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CGAL 4.8 - Triangulated Surface Mesh Shortest Paths

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SurfaceMeshShortestPathTraits Concept Reference
Concepts

## Definition

The concept `SurfaceMeshShortestPathTraits` describes the types, predicates, and constructions required by the traits class parameter of `CGAL::Surface_mesh_shortest_path`.

Has Models:
  `CGAL::Surface_mesh_shortest_path_traits<K,P>`

## Types
typedef unspecified_type | Triangle_mesh
| A model of the concept `FaceListGraph`
typedef unspecified_type | FT
| A model of the concept `FieldWithSqrt` or a model of both `Field` and `RealEmbeddable`.
typedef unspecified_type | Point_2
| The 2-dimensional point type.
typedef unspecified_type | Vector_2
| The 2-dimensional vector type.
typedef unspecified_type | Ray_2
| The 2-dimensional ray type.
typedef unspecified_type | Line_2
| The 2-dimensional line type.
typedef unspecified_type | Segment_2
| The 2-dimensional segment type.
typedef unspecified_type | Triangle_2
| The 2-dimensional triangle type.
typedef unspecified_type | Barycentric_coordinate
| An ordered triple to specify barycentric coordinates in triangles.
typedef unspecified_type | Point_3
| The 3-dimensional point type.
typedef unspecified_type | Vector_3
| The 3-dimensional vector type.
typedef unspecified_type | Ray_3
| The 3-dimensional ray type.
typedef unspecified_type | Triangle_3
| The 3-dimensional triangle type.

