### Contents

- Boost.Polygon Main Page
- Design Overview
- Isotropy
- Coordinate Concept
- Interval Concept
- Point Concept
- Segment Concept
- Rectangle Concept
- Polygon 90 Concept
- Polygon 90 With Holes Concept
- Polygon 45 Concept
- Polygon 45 With Holes Concept
- Polygon Concept
- Polygon With Holes Concept
- Polygon 90 Set Concept
- Polygon 45 Set Concept
- Polygon Set Concept
- Connectivity Extraction 90
- Connectivity Extraction 45
- Connectivity Extraction
- Property Merge 90
- Property Merge 45
- Property Merge
- Voronoi Main Page
- Voronoi Benchmark
- Voronoi Builder
- Voronoi Diagram

### Other Resources

- GTL Boostcon 2009 Paper
- GTL Boostcon 2009 Presentation
- Performance Analysis
- Layout Versus Schematic Tutorial
- Minkowski Sum Tutorial
- Voronoi Basic Tutorial
- Voronoi Advanced Tutorial

### Polygon Sponsor

# Polygon Set Concept

The polygon_set concept tag is polygon_set_concept

The semantic of a polygon_set is zero or more geometry regions. A Polygon Set Concept may be defined with floating point coordinates, but a snap rounding distance of one integer unit will still be applied, furthermore, geometry outside the domain where one integer unit is sufficient to provide robustness may lead to undefined behavior in algorithms. It is recommended to use integer coordinates for robust operations. In the case that data represented contains only Manhattan geometry a runtime check will default to the Manhattan algorithm. The results of which are identical to what the general algorithm would do, but obtained more efficiently. In the case that the data represented contains only Manhattan and 45-degree geometry a runtime check will default to the faster 45-degree algorithm. The results of which may differ slight from what the general algorithm would do because non-integer intersections will be handled differently.

Users are recommended to use std::vector and std::list of user defined polygons or library provided polygon_set_data<coordinate_type> objects. Lists and vectors of models of polygon_concept or polygon_with_holes_concept are automatically models of polygon_set_concept.

Example code custom_polygon_set.cpp demonstrates mapping a user defined class to the library polygon_set_concept

An object that is a model of polygon_set_concept can be viewed as a model of polygon_90_set_concept or polygon_45_set_concept if it is determined at runtime to conform to the restrictions of those concepts. This concept casting is accomplished through the view_as<>() function.

view_as<polygon_90_set_concept>(polygon_set_object)
view_as<polygon_45_set_concept>(polygon_set_object)

The return value of view_as<>() can be passed into any interface that expects an object of the conceptual type specified in its template parameter. Polygon sets cannot be viewed as single polygons or rectangles since it generally cannot be known whether a polygon set contains only a single polygon without converting to polygons.

## Operators

The return type of some operators is the polygon_set_view operator template type. This type is itself a model of the polygon 90 set concept, but furthermore can be used as an argument to the polygon_set_data constructor and assignment operator. The operator template exists to eliminate temp copies of intermediate results when Boolean operators are chained together.

Operators are declared inside the namespace boost::polygon::operators.

template <typename T1, typename T2>
polygon_set_view operator|(const T1& l, const T2& r)

Boolean OR operation (polygon set union). Accepts two objects that model polygon_set or one of its refinements. Returns an operator template that performs the operation on demand when chained or or nested in a library function call such as assign(). Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename T1, typename T2> polygon_set_view operator+(const T1& l, const T2& r) | Same as operator|. The plus sign is also used for OR operations in Boolean logic expressions. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> polygon_set_view operator&(const T1& l, const T2& r) | Boolean AND operation (polygon set intersection). Accepts two objects that model polygon_set or one of its refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> polygon_set_view operator*(const T1& l, const T2& r) | Same as operator&. The multiplication symbol is also used for AND operations in Boolean logic expressions. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> polygon_set_view operator^(const T1& l, const T2& r) | Boolean XOR operation (polygon set disjoint-union). Accepts two objects that model polygon_set or one of its refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> polygon_set_view operator-(const T1& l, const T2& r) | Boolean SUBTRACT operation (polygon set difference). Accepts two objects that model polygon_set or one of its refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator|=(const T1& l, const T2& r) | Same as operator|, but with self assignment, left operand must model polygon_set and not one of it's refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator+=(T1& l, const T2& r) | Same as operator+, but with self assignment, left operand must model polygon_set and not one of it's refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator&=(const T1& l, const T2& r) | Same as operator&, but with self assignment, left operand must model polygon_set and not one of it's refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator*=(T1& l, const T2& r) | Same as operator*, but with self assignment, left operand must model polygon_set and not one of it's refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator^=(const T1& l, const T2& r) | Same as operator^, but with self assignment, left operand must model polygon_set and not one of it's refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator-=(T1& l, const T2& r) | Same as operator-, but with self assignment, left operand must model polygon_set and not one of it's refinements. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1> T1 operator+(const T1&, coordinate_type bloating) | Performs resize operation, inflating by bloating ammount. If negative the result is a shrink instead of bloat. Note: returns result by value. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1 operator-(const T1&, coordinate_type shrinking) | Performs resize operation, deflating by bloating ammount. If negative the result is a bloat instead of shrink. Note: returns result by value. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator+=(const T1&, coordinate_type bloating) | Performs resize operation, inflating by bloating ammount. If negative the result is a shrink instead of bloat. Returns reference to modified argument. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> T1& operator-=(const T1&, coordinate_type shrinking) | Performs resize operation, deflating by bloating ammount. If negative the result is a bloat instead of shrink. Returns reference to modified argument. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

