\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 2D Arrangements

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OverlayTraits Concept Reference
Traits Concepts

## Definition

A model for the `OverlayTraits` should be able to operate on records (namely, vertices, halfedges and faces) of two input Dcel classes, named `Dcel_A` and `Dcel_B`, and construct the records of an output Dcel class, referred to as `Dcel_R`.

Models for the concept are used by the global `overlay()` function to maintain the auxiliary data stored with the Dcel records of the resulting overlaid arrangement, based on the contents of the input records.

Has Models:

  `CGAL::Arr_default_overlay_traits<Arrangement>`

  `CGAL::Arr_face_overlay_traits<Arr1,Arr2,ResArr,OvlFaceData>`

See Also
  `overlay`

## Types
typedef unspecified_type | Vertex_handle_A
| a constant handle a vertex in `Dcel_A`.
typedef unspecified_type | Halfedge_handle_A
| a constant handle to a halfedge in `Dcel_A`.
typedef unspecified_type | Face_handle_A
| a constant handle to a face `Dcel_A`.
typedef unspecified_type | Vertex_handle_B
| a constant handle to a vertex in `Dcel_B`.
typedef unspecified_type | Halfedge_handle_B
| a constant handle to a halfedge in `Dcel_B`.
typedef unspecified_type | Face_handle_B
| a constant handle to a face in `Dcel_B`.
typedef unspecified_type | Vertex_handle_R
| a handle to a vertex in `Dcel_R`.
typedef unspecified_type | Halfedge_handle_R
| a handle to a halfedge in `Dcel_R`.
typedef unspecified_type | Face_handle_R
| a handle to a faces in `Dcel_R`.

## Vertex Creation
Whenever a vertex in the overlaid arrangement is created, one of the following functions is called in order to attach the appropriate auxiliary data to this vertex:
void | create_vertex (Vertex_handle_A v1, Vertex_handle_B v2, Vertex_handle_R v) const
| constructs the vertex `v` induced by the coinciding vertices `v1` and `v2`.
void | create_vertex (Vertex_handle_A v1, Halfedge_handle_B e2, Vertex_handle_R v) const
| constructs the vertex `v` induced by the vertex `v1` that lies on the halfedge `e2`.
void | create_vertex (Vertex_handle_A v1, Face_handle_B f2, Vertex_handle_R v) const
| constructs the vertex `v` induced by the vertex `v1` that lies inside the face `f2`.
void | create_vertex (Halfedge_handle_A e1, Vertex_handle_B v2, Vertex_handle_R v) const
| constructs the vertex `v` induced by the vertex `v2` that lies on the halfedge `e1`.
void | create_vertex (Face_handle_A f1, Vertex_handle_B v2, Vertex_handle_R v) const
| constructs the vertex `v` induced by the vertex `v2` that lies inside the face `f1`.
void | create_vertex (Halfedge_handle_A e1, Halfedge_handle_B e2, Vertex_handle_R v) const
| constructs the vertex `v` induced by the intersection of the halfedges `e1` and `e2`.

## Edge Creation
Whenever an edge in the overlaid arrangement is created, one of the following functions is called in order to attach the appropriate auxiliary data to this vertex. Note that an edge is created after both its end-vertices are created, (and the corresponding `create_vertex()` methods were invoked). In all cases, the edge is represented by a halfedge `e` directed in lexicographic decreasing order (from right to left). The `create_edge()` method should attach auxiliary data to the twin halfedge (namely to `e->twin()`) as well:
void | create_edge (Halfedge_handle_A e1, Halfedge_handle_B e2, Halfedge_handle_R e) const
| constructs the halfedge `e` induced by an overlap between the halfedges `e1` and `e2`.
void | create_edge (Halfedge_handle_A e1, Face_handle_B f2, Halfedge_handle_R e) const
| constructs the halfedge `e` induced by the halfedge `e1` that lies inside the face `f2`.
void | create_edge (Face_handle_A f1, Halfedge_handle_B e2, Halfedge_handle_R e) const
| constructs the halfedge `e` induced by the halfedge `e2` that lies inside the face `f1`.

## Face Creation
The following function is invoked whenever a new face is created. It is guaranteed that all halfedges along the face boundary have already been created an have their auxiliary data fields attached to them:
void | create_face (Face_handle_A f1, Face_handle_B f2, Face_handle_R f) const
| constructs the face `f` induced by the an overlap between the faces `f1` and `f2`.