### 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 90 Set Concept

The polygon_90_set concept tag is polygon_90_set_concept

The semantic of a polygon_90_set is zero or more Manhattan geometry regions.

The motivation for providing the polygon_90_set_concept is that it is a very common special case of planar geometry which afford the implementation of a variety of optimizations on the general planar geometry algorithms. Manhattan geometry processing by the polygon_90_set_concept can be 100X faster than arbitrary angle polygon manipulation. Because the performance benefits are so large and the special case is important enough, the library provides these performance benefits for those application domains that require them.

Users are recommended to use std::vector and std::list of user defined polygons or library provided polygon_90_set_data<coordinate_type> objects. Lists and vectors of models of polygon_90_concept or polygon_90_with_holes_concept or rectangle_concept are automatically models of polygon_90_set_concept.

## Operators

The return type of some operators is the polygon_90_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_90_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_90_set_view operator|(const T1& l, const T2& r)

Boolean OR operation (polygon set union). Accepts two objects that model polygon_90_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(). O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

template <typename T1, typename T2> polygon_90_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. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T1, typename T2> polygon_90_set_view operator&(const T1& l, const T2& r) | Boolean AND operation (polygon set intersection). Accepts two objects that model polygon_90_set or one of its refinements. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T1, typename T2> polygon_90_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. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T1, typename T2> polygon_90_set_view operator^(const T1& l, const T2& r) | Boolean XOR operation (polygon set disjoint-union). Accepts two objects that model polygon_90_set or one of its refinements. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T1, typename T2> polygon_90_set_view operator-(const T1& l, const T2& r) | Boolean SUBTRACT operation (polygon set difference). Accepts two objects that model polygon_90_set or one of its refinements. O( n log n) runtime complexity and O(n) memory 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_90_set and not one of it's refinements. O( n log n) runtime complexity and O(n) memory 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_90_set and not one of it's refinements. O( n log n) runtime complexity and O(n) memory 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_90_set and not one of it's refinements. O( n log n) runtime complexity and O(n) memory 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_90_set and not one of it's refinements. O( n log n) runtime complexity and O(n) memory 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_90_set and not one of it's refinements. O( n log n) runtime complexity and O(n) memory 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_90_set and not one of it's refinements. O( n log n) runtime complexity and O(n) memory 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. O( n log n) runtime complexity and O(n) memory 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. O( n log n) runtime complexity and O(n) memory 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. O( n log n) runtime complexity and O(n) memory 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. O( n log n) runtime complexity and O(n) memory 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_90_set or any of its refinements into an object that models polygon_90_set. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T1, typename T2> bool equivalence(const T1& lvalue, const T2& rvalue) | Returns true if an object that models polygon_90_set or one of its refinements covers the exact same geometric regions as another object that models polygon_90_set or one of its refinements. For example: two of polygon_90 objects. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

template <typename output_container_type, typename T>
void get_rectangles(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_90_set or one of its refinements into non overlapping rectangles and appends them to the output. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

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

Output container is expected to be a standard container. Given an object that models polygon_90_set or one of its refinements finds all overlapping rectangles that are maximal in area and appends them to the output. Expected n log n runtime, worst case quadratic rutnime.

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. O( n log n) runtime complexity and O(n) memory 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_90_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. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

template <typename T> manhattan_area_type area(const T& polygon_set) | Computes the area covered by geometry in an object that models polygon_90_set. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T1, typename T2> T1& interact(T1& a, const T2& b) | Given an object that models polygon_90_set and an object that models polygon_90_set or one of its refinements, modifies a to retain only regions that overlap or touch regions in b. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& self_intersect(T& polygon_set) | Given an object that models polygon_90_set that has self overlapping regions, modifies the argument to contain only the regions of overlap. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& self_xor(T& polygon_set) | Given an object that models polygon_90_set that has self overlapping regions, modifies the argument to contain only the regions that do not overlap. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& bloat(T& polygon_set, unsigned_area_type bloating) | Same as getting all the rectangles, bloating them and putting them back. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& bloat(T& polygon_set, orientation_2d orient, unsigned_area_type bloating) | Same as getting all the rectangles, bloating them and putting them back. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& bloat(T& polygon_set, orientation_2d orient, unsigned_area_type low_bloating, unsigned_area_type high_bloating) | Same as getting all the rectangles, bloating them and putting them back. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& bloat(T& polygon_set, direction_2d dir, unsigned_area_type bloating) | Same as getting all the rectangles, bloating them and putting them back. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& bloat(T& polygon_set, unsigned_area_type west_bloating, unsigned_area_type east_bloating, unsigned_area_type south_bloating, unsigned_area_type north_bloating) | Same as getting all the rectangles, bloating them and putting them back. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& shrink(T& polygon_set, unsigned_area_type shrinking) | Same as getting all the rectangles of the inverse, bloating them and overwriting the polygon set with the resulting regions then negating. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& shrink(T& polygon_set, orientation_2d orient, unsigned_area_type shrinking) | Same as getting all the rectangles of the inverse, bloating them and overwriting the polygon set with the resulting regions then negating. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& shrink(T& polygon_set, orientation_2d orient, unsigned_area_type low_shrinking, unsigned_area_type high_shrinking) | Same as getting all the rectangles of the inverse, bloating them and overwriting the polygon set with the resulting regions then negating. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& shrink(T& polygon_set, direction_2d dir, unsigned_area_type shrinking) | Same as getting all the rectangles of the inverse, bloating them and overwriting the polygon set with the resulting regions then negating. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