## Functions

template <typename T1, typename T2> T1& assign(T1& lvalue, const T2& rvalue) | Eliminates overlaps in geometry and copies from an object that models polygon_set or any of its refinements into an object that models polygon_set. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T1, typename T2> bool equivalence(const T1& lvalue, const T2& rvalue) | Returns true if an object that models polygon_set or one of its refinements covers the exact same geometric regions as another object that models polygon_set or one of its refinements. For example: two of polygon objects. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename output_container_type, typename T>
void get_trapezoids(output_container_type& output,
const T& polygon_set)

Output container is expected to be a standard container. Slices geometry of an object that models polygon_set or one of its refinements into non overlapping trapezoids along a vertical slicing orientation and appends them to the output, which must have a value type that models polygon or polygon_with_holes. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename output_container_type, typename T>
void get_trapezoids(output_container_type& output,
const T& polygon_set,
orientation_2d orient)

Output container is expected to be a standard container. Slices geometry of an object that models polygon_set or one of its refinements into non overlapping trapezoids along a the specified slicing orientation and appends them to the output, which must have a value type that models polygon or polygon_with_holes. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename polygon_set_type> void clear(polygon_set_type& polygon_set) | Makes the object empty of geometry.
template <typename polygon_set_type> bool empty(const polygon_set_type& polygon_set) | Checks whether the object is empty of geometry. Polygons that are completely covered by holes will result in empty returning true. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename T, typename rectangle_type>
bool extents(rectangle_type& extents_rectangle,
const T& polygon_set)

Computes bounding box of an object that models polygon_set and stores it in an object that models rectangle. If the polygon set is empty returns false. If there are holes outside of shells they do not contribute to the extents of the polygon set. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename T> area_type area(const T& polygon_set) | Computes the area covered by geometry in an object that models polygon_set. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T> T& bloat(T& polygon_set, unsigned_area_type bloating) | Same as getting all the polygons, bloating them and putting them back. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T> T& shrink(T& polygon_set, unsigned_area_type shrinking) | Same as getting all the polygons, shrinking them and overwriting the polygon set with the resulting regions. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename T, typename coord_type>
T& resize(T& polygon_set, coord_type resizing,
bool corner_fill_arc = false,
unsigned int num_circle_segments = 0)

Same as bloat if resizing is positive, same as shrink if resizing is negative. Original topology at acute angle vertices is preserved by default, segmented circular arcs are inserted if corner_fill_arc is true. num_circle_segments specifies number of segments to introduce on a full circle when filling acute angle corners with circular arcs. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename T> T& scale_up(T& polygon_set, unsigned_area_type factor) | Scales geometry up by unsigned factor. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T> T& scale_down(T& polygon_set, unsigned_area_type factor) | Scales geometry down by unsigned factor. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename T, typename transformation_type> T& transform(T& polygon_set, const transformation_type& transformation) | Applies transformation.transform() on all vertices. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename T>
T& keep(T& polygon_set,
unsigned_area_type min_area,
unsigned_area_type max_area,
unsigned_area_type min_width,
unsigned_area_type max_width,
unsigned_area_type min_height,
unsigned_area_type max_height)

Retains only regions that satisfy the min/max criteria in the argument list. Note: useful for visualization to cull too small polygons. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

# Polygon Set Data Object

The polygon set data type encapsulates the internal data format that serves as the input to the sweep-line algorithm that implements polygon-clipping Boolean operations. It also internally keeps track of whether that data has been sorted or scanned and maintains the invariant that when its flags indicate that the data is sorted or scanned the data has not been changed to violate that assumption. Using the Polygon Set Data type directly can be more efficient than using lists and vectors of polygons in the functions above because of the invariants it can enforce which provide the opportunity to maintain the data is sorted form rather than going all the way out to polygons then resorting those vertices for a subsequent operation.

The declaration of Polygon Set Data is the following:

template <typename T>
class polygon_set_data;

The class is parameterized on the coordinate data type. Algorithms that benefit from knowledge of the invariants enforced by the class are implemented as member functions to provide them access to information about those invariants.

