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CGAL 4.8 - Triangulated Surface Mesh Simplification

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EdgeCollapsableSurfaceMesh Concept Reference
Concepts

## Definition

The concept `EdgeCollapsableSurfaceMesh` describes the requirements for the type of triangulated surface mesh that can be passed to the simplification algorithm.

The surface mesh must be structurally equivalent to a polyhedral surface having only triangular faces. It can have any number of connected components, boundaries (borders and holes) and handles (arbitrary genus).

Refines:
  `MutableFaceGraph`

Valid Expressions

Let `v0v1` an edge of the triangulated surface mesh `ecm` and `v0` and `v1` being the source and target vertices of that edge. The surface mesh simplification algorithm requires the call to the function `Euler::edge_collapse(e,ecm)` to be valid and to return the vertex not removed after collapsing the edge `e` with the two halfedges `v0v1` and `v1v0`.

For `h` \( \in \{\) `v0v1,v1v0` \( \}\), let `en` and `ep` be the next and previous edges, that is `en = next(h, surface_mesh)`, `ep = prev(h,surface_mesh)`, and let `eno` and `epo` be their opposite edges, that is `eno = opposite(en, surface_mesh)` and `epo = opposite(ep,surface_mesh)`.

Then, after the collapse of `(v0v1,v1v0)` the following holds:

- The edge `e` is no longer in `surface_mesh`.
- One of \( \{\)`v0,v1` \( \}\) is no longer in `surface_mesh` while the other remains. Most of the time v0 is the vertex being removed but in some cases removing the edge e requires v1 to be removed. See Figure CollapseFigure5. Let `vgone` be the removed vertex and `vkept` be the remaining vertex.
- If `e` was a border edge, that is `get(CGAL::is_border, e, surface_mesh) == true`, then `next(ep) == en`, and `prev(en) == ep`.
- If `e` was not a border edge, that is `get(is_border, e, surface_mesh) == false`, then `ep` and `epo` are no longer in `surface_mesh` while `en` and `eno` are kept in `surface_mesh`.
- For all edges `ie` in `in_edges(vgone,surface_mesh)`, `target(ie,surface_mesh) == vkept` and `source(opposite(ie,surface_mesh),surface_mesh) == vkept`.
- No other incidence information has changed in `surface_mesh`.

General case. The following surface mesh elements are removed: triangles ( \( v0,v1,vL\)) and ( \( v1,v0,vR\)), edges \( (e,e')\), \( (ep,epo)\) and \( (ep',epo')\), and vertex \( v0\).
When the collapsing edge is not itself a border, but is incident upon a border edge that is removed, the operation is the same as in the general case.
When the collapsing edge is not itself a border, but is incident upon a border edge that is not removed, the operation is still the same as in the general case.
When the collapsing edge is itself a border, only 1 triangle is removed. Thus, even if \( (ep',epo')\) exists, it's not removed.

This figure illustrates the single exceptional case when removing \( (v0,v1)\) neccesarily implies removing \( (v1)\), thus \( (v0)\) remains.

Has Models:
  `CGAL::Polyhedron_3<Traits>` (If it has only triangular faces), using the specialization `boost::graph_traits< CGAL::Polyhedron_3<Traits> >` .

See Also
  `boost::graph_traits< CGAL::Polyhedron_3<Traits> >`

## Related Functions
(Note that these are not member functions.)
template<class EdgeCollapsableSurfaceMesh >
boost::graph_traits < EdgeCollapsableSurfaceMesh > ::vertex_descriptor | halfedge_collapse (typename boost::graph_traits< EdgeCollapsableSurfaceMesh >::edge_descriptor const &ue, EdgeCollapsableSurfaceMesh &surface_mesh)
| Collapses the undirected edge `(v0v1,v1v0)` replacing it with `v0` or `v1`, as described in the paragraph above. More...

## Friends And Related Function Documentation

template<class EdgeCollapsableSurfaceMesh >

boost::graph_traits< EdgeCollapsableSurfaceMesh >::vertex_descriptor halfedge_collapse | ( | typename boost::graph_traits< EdgeCollapsableSurfaceMesh >::edge_descriptor const & | ue,
|  | EdgeCollapsableSurfaceMesh & | surface_mesh
| ) |  |
| related

Collapses the undirected edge `(v0v1,v1v0)` replacing it with `v0` or `v1`, as described in the paragraph above.

Precondition
  This function requires `surface_mesh` to be an oriented 2-manifold with or without boundaries. Furthermore, the undirected edge `(v0v1,v1v0)` must satisfy the link condition [1], which guarantees that the surface mesh is also 2-manifold after the edge collapse.