## Constructions
typedef unspecified_type | Construct_point_2
| Function object type that provides `Point_2 operator()(FT x, FT y)` and `Point_2 operator()(CGAL::ORIGIN)` to construct points with cartesian coordinates `(x,y)` and `(0,0)` respectively.
typedef unspecified_type | Construct_vector_2
| Function object type that provides `Vector_2 operator()(Point_2 a, Point_2 b)` to construct the vector `b - a`
typedef unspecified_type | Construct_ray_2
| Function object type that provides `Ray_2 operator()(Point_2 a, Point_2 b)` to construct the ray originating from `a` and passing through `b`
typedef unspecified_type | Construct_line_2
| Function object type that provides `Line_2 operator()(Segment_2 s)` and `Line_2 operator()(Ray_2 r)` to construct the supporting line to `s` and `r` respectively.
typedef unspecified_type | Construct_segment_2
| Function object type that provides `Segment_2 operator()(Point_2 a, Point_2 b)` to construct the segment `(a,b)`
typedef unspecified_type | Construct_triangle_2
| Function object type that provides `Triangle_2 operator()(Point_2 a, Point_2 b, Point_2 c)` to construct the triangle `(a,b,c)`
typedef unspecified_type | Construct_vertex_2
| Function object type that provides `Point_2 operator()(Segment_2 s, int i)` to return the source or target of `s`, depending on whether `i` is 0 or 1 respectively, and `Point_2 operator()(Triangle_2 t, int i)` to return the `i`th vertex of `t`.
typedef unspecified_type | Construct_source_2
| Function object type that provides `Point_2 operator()(Segment_2 s)` to return the source of `s`.
typedef unspecified_type | Construct_target_2
| Function object type that provides `Point_2 operator()(Segment_2 s)` to return the target of `s`.
typedef unspecified_type | Construct_barycenter_2
| Function object type that provides `Point_2 operator()(Point_2 p1, FT w1, Point_2 p2, FT w2)` to compute the barycenter of the points `p1` and `p2` with corresponding weights `w1` and `w2`, and `Point_2 operator()(Point_2 p1, FT w1, Point_2 p2, FT w2, Point_2 p3, FT w3)` to compute the barycenter of the points `p1`, `p2`, and `p3` with corresponding weights `w1`, `w2`, and `w3`.
typedef unspecified_type | Compute_squared_distance_2
| Function object type that provides `FT operator()(A obj1, B obj2)` to compute the squared distance between `obj1` and `obj2`, `A` and `B` can be any one of `Point_2`, `Segment_2`, type objects.
typedef unspecified_type | Intersect_2
| Function object type. More...
typedef unspecified_type | Construct_point_on_2
| Function object type that provides `Point_2 operator()(Ray_2 r, int i)` to construct a point on `r`, such that `i = 0` is the source, and any `i > 0` is not the source.
typedef unspecified_type | Construct_vector_3
| Function object type that provides `Vector_3 operator()(Point_3 a, Point_3 b)` to construct the vector `b - a`
typedef unspecified_type | Construct_triangle_3
| Function object type that provides `Triangle_3 operator()(Point_3 a, Point_3 b, Point_3 c)` to construct the triangle `(a,b,c)`
typedef unspecified_type | Construct_vertex_3
| Function object type that provides `Point_3 operator()(Segment_3 s, int i)` to return the source or target of `s`, depending on whether `i` is 0 or 1 respectively, and `Point_3 operator()(Triangle_3 t, int i)` to return the `i`th vertex of `t`.
typedef unspecified_type | Construct_source_3
| Function object type that provides `Point_3 operator()(Segment_3 s)` to return the source of `s`.
typedef unspecified_type | Construct_target_3
| Function object type that provides `Point_3 operator()(Segment_3 s)` to return the target of `s`.
typedef unspecified_type | Construct_barycenter_3
| Function object type that provides `Point_3 operator()(Point_3 p1, FT w1, Point_3 p2)` to compute the barycenter of the points `p1` and `p2` with corresponding weights `w1` and `1-w1`, and `Point_3 operator()(Point_3 p1, FT w1, Point_3 p2, FT w2, Point_3 p3, FT w3)` to compute the barycenter of the points `p1`, `p2`, and `p3` with corresponding weights `w1`, `w2`, and `w3`.
typedef unspecified_type | Compute_squared_distance_3
| Function object type that provides `FT operator()(Point_3 p1, Point_3 p2)` to compute the squared distance between `p1` and `p2`.
typedef unspecified_type | Construct_triangle_3_to_triangle_2_projection
| Function object type that provides `Triangle_2 operator()(Triangle_3 t)` which computes a 2-dimensional projection of `t` that preserves edge lengths.
typedef unspecified_type | Construct_triangle_3_along_segment_2_flattening
| Function object type that provides `Triangle_2 operator()(Triangle_3 t, std::size_t i, Segment_2 base)` which computes a 2-dimensional projection of t that preserves edge lengths, such that the `i`th edge of the projection of `t` is incident to `base`. More...
typedef unspecified_type | Compute_parametric_distance_along_segment_2
| Function object type that provides `FT operator()(Point_2 x0, Point_2 x1, Point_2 p)` which computes the parametric distance of `p` along the segment `[x0,x1]`. More...
typedef unspecified_type | Construct_barycentric_coordinate
| Function object type that provides `Barycentric_coordinate operator()(FT a, FT b, FT c)` to introduce a new triangular barycentric coordinate.
typedef unspecified_type | Construct_barycentric_coordinate_weight
| Function object type that provides `FT operator(Barycentric_coordinate b, std::size_t i)` to get the `i`th weight of barycentric coordinate `b`.
typedef unspecified_type | Construct_barycentric_coordinate_in_triangle_2
| Function object type that provides `Barycentric_coordinate operator()(Triangle_2 t, Point_2 p)` which computes the Barycentric location of `p` in `t`.
typedef unspecified_type | Construct_barycentric_coordinate_in_triangle_3
| Function object type that provides `Barycentric_coordinate operator()(Triangle_3 t, Point_3 p)` which computes the Barycentric location of `p` in `t`.

## Predicates
typedef unspecified_type | Classify_barycentric_coordinate
| Function object type that provides `std::pair<CGAL::Surface_mesh_shortest_paths_3::Barycentric_coordinate_type,std::size_t> operator()(Barycentric_coordinate b)`, which computes the classification and the associated edge (if applicable) of the coordinate `b` More...
typedef unspecified_type | Orientation_2
| Function object type that provides `CGAL::Orientation operator()(Point_2 s1, Point_2 s2, Point_2 s3)`, which performs an orientation test for the given points.
typedef unspecified_type | Compare_distance_2
| Function object type that provides `CGAL::Comparison_result operator()(Point_2 p1, Point_2 p2, Point_2 p3, Point_2 p4)`, which compares the squared distance between `p1` and `p2` with the squared distance between `p3` and `p4`.
typedef unspecified_type | Compare_relative_intersection_along_segment_2
| Function object type that provides `CGAL::Comparison_result operator()(Segment_2 s1, Line_2 l1, Segment_2 s2, Line_2 l2)`. More...
typedef unspecified_type | Is_saddle_vertex
| Function object type that provides `template <class VertexPointMap> bool operator()(boost::graph_traits<Triangle_mesh>::vertex_descriptor v, Triangle_mesh& g, VertexPointMap vpm)` that returns true if the vertex is a saddle vertex (more than \( 2 \pi \) surface area over all adjacent faces), and false otherwise. More...