Example code polygon_set_usage.cpp demonstrates using the library provided polygon set data types and functions

## Member Functions

polygon_set_data() | Default constructor.
template <typename iT> polygon_set_data(iT input_begin, iT input_end) | Construct with scanning orientation from an iterator range of insertable objects.
polygon_set_data(const polygon_set_data& that) | Copy construct.
template <typename l, typename r, typename op> polygon_set_data(const polygon_set_view<l,r,op>& t) | Copy construct from a Boolean operator template.
polygon_set_data& operator=(const polygon_set_data& that) | Assignment from another polygon set, may change scanning orientation.
template <typename l, typename r, typename op> polygon_set_data& operator=(const polygon_set_view<l, r, op>& that) | Assignment from a Boolean operator template.
template <typename geometry_object> polygon_set_data& operator=(const geometry_object& geo) | Assignment from an insertable object.
template <typename iT> void insert(iT input_begin, iT input_end) | Insert objects of an iterator range. Linear wrt vertices inserted.
void insert(const polygon_set_data& polygon_set) | Insert a polygon set. Linear wrt vertices inserted.
template <typename geometry_type> void insert(const geometry_type& geometry_object, bool is_hole = false) | Insert a geometry object, if is_hole is true then the inserted region is subtractive rather than additive. Linear wrt vertices inserted.

template <typename output_container>
void get(output_container& output) const

Expects a standard container of polygons objects. Will scan and eliminate overlaps. Converts polygon set geometry to objects of the polygon type and appends them to the container. Polygons will be output with counterclockwise winding, hole polygons will be output with clockwise winding. The last vertex of an output polygon is the duplicate of the first, and the number of points is equal to the number of edges plus 1. If required by the output data type, polygons will have holes fractured out to the outer boundary along the positive y direction and off grid intersections on the outer boundary introduced by this fracture will be truncated downward. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename output_container> void get_trapezoids(output_container& output) const | Expects a standard container of polygon objects. Will scan and eliminate overlaps. Slices polygon set geometry to trapezoids vertically and appends them to the container. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename output_container> void get_trapezoids(output_container& output, orientation_2d slicing_orientation) const | Expects a standard container of polygon objects. Will scan and eliminate overlaps. Slices polygon set geometry to trapezoids along the given orientation and appends them to the container. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

bool operator==(const polygon_set_data& p) const

Once scanned the data representation of geometry within a polygon set is in a mathematically canonical form. Comparison between two sets is therefore a linear time operation once they are in the scanned state. Will scan and eliminate overlaps in both polygon sets. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

bool operator!=(const polygon_set_data& p) const | Inverse logic of equivalence operator.
void clear() | Make the polygon set empty. Note: does not de-allocate memory. Use shrink to fit idiom and assign default constructed polygon set to de-allocate.
bool empty() const | Check whether the polygon set contains no geometry. Will scan and eliminate overlaps because subtractive regions might make the polygon set empty. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
void clean() const | Scan and eliminate overlaps. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections the first time, constant time subsequently.
template <typename input_iterator_type> void set(input_iterator_type input_begin, input_iterator_type input_end) | Overwrite geometry in polygon set with insertable objects in the iterator range.

template <typename rectangle_type>
bool extents(rectangle_type& extents_rectangle) const

Given an object that models rectangle, scans and eliminates overlaps in the polygon set because subtractive regions may alter its extents then computes the bounding box and assigns it to extents_rectangle. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections the first time, linear subsequently.

polygon_set_data&
resize(coord_type resizing,
bool corner_fill_arc = false,
unsigned int num_circle_segments = 0)

Inflates if resizing is positive, deflates if resizing is negative. Original topology at acute angle vertices is preserved by default, segmented circular arcs are inserted if corner_fill_arc is true. num_circle_segments specifies number of segments to introduce on a full circle when filling acute angle corners with circular arcs. Specifying zero for num_circle_segments results in only a single segment being inserted at acute corners. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

template <typename transformation_type> polygon_set_data& transform(const transformation_type& transformation) | Applies transformation.transform() on vertices stored within the polygon set. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
polygon_set_data& scale_up(unsigned_area_type factor) | Scales vertices stored within the polygon set up by factor. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
polygon_set_data& scale_down(unsigned_area_type factor) | Scales vertices stored within the polygon set down by factor. Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.
template <typename scaling_type> polygon_set_data& scale(const scaling_type& f) | Scales vertices stored within the polygon set by applying f.scale(). Expected n log n runtime, worst case quadratic runtime wrt. vertices + intersections.

Copyright: | Copyright © Intel Corporation 2008-2010.
License: | Distributed under the Boost Software License, Version 1.0. (See accompanying file `LICENSE_1_0.txt` or copy at http://www.boost.org/LICENSE_1_0.txt)