### 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
- Voronoi Predicates
- Voronoi Robust FPT

### 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 45 Set Concept

The polygon_45_set concept tag is polygon_45_set_concept

The semantic of a polygon_45_set is zero or more geometry regions which have angles at the vertices that are multiples of 45-degrees relative to the coordinate axis. A Polygon 45 Set Concept makes no sense in floating point, but currently does not provide a static assert to prevent it from being used with floating point coordinates. The result of such use is undefined. When the intersection of two 45 degree edges results in a vertex that is off the integer grid that case is handled by inserting a unit length edge between the two 45 degree edges near the off grid intersection point. In the case that data represented contains no 45-degree angles and is Manhattan a runtime check will default to the Manhattan algorithm. The results of which are identical to what the 45-degree algorithm would do, but obtained more efficiently.

The motivation for providing the polygon_45_set is to extend the special case of Manhattan geometry capability of the library to encompass the slightly less common, but still important special case of geometry that is described by angles that are multiples of 45-degress with respect to the coordinate axis. This simplifies the implementation of geometry algorithms and affords many opportunities for optimization. 45-degree algorithms can be 50X faster than arbitrary angle algorithms and are required to provide a complete feature set that meets the performance requirements of application domains in which Manhattan and 45-degree geometry are a common special case.

Users are recommended to use std::vector and std::list of user defined polygons or library provided polygon_45_set_data<coordinate_type> objects. Lists and vectors of models of polygon_45_concept or polygon_45_with_holes_concept are automatically models of polygon_45_set_concept.

An object that is a model of polygon_45_set_concept can be viewed as a model of polygon_90_set_concept if it is determined at runtime to conform to the restriction that all edges are axis-parallel. This concept casting is accomplished through the view_as<>() function.

view_as<polygon_90_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_45_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_45_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_45_set_view operator|(const T1& l, const T2& r)

Boolean OR operation (polygon set union). Accepts two objects that model polygon_45_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_45_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_45_set_view operator&(const T1& l, const T2& r) | Boolean AND operation (polygon set intersection). Accepts two objects that model polygon_45_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_45_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_45_set_view operator^(const T1& l, const T2& r) | Boolean XOR operation (polygon set disjoint-union). Accepts two objects that model polygon_45_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_45_set_view operator-(const T1& l, const T2& r) | Boolean SUBTRACT operation (polygon set difference). Accepts two objects that model polygon_45_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_45_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_45_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_45_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_45_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_45_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_45_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_45_set or any of its refinements into an object that models polygon_45_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_45_set or one of its refinements covers the exact same geometric regions as another object that models polygon_45_set or one of its refinements. For example: two of polygon_45 objects. O( n log n) runtime complexity and O(n) memory 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_45_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_45, polygon_45_with_holes, polygon or polygon_with_holes. O( n log n) runtime complexity and O(n) memory wrt vertices.

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_45_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_45, polygon_45_with_holes, polygon or polygon_with_holes. O( n log n) runtime complexity and O(n) memory wrt vertices.

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.

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_45_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.

template <typename T> area_type area(const T& polygon_set) | Computes the area covered by geometry in an object that models polygon_45_set. O( n log n) runtime complexity and O(n) memory wrt vertices.
template <typename T1, typename T2> T1& interact(T1& a, const T2& b) | Given an object that models polygon_45_set and an object that models polygon_45_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 plus intersections.
template <typename T> T& self_intersect(T& polygon_set) | Given an object that models polygon_45_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_45_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 polygons, 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 polygons, shrinking them and overwriting the polygon set with the resulting regions. 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,
RoundingOption rounding = CLOSEST,
CornerOption corner = INTERSECTION)

Same as bloat if resizing is positive, same as shrink if resizing is negative. RoundingOption is an enum that controls snapping of non-integer results of resizing 45 degree edges. CornerOption is an enum that controls how corner filling is performed. polygon_45_set_data.hpp defines these enums. 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& 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.
template <typename T> T& scale_down(T& polygon_set, unsigned_area_type factor) | Scales geometry down by unsigned factor. Snaps 45 degree edges back to 45 degrees after division truncation leads to small changes in angle. O( n log n) runtime complexity and O(n) memory wrt vertices.
template <typename T, typename scaling_type> T& scale(polygon_set_type& polygon_set, double scaling) | Scales geometry by multiplying by floating point factor. Snaps 45 degree edges back to 45 degrees after truncation of fractional results of multiply leads to small changes in angle. O( n log n) runtime complexity and O(n) memory wrt vertices.
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.