## Creation
| SurfaceMeshShortestPathTraits (SurfaceMeshShortestPathTraits &copy)

## Operations
For all of the above predicate and construction types, e.g. `Func_obj_type`, a function must exist with the name `func_obj_type_object()` that creates an instance of the construction or predicate object type. For example:
Construct_point_2 | construct_point_2_object ()

## Member Typedef Documentation

typedef unspecified_type SurfaceMeshShortestPathTraits::Classify_barycentric_coordinate

Function object type that provides `std::pair<CGAL::Surface_mesh_shortest_paths_3::Barycentric_coordinate_type,std::size_t> operator()(Barycentric_coordinate b)`, which computes the classification and the associated edge (if applicable) of the coordinate `b`

Returns the pair (`type`, `i`), such that `type` is one of the values of `CGAL::Surface_mesh_shortest_paths_3::Barycentric_coordinate_type`

- If `type` is `CGAL::Surface_mesh_shortest_paths_3::BARYCENTRIC_COORDINATE_ON_VERTEX`, `i` is the index of that vertex
- If `type` is `CGAL::Surface_mesh_shortest_paths_3::BARYCENTRIC_COORDINATE_ON_BOUNDARY`, `i` is the index of the non-zero edge
  - 0 if (0,1) are the non-zero coordinates
  - 1 if (1,2) are the non-zero coordinates
  - 2 if (2,0) are the non-zero coordinates
- Otherwise, the value of `i` is undefined.

typedef unspecified_type SurfaceMeshShortestPathTraits::Compare_relative_intersection_along_segment_2

Function object type that provides `CGAL::Comparison_result operator()(Segment_2 s1, Line_2 l1, Segment_2 s2, Line_2 l2)`.

This compares the relative parametric intersections of `s1` with `l1` against `s2` with `l2`. That is, compare the distance of the intersection of `s1` with `l1` from the source of `s1`, scaled by the length of `s1`, to the distance of the intersection of `s2` with `l2` from the source of `s2`, scaled by the length of `s2`.

Precondition
  the intersection of `s1` and `l1` is a point
  the intersection of `s2` and `l2` is a point

typedef unspecified_type SurfaceMeshShortestPathTraits::Compute_parametric_distance_along_segment_2

Function object type that provides `FT operator()(Point_2 x0, Point_2 x1, Point_2 p)` which computes the parametric distance of `p` along the segment `[x0,x1]`.

That is, it computes `t`, such that `p = (1.0 - t)*x0 + t*x1`

Precondition
  `p` is a point in the segment `[x0,x1]`

typedef unspecified_type SurfaceMeshShortestPathTraits::Construct_triangle_3_along_segment_2_flattening

Function object type that provides `Triangle_2 operator()(Triangle_3 t, std::size_t i, Segment_2 base)` which computes a 2-dimensional projection of t that preserves edge lengths, such that the `i`th edge of the projection of `t` is incident to `base`.

Precondition
  The length of the `i`th edge of `t` is equal to the length of `base`

typedef unspecified_type SurfaceMeshShortestPathTraits::Intersect_2

Function object type.

Must provide `CGAL::cpp11::result_of<Intersect_2(A,B)>::type operator()(A obj1, B obj2)` to compute the intersection between `obj1` and `obj2`, where `A` and `B` can be any type amongst `Line_2`, `Ray_2`, `Segment_2`.

typedef unspecified_type SurfaceMeshShortestPathTraits::Is_saddle_vertex

Function object type that provides `template <class VertexPointMap> bool operator()(boost::graph_traits<Triangle_mesh>::vertex_descriptor v, Triangle_mesh& g, VertexPointMap vpm)` that returns true if the vertex is a saddle vertex (more than \( 2 \pi \) surface area over all adjacent faces), and false otherwise.

`vpm` must be a model of concept `ReadablePropertyMap` that maps from `vertex_descriptor` to `Point_3` objects.