template <typename T>
T& shrink(T& polygon_set,
unsigned_area_type west_shrinking,
unsigned_area_type east_shrinking,
unsigned_area_type south_shrinking,
unsigned_area_type north_shrinking)

Same as getting all the rectangles of the inverse, bloating them and overwriting the polygon set with the resulting regions then negating. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

template <typename T, typename coord_type> T& resize(T& polygon_set, coord_type resizing) | Same as bloat if resizing is positive, same as shrink if resizing is negative.
template <typename T, typename coord_type> T& resize(polygon_set_type& polygon_set, coord_type west, coord_type east, coord_type south, coord_type north) | Same as bloat if resizing is positive, same as shrink if resizing is negative. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& grow_and(T& polygon_set, unsigned_area_type bloating) | Same as bloating non-overlapping regions and then applying self intersect to retain only the overlaps introduced by the bloat. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& grow_and(T& polygon_set, orientation_2d orient, unsigned_area_type bloating) | Same as bloating non-overlapping regions and then applying self intersect to retain only the overlaps introduced by the bloat. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& grow_and(T& polygon_set, orientation_2d orient, unsigned_area_type low_bloating, unsigned_area_type high_bloating) | Same as bloating non-overlapping regions and then applying self intersect to retain only the overlaps introduced by the bloat. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& grow_and(T& polygon_set, direction_2d dir, unsigned_area_type bloating) | Same as bloating non-overlapping regions and then applying self intersect to retain only the overlaps introduced by the bloat. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& grow_and(T& polygon_set, unsigned_area_type west_bloating, unsigned_area_type east_bloating, unsigned_area_type south_bloating, unsigned_area_type north_bloating) | Same as bloating non-overlapping regions and then applying self intersect to retain only the overlaps introduced by the bloat. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& scale_up(T& polygon_set, unsigned_area_type factor) | Scales geometry up by unsigned factor. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T> T& scale_down(T& polygon_set, unsigned_area_type factor) | Scales geometry down by unsigned factor. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T, typename scaling_type> T& scale(polygon_set_type& polygon_set, const scaling_type& scaling) | Scales geometry by applying scaling.scale() on all vertices. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T, typename coord_type> T& move(T& polygon_set, orientation_2d orient, coord_type displacement) | Moves geometry by displacement amount in the orientation. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T, typename coord_type> T& move(T& polygon_set, coord_type x_displacement, coord_type y_displacement) | Moves the geometry by x_dispacement in x and y_displacement in y. Note: for consistency should be convolve(polygon_set, point). O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename T, typename transformation_type> T& transform(T& polygon_set, const transformation_type& transformation) | Applies transformation.transform() on all vertices. O( n log n) runtime complexity and O(n) memory 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. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

# Polygon 90 Set Data Object

The polygon 90 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 90 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 90 Set Data is the following:

template <typename T>
class polygon_90_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.

## Member Functions

polygon_90_set_data() | Default constructor. Scanning orientation defaults to HORIZONTAL
polygon_90_set_data(orientation_2d orient) | Construct with scanning orientation.
template <typename iT> polygon_90_set_data(orientation_2d orient, iT input_begin, iT input_end) | Construct with scanning orientation from an iterator range of insertable objects.
polygon_90_set_data(const polygon_90_set_data& that) | Copy construct.
template <typename l, typename r, typename op> polygon_90_set_data(const polygon_90_set_view<l,r,op>& t) | Copy construct from a Boolean operator template.
polygon_90_set_data(orientation_2d orient, const polygon_90_set_data& that) | Construct with scanning orientation and copy from another polygon set.
polygon_90_set_data& operator=(const polygon_90_set_data& that) | Assignment from another polygon set, may change scanning orientation.
template <typename l, typename r, typename op> polygon_90_set_data& operator=(const polygon_90_set_view<l, r, op>& that) | Assignment from a Boolean operator template.
template <typename geometry_object> polygon_90_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. inserted vertices.
void insert(const polygon_90_set_data& polygon_set) | Insert a polygon set. Linear wrt. inserted vertices.
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. inserted vertices.

