# Graphs

Plexus provides a flexible representation of meshes as a half-edge graph via the graph module and MeshGraph type. Graphs can store arbitrary geometric data associated with any topological structure. Unlike iterator expressions and buffers, graphs provide efficient traversals and complex manipulation of meshes.

Note

Plexus refers to half-edges as arcs. This borrows from graph theory, where arc typically refers to a directed adjacency.

MeshGraphs can be created in various ways, including from raw buffers and iterator expressions.

  1 2 3 4 5 6 7 8 9 10 11 // Create a graph of a two-dimensional quadrilateral from raw buffers. let mut graph = MeshGraph::>::from_raw_buffers( vec![Tetragon::new(0usize, 1, 2, 3)], vec![(0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)], ) .unwrap(); // Create a graph with positional data from a unit cube. let mut graph = Cube::new() .polygons::>>() .collect>>(); 

## Representation¶

A MeshGraph is conceptually composed of vertices, arcs, edges, and faces. The figure below summarizes the connectivity in a MeshGraph. Arcs are directed and connect vertices. An arc that is directed toward a vertex $A$ is an incoming arc with respect to $A$. Similarly, an arc directed away from such a vertex is an outgoing arc. Every vertex is associated with exactly one leading arc, which is always an outgoing arc. The vertex toward which an arc is directed is the arc's destination vertex and the other is its source vertex.

Every arc is paired with and connected to an opposite arc with an opposing direction. Given an arc from a vertex $A$ to a vertex $B$, that arc will have an opposite arc from $B$ to $A$. Such arcs are typically notated $\overrightarrow{AB}$ and $\overrightarrow{BA}$. Together, these arcs form an edge, which is not directed. An edge and its two arcs are together called a composite edge.

Arcs are connected to their neighbors, known as next and previous arcs. A traversal along a series of arcs is a path. The path formed by traversing from an arc to its next arc and so on is a ring. When a face is present within a ring, the arcs will refer to that face and the face will refer to exactly one of the arcs in the ring (its leading arc). An arc with no associated face is known as a boundary arc. If either of an edge's arcs is a boundary arc, then that edge is a boundary edge.

A path is closed if it forms a loop and is open if it terminates. A path over vertices $A$, $B$, and $C$ is notated $\overrightarrow{\{A,B,C\}}$. This path notation is used to describe rings and faces. It may also be used to describe arcs, but arcs more often use the shorthand notation shown above.

Together with vertices and faces, the connectivity of arcs allows for efficient traversals of topology. For example, it becomes trivial to find neighboring topologies, such as the faces that share a given vertex or the neighboring faces of a given face.

Warning

The MeshGraph data structure has some limitations. With few exceptions, only orientable compact manifolds can be represented. Unorientable manifolds such as Möbius strips and non-manifold structures such as singularities and edge-fans cannot be modeled using MeshGraph.

MeshGraphs store topological data using associative collections and mesh data is accessed using keys into this storage. Keys are exposed as strongly typed and opaque values, which can be used to refer to a topological structure.

## Geometry¶

MeshGraph exposes a type parameter that determines the representation of geometry in a graph. This type must implement the GraphGeometry trait, which specifies which types are used for geometry in vertices, arcs, edges, and faces.

  1 2 3 4 5 6 7 8 9 10 11 12 13 pub struct Vertex { pub position: Point3, pub normal: Vector3, } impl GraphGeometry for Vertex { type Vertex = Self; type Arc = (); type Edge = (); type Face = (); } let mut graph = MeshGraph::::new(); 

Note

Most examples on this page use the N64 and R64 types from the decorum crate and the Point2 and Point3 types from the nalgebra crate for graph geometry. When the geometry-nalgebra feature is enabled, these types implement GraphGeometry.

The associated types specified by a GraphGeometry implementation determine the type of the geometry field exposed by views. When set to (), no geometry is present.

Geometry is vertex-based, meaning that geometric operations depend on vertex geometry exposing some notion of positional data via the AsPosition trait. If geometry does not have this property, then geometric operations will not be available. Read more about geometric traits and spatial operations here.

