All About Halfedge_Mesh

In our code, a halfedge mesh -- represented by the Halfedge_Mesh structure -- is a pointer-based repository for manifold mesh connectivity and data. This section provides an overview of some key ideas about halfedge meshes and the Halfedge_Mesh structure in particular. Consider it a supplement to a thorough reading of src/geometry/halfedge.h.

Mesh Elements and Memory Management

A Halfedge_Mesh wrangles four mesh element types: Vertices, Edges, Faces, and Halfedges:

overview of halfedge mesh elements

As a pointer-based structure, Halfedge_Mesh operations will often require the allocation and deallocation of new elements. Rather than using the C++ new and delete operators directly, Halfedge_Mesh instead provides emplace_* and erase_* functions to allocate and free (respectively) mesh elements. This allows the mesh to track all of the elements that belong to it in std::lists (called vertices, edges, faces, and halfedges).

The functions also maintain a unique-within-the-mesh id value for mesh elements, which can be useful for debugging (printing v->id for debug messages is a lot more readable than &*v). As an additional feature, the high-order bit of the id is set by the erase_* functions. So if you start seeing very very large id values, probably you have a use-after-free bug.

Mesh Connectivity

In our halfedge mesh structure, elements store both connectivity information as references (really, std::list< > iterators) to other elements.

Vertices store a reference to one of the halfedges directed away from them in Vertex::halfedge:

vertex connectivity

Edges store a reference to one of the halfedges that make them up in Edge::halfedge:

edge connectivity

Faces store a reference to one of the halfedges that circulate around them in Face::halfedge:

face connectivity

Halfedges store the bulk of the connectivity information. They store a reference to the halfedge on the other side of their edge in Halfedge::twin, a reference to the halfedge that follows them in their current face in Halfedge::next, a reference to the vertex they leave in Halfedge::vertex, a reference to the edge they border in Halfedge::edge, and a reference to the face they circulate in Halfedge::face:

halfedge connectivity

Mesh Data

In addition to the mesh connectivity, the mesh stores data on each of the mesh features.

mesh data overview

Vertices store their location in position, and information about skinning (A4!) in bone_weights.

Edges store a sharp flag which is used in shading normal computation.

Faces store a boundary flag which tells you if the face is a solid face or a face that exists to book-keep holes in the model:

face data

Halfedges store corner_uv and corner_normal values, which are used to set texture coordinates and shading normals for the corner of the face associated with the halfedge.

Splitting and Merging Data

Since it is often the case that you are adding or removing features from the mesh, we include an overloaded interpolate_data helper function to help in interpolating/merging Vertex::bone_weights and Halfedge::corner_* data. (Handling of Edge::sharp is probably either obvious [as in splits] or arbitrary [as in merges with different flags]. Handling of Face::boundary is generally specified by the operation.)

Check the comments near interpolate_data in halfedge.h for some usage examples.

Temporary Storage

Sometimes it is useful to associate temporary storage with the various mesh elements. E.g., to store a position or flag on a mesh element during an algorithm. You can use the fact that mesh elements have stable addresses (and we provide a std::hash specialization that hashes their addresses) to do this:

std::unordered_map< VertexRef, Vec3 > next_position; //store data per-element
std::unordered_set< VertexRef > special; //store a set of

for (VertexRef v = vertices.begin(); v != vertices.end(); ++v) {
  next_position[v] = Vec3(0.0f, 0.0f, 0.0f);
  if (check_something(v)) {

//... later ...

for (VertexRef v = vertices.begin(); v != vertices.end(); ++v) {
  v->position =;
  if (was_special.count(v)) {

Keep in mind that neither addresses or the order of iteration of hashed containers (unordered_*) will be stable between OS's, standard library versions, or even (if using address-space layout randomization) runs of the same program. This probably won't matter for you, but I'm mentioning it now just in case.


A Halfedge_Mesh structure is a sea of pointers. It is surprisingly easy to tangle these pointers into a mess that fails to represent a manifold mesh (or, really, a consistent mesh at all). In order to provide guidance on what constitutes a reasonable setting for a mesh, we define a valid mesh as one for which the following properties hold:

  1. The mesh is self-contained. All pointers are to elements in the vertices, edges, faces, or halfedges lists.
  2. Edges correspond to twinned halfedges. That is, for every edge e, e->halfedge(->twin)^n is a cycle of exactly two halfedges, and these are exactly the halfedges with h->edge equal to e.
  3. Faces correspond to cycles of halfedges. That is, for every face f, f->halfedge(->next)^n is a cycle of at least three halfedges, and these are exactly the halfedges with h->face equal to f.
  4. Vertices correspond to stars of halfedges. That is, for every vertex v, v->halfedge(->twin->next)^n is a cycle of at least two halfedges, and these are exactly the halfedges with h->vertex equal to v.
  5. Vertices are not orphaned, nor is the surface adjacent to them hourglass-shaped. That is, vertices are adjacent to at least one non-boundary face and at most one boundary face.
  6. Edges not orphaned. That is, edges are adjacent to at least one non-boundary face.
  7. Faces are simple. That is, faces touch each vertex and edge at most once.

These properties are checked by the Halfedge_Mesh::validate() function, which returns nullopt if the mesh is valid or an element reference and an explanatory message if something is wrong with the mesh.

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