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

Expects a standard container of geometry objects. Will scan and eliminate overlaps. Converts polygon set geometry to objects of that 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 not the duplicate of the first, and the number of points is equal to the number of edges. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

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

Expects a standard container of geometry objects. Will scan and eliminate overlaps. Converts polygon set geometry to objects of that type and appends them to the container. The resulting polygons will have at most k vertices. For Manhattan data k should be at least 4 . Polygons will be output with counterclockwise winding, hole polygons will be output with clockwise winding. The last vertex of an output polygon is not the duplicate of the first, and the number of points is equal to the number of edges. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

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

Expects a standard container of polygon objects. Will scan and eliminate overlaps. Converts polygon set geometry to polygons and appends them to the container. Polygons will have holes fractured out to the outer boundary along the positive direction of the scanline orientation of the polygon set. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

template <typename output_container> void get_rectangles(output_container& output) const | Expects a standard container of rectangle objects. Will scan and eliminate overlaps. Slices polygon set geometry to rectangles along the scanning orientation and appends them to the container. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
template <typename output_container> void get_rectangles(output_container& output, orientation_2d slicing_orientation) const | Expects a standard container of rectangle objects. Will scan and eliminate overlaps. Slices polygon set geometry to rectangles along the given orientation and appends them to the container. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

bool operator==(const polygon_90_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. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections the first time, linear subsequently.

bool operator!=(const polygon_90_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. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections the first time, linear subsequently.
orientation_2d orient() const | Get the scanning orientation. Depending on the data it is sometimes more efficient to scan in a specific orientation. This is particularly true of Manhattan geometry data. Constant time.
void clean() const | Scan and eliminate overlaps. O( n log n) runtime complexity and O(n) memory 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, orientation_2d orient) | Overwrite geometry in polygon set with insertable objects in the iterator range. Also sets the scanning orientation to that specified.

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. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections the first time, linear subsequently.

polygon_90_set_data&
bloat(unsigned_area_type west_bloating,
unsigned_area_type east_bloating,
unsigned_area_type south_bloating,
unsigned_area_type north_bloating)

Scans to eliminate overlaps and subtractive regions. Inserts rectangles of width specified by bloating values to the indicated side of geometry within the polygon set and fills corners with rectangles of the length and width specified for the adjacent sides. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

polygon_90_set_data&
shrink(unsigned_area_type west_shrinking,
unsigned_area_type east_shrinking,
unsigned_area_type south_shrinking,
unsigned_area_type north_shrinking)

Scans to eliminate overlaps and subtractive regions. Inserts subtractiive rectangles of width specified by bloating values to the indicated side of geometry within the polygon set and subtractive rectangle at convex corners of the length and width specified for the adjacent sides. Scans to eliminate overlapping subtractive regions. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.

polygon_90_set_data& resize(coordinate_type west, coordinate_type east, coordinate_type south, coordinate_type north) | Call bloat or shrink or shrink then bloat depending on whether the resizing values are positive or negative. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
polygon_90_set_data& move(coordinate_type x_delta, coordinate_type y_delta) | Add x_delta to x values and y_delta to y values of vertices stored within the polygon set. Linear wrt. vertices.
template <typename transformation_type> polygon_90_set_data& transform(const transformation_type& transformation) | Applies transformation.transform() on vertices stored within the polygon set. Linear wrt. vertices.
polygon_90_set_data& scale_up(unsigned_area_type factor) | Scales vertices stored within the polygon set up by factor. Linear wrt. vertices.
polygon_90_set_data& scale_down(unsigned_area_type factor) | Scales vertices stored within the polygon set down by factor. Linear wrt. vertices.
template <typename scaling_type> polygon_90_set_data& scale(const anisotropic_scale_factor<scaling_type>& f) | Scales vertices stored within the polygon set by applying f.scale(). Linear wrt. vertices.
polygon_90_set_data& scale(double factor) | Scales vertices stored within the polygon set by floating point factor. Linear wrt. vertices.
polygon_90_set_data& self_xor() | Retain only non-overlapping regions of geometry within polygon set. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
polygon_90_set_data& self_intersect() | Retain only overlapping regions of geometry within a polygon set. O( n log n) runtime complexity and O(n) memory wrt vertices + intersections.
polygon_90_set_data& interact(const polygon_90_set_data& that) | Retain only regions that touch or overlap regions in argument. O( n log n) runtime complexity and O(n) memory 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)