## Topological Views¶

MeshGraphs expose views over their topological structure (vertices, arcs, edges, and faces). Views are obtained using keys or iteration and behave similarly to references. They provide the primary API for interacting with a MeshGraph's topology and geometry. There are three types of views summarized below:

Type Traversal Exclusive Geometry Topology
Immutable Yes No Immutable Immutable
Mutable Yes Yes Mutable Mutable
Orphan No No Mutable N/A

Immutable and mutable views behave similarly to references: immutable views cannot mutate a graph and are not exclusive while mutable views may mutate both the geometry and topology of a graph but are exclusive. This example uses a view to traverse a graph:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 type E3 = Point3; // Create a graph with positional and normal data from a unit cube. let cube = Cube::new(); let mut graph = primitive::zip_vertices(( cube.polygons::>(), cube.polygons::>(), )) .map_vertices(|(position, normal)| Vertex::new(position, normal)) .collect::>(); // Get a view of a face and its opposite face. let face = graph.faces().nth(0).expect("cube"); let opposite = face .into_arc() .into_opposite_arc() .into_next_arc() .into_next_arc() .into_opposite_arc() .into_face() .expect("cube"); 

Orphan views are similar to mutable views in that they may mutate the geometry of a graph, but they do not have access to the topology of a graph. Because they do not know about other vertices, arcs, etc., orphan views cannot traverse a graph in any way. These views are most useful for modifying the geometry of a graph and, unlike mutable views, are not exclusive. Iterators over topological structures in a graph sometimes emit orphan views.

 1 2 3 4 5 6 7 8 9 // Create a graph with positional data from a UV-sphere. let mut graph = UvSphere::new(8, 8) .polygons::>>() .collect_with_indexer::>, _>(LruIndexer::default()); // Scale the position data in all vertices. for mut vertex in graph.vertex_orphans() { vertex.geometry *= 2.0; } 

Immutable and mutable views are both represented by view types, such as FaceView. Orphan views are represented by orphan view types, such as FaceOrphan.

## Traversals¶

Topological views can be used to traverse a graph. Traversals are an important feature of graphs that are not supported by iterator expressions or buffers. Most traversals involve some notion of adjacency, but MeshGraph also provides categorical traversals.

### One-to-One¶

Conversions and accessors provide one-to-one traversals from one topological structure to another. Conversions consume a view and emit another view while accessors borrow a view and use that borrow to produce another view. Accessors only expose immutable views with a limited lifetime while conversions expose views with the same lifetime and mutability as the source view.

 1 2 3 4 5 // Get a mutable view of a vertex in a graph. let vertex = graph.vertex_mut(key).unwrap(); let arc = vertex.into_outgoing_arc(); // Consumes vertex. arc is mutable. let opposite = arc.opposite_arc(); // Borrows arc. opposite is immutable. 

Conversions and accessors are distinguished using standard Rust naming conventions. For example, into_outgoing_arc is consuming and outgoing_arc is borrowing; both traverse from a vertex to its outgoing arc.

Note

It is not possible to perform topological mutations using a view obtained via a borrowing traversal, because these views are always immutable.

### One-to-Many¶

A circulator is a type of iterator that provides a one-to-many traversal that examines all of the immediate neighbors of a topological structure. For example, the face circulator of a vertex yields all faces that share that vertex in order.

 1 2 3 4 5 for face in vertex.neighboring_faces() { for arc in face.interior_arcs() { // ... } } 

Circulators generally begin iteration from a leading arc and then traverse topology in a deterministic order from that arc. Because mutability requires orphan views, only the geometry of immediate neighbors can be mutated using circulators.

 1 2 3 for mut face in vertex.neighboring_face_orphans() { face.geometry = Color4::white(); } 

Search traversals visit all connected topological structures with some notion of adjacency. Both vertices and faces can be traversed in this way to perform searches using breadth- and depth-first ordering.

 1 2 3 4 5 6 if let Some(vertex) = vertex .traverse_by_depth() .find(|vertex| vertex.geometry == target) { // ... } 

Warning

It is possible for vertices and faces to be disjoint, meaning that they do not share a path with all other vertices or faces. Therefore, these traversals are only exhaustive with respect to the topologically connected group with which the initiating view is associated; they do not necessarily visit every vertex or face that is a member of a particular graph.

