halfedge 0.4.0

A typical half-edge data structure with some padding

A typical halfedges data structure with some padding

Use

You will most likely want to create meshes with the from_vlfi constructor. It's got a few tricks, so be sure to read the docstring.

from halfedges import HalfEdges, Vert

vertices = [Vert() for _ in range(4)]
face_indices: list[tuple[int, ...]] = [(0, 1, 2), (0, 2, 3)]

mesh = HalfEdges.from_vlfi(vertices, face_indices)

edge = next(iter(mesh.edges))
mesh.remove_edge(edge)

Particulars

The idea (for 18 years and counting) has been to create an interface that is neither too complex, too verbose, nor too magical. I've been all over the place as to where that line should be. This is my current thinking.

Reflection

If you set edge_a.orig = vert_a, then the Edge.vert setter will automagically set vert_a.edge = edge_a. This is true for any setter that might otherwise break the mesh. Sometimes, this reflection will happen when it isn't strictly necessary. Imagine you have face_a with three edges: edge_a, edge_b and edge_c. face_a has a pointer to edge_a, but it could point to any of the three edges and still be correct per the requirements of the halfedge data structure. If you directly set edge_b.face = face_a, everything would still be correct (face_a.edge would still be edge_a and that would still be correct), but the Edge.edge setter will nevertheless set face_a.edge = edge_b.

id

Each Vert, Edge, and Face instance has an id attribute. This is a unique, sequential identifier for the instance. Sorted vert, edge, and face lists (HalfEdges.vl, HalfEdges.el, HalfEdges.fl) are sorted by id. All Vert, Edge, and Face instances across all HalfEdges instances will share the id counter, so don't expect the id of your verts to start at 0.

Holes

A triangle mesh in 2D will never be entirely triangular (and also manifold). There will be a boundary around the triangles. This library treats the outside of that boundary as a face (with an is_hole == True attribute). This way, every edge has a pair and a face, and the is_hole flags can be used to keep these hole faces out of your way.

Holes are useful for more than just 2D mesh boundaries, you can explicitly create holes (Face instances with an as is_hole == True property) to maintain manifold mesh conditions in many circumstances. The constructor HalfEdges.from_vlfi() will try to insert hole faces to maintain manifold conditions.

Four main types: Vert, Edge, Face, and HalfEdges. Vert and Face instances have *.faces properties which will return all adjacent faces. These properties will not return faces flagged as holes. The holes are there to keep things simple and manifold, but otherwise stay out of your way.

Face(
    *attributes: Attrib[Any],
    mesh: BlindHalfEdges | None = None,
    edge: Edge | None = None,
    is_hole: bool = False,
) -> None:

The is_hole __init__ kwarg is shorthand for

class IsHole(ContagionAttribute):
    pass

vert = Vert()
vert.add_attrib(IsHole())

The face_instance.is_hole property getter is shorthand for

vert.get_attrib(IsHole())

More on Attrib classes below and in type_attrib.py.

Element Attributes

By halfedge convention, each Vert instance holds a pointer to one Edge instance, each Face instance holds a pointer to one Edge instance, and each Edge instance holds four pointers (orig, pair, face, next). These describe the geometry of a mesh, but there may be other attributes you would like to assign to these instances. For example, each Face instance might have a color or each Vert instance an (x, y) coordinate. There is no objectively correct way define these attributes or to combine them when two elements are merged. If you assign position vertices to your verts, will they be xy tuples or numpy arrays? Do a red and a blue face behave like paint and combine to make a purple face? Or do they behave like DNA to make a red or blue face depending on which is dominant?

These cannot be stored as simple attributes (e.g., face.color), because it wouldn't be clear what to do when two faces were combined--by, for instance, deleting a shared edge. Somewhere, you have to define a rule for how different colored faces merge or how coordinate locations combine when merging verts. So, properties like color must be defined here as Attrib instances. To create an attribute, inherit from Attrib or one of its children defined in type_attrib.py. Define merge, split, and _infer_value methods to determine how (for instance, face color) will behave when merged or cached.

You cannot assign these with instance.attribute. Instead assign with vert.add_attrib(attrib_instance). This will add the Attrib to a attrib dict in Vert instance dict. Retrieve the value with vert_instance.get_attrib(attrib_class). Everything will be keyed to the class name, so you will need a new ElemAttribBase descendant for each attribute type.

class Coordinate(IncompatibleAttribute[Tuple[float, float]]):
    pass

vert = Vert()
vert.add_attrib(Coordinate((1, 2)))
assert vert.get_attrib(Coordinate).value == (1, 2)

These element attributes can also be passed at __init__

vert = Vert(Coordinate(1, 2))
assert vert.get_attrib(Coordinate).value == (1, 2)

The Attrib classes and merge and split methods that will be called when two elements are merged (e.g., merge two faces when removing the edge between them) or split (e.g., split an edge into two edges).

You Should Know

A canonical half-edge data structure stores:

• a set of verts (redundant)
• for each vert, a pointer to an edge
• a set of edges
• for each edge, pointers to vert, pair, face, next
• a set of faces (redundant)
• for each face, a pointer to an edge

This implementation only stores a set of edges. Sets of verts and faces are generated by iterating through references in edge instances. This makes for slower code, but does not violate DRY and makes for dramatically cleaner code.

This means that verts and faces disappear from a mesh when the edges referencing them are removed.

Also means that this won't be useful for more than hundreds (maybe thousands) of edges. Good enough for my purposes.

Project Structure

The HalfEdges class is defined across three modules:

half_edge_constructors.py

class BlindHalfEdges defines the from_vlfi constructor and just enough methods to allow it to work. These methods define how pointers and Attrib instances are assigned.

half_edge_querries.py

class StaticHalfEdges(BlindHalfEdges) defines all the look-up tricks of the halfedge data structure. What faces are adjacent to an edge? What edges radiate from a vert? etc.

half_edge_object.py

class HalfEdges(StaticHalfEdges) defines all of the methods that change the structure of the mesh: insert_edge, collapse_edge, etc.