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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
45degrees 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
45degree angles and is Manhattan a runtime check will default to the Manhattan
algorithm. The results of which are identical to what the 45degree
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 45degress with respect to the
coordinate axis. This simplifies the implementation of geometry algorithms
and affords many opportunities for optimization. 45degree 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 45degree 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 axisparallel. 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 disjointunion). 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
noninteger 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 nonoverlapping 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 sweepline algorithm that implements polygonclipping
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 deallocate memory.
Use shrink to fit idiom and assign default constructed polygon set to
deallocate. 
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 (axisparallel 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
noninteger 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 nonoverlapping 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 noninteger 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 noninteger 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 noninteger
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. 