MeshGraphs also directly expose topological structures via iterators, but without a deterministic ordering. These categorical iterators are always exhaustive, and visit all topological structures in a graph regardless of their connectivity.

  1 2 3 4 5 6 7 8 9 10 let (graph, _) = MeshGraph::>::from_ply( PositionEncoding::default(), File::open("flower.ply").unwrap(), ) .unwrap(); // Modify the geometry of every vertex. for mut vertex in graph.vertex_orphans() { vertex.geometry *= 2.0; } 

Mutable iterators (including circulators) emit orphan views, because mutable views require exclusive access. To mutate topology using multiple mutable views, use an immutable circulator to collect the keys of the target topology and then lookup each mutable view using those keys.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 // Create a graph with positional data from a unit cube. let mut graph = Cube::new() .polygons::>>() .collect::>>(); // Collect the keys of the faces in the graph. let keys = graph .faces() .map(|face| face.key()) .collect::>(); for key in keys { // Get a mutable face view for each key. // Poke each face and translate the centroid along the face's normal. let _ = graph.face_mut(key).expect("independent").poke_with_offset(0.5); } 

Note

Mutations may be dependent and invalidate keys. Some mutations may not be able to operate on the given set of keys as trivially as seen in the example above.

### Rekeying¶

Topological views pair a key with a reference to the underlying storage of a graph. Given a view, it is possible to rekey the view to construct a new view using the same underlying storage. It is even possible to rekey between different topological structures, such as rekeying a FaceView into a VertexView.

Rekeying is useful for fallible traversals that maintain mutability. A mutable view can be used to look up a key and, if such a key is found, be rekeyed into that topology. This avoids performing the same traversal more than once in order to query and then convert or mutate.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 let face = graph.face_mut(key).unwrap(); // Find a face along a boundary. If no such face is found, continue to use the // initiating face. let mut face = { let key = face .traverse_by_depth() .find(|face| { face.interior_arcs() .map(|arc| arc.into_opposite_arc()) .any(|arc| arc.is_boundary_arc()) }) .map(|face| face.key()); if let Some(key) = key { face.rekey(key).unwrap() // Rekey into the boundary face. } else { face // Use the initiating face. } }; 

Rekeying can also be useful for code that only operates on topological views and does not have access to the associated MeshGraph, because it allows arbitrary access to the graph's structure.

## Topological Mutations¶

Mutable views expose topological mutations that alter the structure of a graph. These operations are always consuming, because they often invalidate the view that initiates them.

Most mutations return a view over a modified or newly inserted topological structure that can be used to further traverse the graph. For example, splitting an arc $\overrightarrow{AB}$ returns a vertex $M$ that subdivides the composite edge. The leading arc of $M$ is $\overrightarrow{MB}$ and is a part of the same ring as the initiating arc.

 1 let vertex = arc.split_at_midpoint(); // Consumes arc. vertex is mutable. 

It is possible to downgrade mutable views into immutable views using into_ref. This can be useful when performing topological mutations, as it allows for any number of traversals immediately after the mutation is performed.

  1 2 3 4 5 6 7 8 9 10 // Split an arc and use into_ref to get an immutable view. let vertex = arc.split_at_midpoint().into_ref(); // vertex is immutable and implements Copy. let source = vertex .into_outgoing_arc() .into_previous_arc() .into_source_vertex(); let destination = vertex.into_outgoing_arc().into_destination_vertex(); let span = (source, destination); 

Graphs also provide topological mutations that may operate over an entire graph.

  1 2 3 4 5 6 7 8 9 10 type E3 = Point3; let cube = Cube::new(); let mut graph = primitive::zip_vertices(( cube.polygons::>(), cube.polygons::>(), )) .collect::>(); graph.triangulate(); // Triangulates all faces in the graph. 

