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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
disjointunion). 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 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& grow_and(T& polygon_set, orientation_2d orient,
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& grow_and(T& polygon_set, orientation_2d orient,
unsigned_area_type low_bloating,
unsigned_area_type high_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& grow_and(T& polygon_set, direction_2d dir,
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& 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 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 + 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 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 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
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
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 nonoverlapping 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. 
