CGAL 6.1.1 - 2D Generalized Barycentric Coordinates

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CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap > Class Template Reference
Reference Manual » Harmonic Coordinates

`#include <CGAL/Barycentric_coordinates_2/Harmonic_coordinates_2.h>`

## Definition

template<typename VertexRange, typename DiscretizedDomain, typename GeomTraits, typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>
class CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >

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.

Template Parameters

  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`.

Examples
  Barycentric_coordinates_2/harmonic_coordinates.cpp, and Barycentric_coordinates_2/shape_deformation.cpp.

## 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.

## Access
template<typename OutIterator >
OutIterator | operator() (const Point_2 &query, OutIterator c_begin)
| evaluates 2D harmonic coordinates.
template<typename OutIterator >
OutIterator | operator() (const std::size_t query_index, OutIterator c_begin)
| returns 2D harmonic coordinates at one domain vertex.
template<typename OutIterator >
OutIterator | operator() (OutIterator c_begin)
| returns 2D harmonic coordinates at all domain vertices.

## 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.
void | factorize ()
| factorizes the matrix A.
void | solve ()
| solves the linear system Ax = b.

## Memory Management
void | clear ()
| clears all internal data structures.
void | release_memory ()
| releases all memory that is used internally.

## Constructor & Destructor Documentation

## ◆ Harmonic_coordinates_2()

template<typename VertexRange , typename DiscretizedDomain , typename GeomTraits , typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>

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.

Parameters

  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

Precondition
  polygon.size() >= 3
  polygon is simple

## Member Function Documentation

## ◆ factorize()

template<typename VertexRange , typename DiscretizedDomain , typename GeomTraits , typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>

void CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::factorize | ( |  | ) |

factorizes the matrix A.

Precondition
  setup() is called

## ◆ operator()() [1/3]

template<typename VertexRange , typename DiscretizedDomain , typename GeomTraits , typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>
template<typename OutIterator >

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.

Template Parameters

  OutIterator | a model of `OutputIterator` that accepts values of type `FT`

Parameters

  query | a query point
  c_begin | the beginning of the destination range with the computed coordinates

Returns
  an output iterator to the element in the destination range, one past the last coordinate stored

## ◆ operator()() [2/3]

template<typename VertexRange , typename DiscretizedDomain , typename GeomTraits , typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>
template<typename OutIterator >

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.

Template Parameters

  OutIterator | a model of `OutputIterator` that accepts values of type `FT`

Parameters

  query_index | a domain's vertex index
  c_begin | the beginning of the destination range with the computed coordinates

Returns
  an output iterator to the element in the destination range, one past the last coordinate stored

Precondition
  query_index >= 0 && query_index < domain.number_of_vertices()

## ◆ operator()() [3/3]

template<typename VertexRange , typename DiscretizedDomain , typename GeomTraits , typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>
template<typename OutIterator >

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.

Template Parameters

  OutIterator | a model of `OutputIterator` that accepts values of type `std::vector<FT>`

Parameters

  c_begin | the beginning of the destination range with the computed coordinates

Returns
  an output iterator to the element in the destination range, one past the last coordinate set stored

## ◆ setup()

template<typename VertexRange , typename DiscretizedDomain , typename GeomTraits , typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>

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.

## ◆ solve()

template<typename VertexRange , typename DiscretizedDomain , typename GeomTraits , typename PointMap = CGAL::Identity_property_map<typename GeomTraits::Point_2>>

void CGAL::Barycentric_coordinates::Harmonic_coordinates_2< VertexRange, DiscretizedDomain, GeomTraits, PointMap >::solve | ( |  | ) |

solves the linear system Ax = b.

Precondition
  factorize() is called