Graphs may contain disjoint sub-graphs, which are topological groups that cannot reach each other. It is possible to split a graph into sub-graphs.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 // Create a graph of two squares sharing a single edge. let mut graph = MeshGraph::>::from_raw_buffers( vec![Tetragon::new(0usize, 1, 2, 3), Tetragon::new(0, 3, 4, 5)], vec![ (0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0), (-1.0, 1.0), (-1.0, 0.0), ], ) .unwrap(); // Find the shared edge and a get a path along one of its arcs. let key = graph .edges() .find(|edge| !edge.is_boundary_edge()) .map(|edge| edge.key()) .unwrap(); let path = graph.edge_mut(key).unwrap().into_arc().into_path(); // Split the graph into two disjoint sub-graphs. let _ = MeshGraph::split_at_path(path).unwrap(); let graphs = graph.into_disjoint_subgraphs(); 

Topological mutations expose spatial functions for types that implement geometric traits, such as split_at_midpoint. Views also expose purely topological functions, which can always be used (even if the geometry is non-spatial).

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 pub enum Weight {} impl GraphGeometry for Weight { type Vertex = f64; type Arc = (); type Edge = (); type Face = (); } let mut graph = MeshGraph::::from_raw_buffers( vec![Trigon::new(0usize, 1, 2)], vec![1.0, 2.0, 0.5], ) .expect("triangle"); let key = graph.arcs().nth(0).expect("triangle").key(); let vertex = graph.arc_mut(key).unwrap().split_with(|| 0.1); 

In the above example, split_with accepts a function that returns geometry for the subdividing vertex of the split. Similar functions exist for other topological mutations as well, such as poke_with.

When graph geometry implements geometric traits, views expose methods to compute related attributes like normals and centroids.

 1 2 3 4 5 6 7 8 let (graph, _) = MeshGraph::>::from_ply( PositionEncoding::default(), File::open("teapot.ply").unwrap(), ) .unwrap(); // Computes the centroid of the face. let centroid = graph.faces().nth(0).unwrap().centroid(); 

These computations are based on the positional data in vertices. However, it is also possible to include these attributes in the geometry (payload) of a graph and assign arbitrary values as needed. For example, it is sometimes desirable to establish vertex normals independently of surrounding face or edge geometry.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 pub struct Vertex { pub position: Point3, pub normal: Vector3, } impl GraphGeometry for Vertex { type Vertex = Self; type Arc = (); type Edge = (); type Face = (); } let mut graph = Cube::new() .polygons::>>() .map_vertices(|position| Vertex { position, normal: Unit::x(), }) .collect>(); // Write arbitrary data to the payload. let mut vertex = graph.vertex_orphans().nth(0).unwrap(); vertex.geometry.normal = Unit::z(); 

The above example uses the Vertex type to store a position and normal in each vertex. This is distinct from computed attributes, as this normal is arbitrary and is not recomputed when it is accessed. Applications may choose either approach, though computation is typically preferable.

## Generic Programming¶

The graph module provides traits that express the geometric capabilities of a MeshGraph. These can be used to write generic code that requires particular geometric operations, such as computing edge midpoints. This example subdivides a face in a mesh by splitting arcs at their midpoints:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 pub fn circumscribe(face: FaceView<&mut MeshGraph, G>) -> FaceView<&mut MeshGraph, G> where G: EdgeMidpoint + GraphGeometry, G::Vertex: AsPosition, { // Split each edge, stashing the vertex key and moving to the next arc. let arity = face.arity(); let mut arc = face.into_arc(); let mut splits = SmallVec::<[_; 4]>::with_capacity(arity); for _ in 0..arity { let vertex = arc.split_at_midpoint(); splits.push(vertex.key()); arc = vertex.into_outgoing_arc().into_next_arc(); } // Split faces along the vertices from each arc split. let mut face = arc.into_face().unwrap(); for (a, b) in splits.into_iter().perimeter() { face = face.split(ByKey(a), ByKey(b)).unwrap().into_face().unwrap(); } // Return the terminating face of the decomposition. face } 

These traits avoid the need to specify very complex type bounds, but it is also possible to express type bounds directly using traits from the decorum and theon crates.

The following example expresses type bounds for a function that computes the area of faces in two-dimensional graphs:

 1 2 3 4 5 6 7 8 pub fn area(face: FaceView<&MeshGraph, G>) -> Scalar> where G: GraphGeometry, G::Vertex: AsPosition, VertexPosition: EuclideanSpace + FiniteDimensional, { // ... }