coordinate data should be integers.

    $area = area($polygon);

orientation

Determine the winding order of a polygon. It returns a true value if the polygon is counter-clockwise and you're assuming a display where the Y-axis coordinates are positive upward, or if the polygon is clockwise and you're assuming a positive-downward Y-axis. Coordinate data should be integers. The majority of 2D graphic display libraries have their origin (0,0) at the top left corner, thus Y increases downward; however some libraries (Quartz, OpenGL) as well as non-display applications (CNC) assume Y increases upward.

    my $poly = [ [0, 0] , [2, 0] , [1, 1] ]; # a counter clockwise wound polygon (assuming Y upward)
    my $direction = orientation($poly);
    # now $direction == 1

This function was previously named "is_counter_clockwise()". This symbol is still exported for backwards compatibility; however you're encouraged to switch it to "orientation()" as the underlying Clipper library switched to it too to clarify the Y axis convention issue.

simplify_polygon =head2 simplify_polygons

These functions convert self-intersecting polygons (known as complex polygons) to simple polygons. "simplify_polygon()" takes a single polygon as first argument, while "simplify_polygons()" takes multiple polygons in a single arrayref. The second argument must be a polyfilltype constant (PFT_*, see above). Both return an arrayref of polygons.

MAXIMUM COORDINATE VALUES AND 64 BIT SUPPORT

Clipper accepts 64 bit integer input, but limits the domain of input coordinate values to +/-4,611,686,018,427,387,902, to allow enough overhead for certain calculations. Coordinate values up to these limits are possible with Perls built to support 64 bit integers.

A typical Perl that supports 32 bit integers can alternatively store 53 bit integers as floating point numbers. In this case, the coordinate domain is limited to +/-9,007,199,254,740,992.

When optionally constraining coordinate values to 32 bit integers, the domain is +/-1,073,741,822.

The "integerize_coordinate_sets" utility function automatically respects whichever limit applies to your Perl build.

NONZERO vs. EVENODD

Consider the following example:

    my $p1 = [ [0,0], [200000,0], [200000,200000]             ];   # CCW
    my $p2 = [ [0,200000], [0,0], [200000,200000]             ];   # CCW
    my $p3 = [ [0,0], [200000,0], [200000,200000], [0,200000] ];   # CCW
    my $clipper = Math::Clipper->new;
    $clipper->add_subject_polygon($p1);
    $clipper->add_clip_polygons([$p2, $p3]);
    my $result = $clipper->execute(CT_UNION, PFT_EVENODD, PFT_EVENODD);

$p3 is a square, and $p1 and $p2 are triangles covering two halves of the $p3 area. The "CT_UNION" operation will produce different results, depending on whether "PFT_EVENODD" or "PFT_NONZERO" is used. These are the two different strategies used by Clipper to identify filled vs. empty regions.

Let's see the thing in detail: $p2 and $p3 are the clip polygons. $p2 overlaps half of $p3. With the "PFT_EVENODD" fill strategy, the number of polygons that overlap in a given area determines whether that area is a hole or a filled region. If an odd number of polygons overlap there, it's a filled region. If an even number, it's a hole/empty region. So with "PFT_EVENODD", winding order doesn't matter. What matters is where areas overlap.

So, using "PFT_EVENODD", and considering $p2 and $p3 as the set of clipping polygons, the fact that $p2 overlaps half of $p3 means that the region where they overlap is empty. In effect, in this example, the set of clipping polygons ends up defining the same shape as the subject polygon $p1. So the union is just the union of two identical polygons, and the result is a triangle equivalent to $p1.

If, instead, the "PFT_NONZERO" strategy is specified, the set of clipping polygons is understood as two filled polygons, because of the winding order. The area where they overlap is considered filled, because there is at least one filled polygon in that area. The set of clipping polygons in this case is equivalent to the square $p3, and the result of the "CT_UNION" operation is also equivalent to the square $p3.

This is a good example of how "PFT_NONZERO" is more explicit, and perhaps more intuitive.

VERSION

This module was built around, and includes, Clipper version 5.1.5.

AUTHOR

The Perl module was written by:

Steffen Mueller (<[email protected]>), Mike Sheldrake and Alessandro Ranellucci (aar/alexrj)

But the underlying library "Clipper" was written by Angus Johnson. Check the SourceForge project page for contact information.

COPYRIGHT AND LICENSE

The "Math::Clipper" module is

but we are shipping a copy of the "Clipper" C++ library, which is

"Math::Clipper" is available under the same license as "Clipper" itself. This is the "boost" license:

  Boost Software License - Version 1.0 - August 17th, 2003
  http://www.boost.org/LICENSE_1_0.txt
  
  Permission is hereby granted, free of charge, to any person or organization
  obtaining a copy of the software and accompanying documentation covered by
  this license (the "Software") to use, reproduce, display, distribute,
  execute, and transmit the Software, and to prepare derivative works of the
  Software, and to permit third-parties to whom the Software is furnished to
  do so, all subject to the following:
  
  The copyright notices in the Software and this entire statement, including
  the above license grant, this restriction and the following disclaimer,
  must be included in all copies of the Software, in whole or in part, and
  all derivative works of the Software, unless such copies or derivative
  works are solely in the form of machine-executable object code generated by
  a source language processor.
  
  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
  SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
  FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
  ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
  DEALINGS IN THE SOFTWARE.