CGAL 5.4 - 2D Generalized Barycentric Coordinates
|
#include <CGAL/Barycentric_coordinates_2/Harmonic_coordinates_2.h>
2D harmonic coordinates.
This class implements 2D harmonic coordinate functions ( [2], [7] ), which can be evaluated at any point inside a simple polygon.
Harmonic coordinates are well-defined and non-negative in the closure of any simple polygon, however they cannot be computed analytically and hence they are approximated. The classical way to approximate these coordinates is by discretizing over the space of piecewise linear functions with respect to a partition of the polygon's interior domain, e.g. a triangulation.
Once computed at the vertices of the discretized domain, the coordinate functions can be evaluated at any point inside a polygon by locating the finite element that contains a query point and linearly interpolating within this element. See more details in the user manual here.
VertexRange | a model of ConstRange whose iterator type is RandomAccessIterator |
DiscretizedDomain | a model of DiscretizedDomain_2 . For the moment, we only support domains whose partition's finite elements are triangles. |
GeomTraits | a model of BarycentricTraits_2 |
PointMap | a model of ReadablePropertyMap whose key type is VertexRange::value_type and value type is Point_2 . The default is CGAL::Identity_property_map . |
Types | |
typedef GeomTraits::FT | FT |
Number type. | |
typedef GeomTraits::Point_2 | Point_2 |
Point type. | |
Initialization | |
Harmonic_coordinates_2 (const VertexRange &polygon, const DiscretizedDomain &domain, const GeomTraits traits=GeomTraits(), const PointMap point_map=PointMap()) | |
initializes all internal data structures. More... | |
Access | |
template<typename OutIterator > | |
OutIterator | operator() (const Point_2 &query, OutIterator c_begin) |
evaluates 2D harmonic coordinates. More... | |
template<typename OutIterator > | |
OutIterator | operator() (const std::size_t query_index, OutIterator c_begin) |
returns 2D harmonic coordinates at one domain vertex. More... | |
template<typename OutIterator > | |
OutIterator | operator() (OutIterator c_begin) |
returns 2D harmonic coordinates at all domain vertices. More... | |
Computation | |
void | compute () |
computes 2D harmonic coordinates at the vertices of the input domain. | |
Step by Step Computation | |
void | setup () |
computes all necessary harmonic data. More... | |
void | factorize () |
factorizes the matrix A. More... | |
void | solve () |
solves the linear system Ax = b. More... | |
Memory Management | |
void | clear () |
clears all internal data structures. | |
void | release_memory () |
releases all memory that is used internally. | |
CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::Harmonic_coordinates_2 | ( | const VertexRange & | polygon, |
const DiscretizedDomain & | domain, | ||
const GeomTraits | traits = GeomTraits() , |
||
const PointMap | point_map = PointMap() |
||
) |
initializes all internal data structures.
This class implements the behavior of harmonic coordinates for 2D query points.
polygon | an instance of VertexRange with the vertices of a simple polygon |
domain | an instance of DiscretizedDomain with a partition of the interior part of a simple polygon |
traits | a traits class with geometric objects, predicates, and constructions; the default initialization is provided |
point_map | an instance of PointMap that maps a vertex from polygon to Point_2 ; the default initialization is provided |
void CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::factorize | ( | ) |
factorizes the matrix A.
OutIterator CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::operator() | ( | const Point_2 & | query, |
OutIterator | c_begin | ||
) |
evaluates 2D harmonic coordinates.
This function fills c_begin
with harmonic coordinates evaluated at the query
point with respect to the vertices of the input polygon. Evaluation is performed by locating the finite element in the input domain that contains query
and then linearly interpolating harmonic coordinates within this element.
If query
does not belong to the input domain or the located element has more than 3 vertices, all coordinates are set to zero.
The number of returned coordinates equals to the number of polygon vertices.
After the coordinates \(b_i\) with \(i = 1\dots n\) are computed, where \(n\) is the number of polygon vertices, the query point \(q\) can be obtained as \(q = \sum_{i = 1}^{n}b_ip_i\), where \(p_i\) are the polygon vertices.
OutIterator | a model of OutputIterator that accepts values of type FT |
query | a query point |
c_begin | the beginning of the destination range with the computed coordinates |
OutIterator CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::operator() | ( | const std::size_t | query_index, |
OutIterator | c_begin | ||
) |
returns 2D harmonic coordinates at one domain vertex.
This function fills c_begin
with harmonic coordinates computed at the vertex of the input domain with the index query_index
.
The number of returned coordinates equals to the number of polygon vertices.
After the coordinates \(b_i\) with \(i = 1\dots n\) are computed, where \(n\) is the number of polygon vertices, the partition vertex \(q\) with the index query_index
can be obtained as \(q = \sum_{i = 1}^{n}b_ip_i\), where \(p_i\) are the polygon vertices.
OutIterator | a model of OutputIterator that accepts values of type FT |
query_index | a domain's vertex index |
c_begin | the beginning of the destination range with the computed coordinates |
OutIterator CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::operator() | ( | OutIterator | c_begin | ) |
returns 2D harmonic coordinates at all domain vertices.
This function fills c_begin
with harmonic coordinates computed at the vertices of the input domain.
The number of returned coordinates equals to the number of input domain vertices.
OutIterator | a model of OutputIterator that accepts values of type std::vector<FT> |
c_begin | the beginning of the destination range with the computed coordinates |
void CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::setup | ( | ) |
computes all necessary harmonic data.
This function fills in the left side matrix A and the right side vector b of the linear system Ax = b. The matrix A is a sparse symmetric positive definite matrix. The solution vector x of this system gives the harmonic coordinates at the vertices of the input domain.
void CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::solve | ( | ) |
solves the linear system Ax = b.