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.

# Polygon 45 Set Data Object

The polygon 45 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 45 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 45 Set Data is the following:

template <typename T>
class polygon_45_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_45_set_data() | Default constructor.
template <typename iT> polygon_45_set_data(iT input_begin, iT input_end) | Construct from an iterator range of insertable objects.
polygon_45_set_data(const polygon_45_set_data& that) | Copy construct.
template <typename l, typename r, typename op> polygon_45_set_data(const polygon_45_set_view<l,r,op>& t) | Copy construct from a Boolean operator template.
polygon_45_set_data& operator=(const polygon_45_set_data& that) | Assignment from another polygon set, may change scanning orientation.
template <typename l, typename r, typename op> polygon_45_set_data& operator=(const polygon_45_set_view<l, r, op>& that) | Assignment from a Boolean operator template.
template <typename geometry_object> polygon_45_set_data& operator=(const geometry_object& geo) | Assignment from an insertable object.
template <typename iT> void insert(iT input_begin, iT input_end, bool is_hole = false) | Insert objects of an iterator range. If is_hole is true inserts subtractive regions. Linear wrt the number of vertices added.
void insert(const polygon_45_set_data& polygon_set, bool is_hole = false) | Insert a polygon set. Linear wrt the number of vertices added.
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 the number of vertices added.

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 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 y direction. O(n log n) runtime and O(n) memory wrt. vertices + intersections.

template <typename output_container> void get_polygons_with_holes(output_container& o) const | Expects a standard container of polygon with holes objects. Will scan and eliminate overlaps. Converts polygon set geometry to polygons and appends them to the container. O(n log n) runtime and O(n) memory 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. O(n log n) runtime and O(n) memory 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. O(n log n) runtime and O(n) memory wrt. vertices + intersections.

bool operator==(const polygon_45_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 and O(n) memory wrt. vertices + intersections the first time and linear runtime and constant memory subsequently.

bool operator!=(const polygon_45_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 and O(n) memory wrt. vertices + intersections the first time and linear runtime and constant memory subsequently.
bool is_manhattan() const | Returns in constant time the information about whether the geometry contains only Manhattan (axis-parallel rectilinear) edges. Constant time.
void clean() const | Scan and eliminate overlaps. O(n log n) runtime and O(n) memory wrt. vertices + intersections the first time and linear runtime and constant memory 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. O(n log n) runtime and O(n) memory wrt. vertices the first time and linear runtime and constant memory subsequently.

polygon_45_set_data&
resize(coord_type resizing,
RoundingOption rounding = CLOSEST,
CornerOption corner = INTERSECTION)

Same as bloat if resizing is positive, same as shrink if resizing is negative. RoundingOption is an enum that controls snapping of non-integer results of resizing 45 degree edges. CornerOption is an enum that controls how corner filling is performed. polygon_45_set_data.hpp defines these enums. O(n log n) runtime and O(n) memory wrt. vertices + intersections.

template <typename transformation_type> polygon_45_set_data& transform(const transformation_type& transformation) | Applies transformation.transform() on vertices stored within the polygon set. O(n log n) runtime and O(n) memory wrt. vertices + intersections.
polygon_45_set_data& scale_up(unsigned_area_type factor) | Scales vertices stored within the polygon set up by factor. Linear wrt vertices.
polygon_45_set_data& scale_down(unsigned_area_type factor) | Scales vertices stored within the polygon set down by factor. Linear wrt vertices.
polygon_45_set_data& scale(double factor) | Scales vertices stored within the polygon set by floating point factor. Linear wrt vertices.
polygon_45_set_data& self_xor() | Retain only non-overlapping regions of geometry within polygon set. O(n log n) runtime and O(n) memory wrt. vertices + intersections.
polygon_45_set_data& self_intersect() | Retain only overlapping regions of geometry within a polygon set. O(n log n) runtime and O(n) memory wrt. vertices + intersections.
bool has_error_data() const | Returns true if non-integer intersections resulted in small artifacts in the output of a boolean. Constant time.
std::size_t error_count() const | Returns the number of artifacts that may potentially be present in the output due to non-integer intersections. Constant time.
void get_error_data(polygon_45_set_data& p) const | Populates the input polygon set with 1x1 unit squares that bound the error that may be present in the output due to non-integer intersections. Linear wrt. vertices of error data.
polygon_45_set_data& self_intersect() | Retain only overlapping regions of geometry within a polygon set. O(n log n) runtime 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)