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Incoming links: Boundary representation
Boundary representation 2026-10-31
A boundary representation specifies a solid through its oriented bounding faces and their edge and vertex of a polyhedron incidence. Supporting planes and coordinates supply geometry; face loops and adjacency supply topology. A valid closed manifold model requires compatible orientation, two incident faces per edge and a single local fan at each vertex of a polyhedron, together with an embedded boundary.
For an ordinary closed manifold boundary representation, three requirements are:
- Valid embedded faces. Each face is planar and bounded by simple closed edge loops with nonzero area; distinct face interiors do not cross, and faces meet only in their declared common edges or vertices of a polyhedron. A quadrilateral with four noncoplanar vertices of a polyhedron is not one planar face; a bow-tie loop does not bound a simple face. Holes, when allowed, need separate correctly oriented loops.
- Closed manifold incidence. Exactly two faces meet at each edge, and the incident faces around each vertex of a polyhedron form one cyclic fan. A single missing face leaves edges incident with only one face. Three faces sharing an edge, or two closed shells touching only at one vertex of a polyhedron, violate the local surface condition. The second example shows why the edge count alone is insufficient.
- Consistent orientation. Adjacent oriented faces traverse their common edge in opposite directions, giving compatible outward normal vectors. Reversing just one face of a tetrahedron breaks this condition. For a cavity, the material's outward normal vector points into the cavity, so shell orientation must also respect which region is solid.
Together these support the answerFor a connected orientable closed boundary of genus , the Euler characteristic provides an additional check, , when faces are disk-like. The familiar value applies to a sphere-like shell; it must not be imposed blindly on every polyhedron. Correct counts alone do not exclude intersections or invalid local incidence.
A boundary representation separates geometry, which supplies locations and supporting planes, from topology, which specifies incidence and orientation. A practical winged-edge data structure uses three related record types:
- A vertex of a polyhedron record holds its three coordinates and a reference to one incident edge, sufficient to begin traversing the local fan.
- A face record holds its supporting plane, an outward normal vector or equivalent orientation, and a reference to a boundary edge. Faces with holes need one entry per boundary loop.
- An edge record holds its two endpoint vertices of a polyhedron, its left and right incident faces, and predecessor and successor edges around each of those faces: four “wing” references. “Left” and “right” refer to a chosen orientation of the edge and compatible face traversal conventions.
This explicitly stores adjacency rather than recovering it by searching all faces. A half-edge data structure expresses much the same information with two oppositely directed records per undirected edge: each has an origin vertex of a polyhedron, an incident face, its next boundary record and its opposite record; a previous record is often stored as well. The destination can be obtained from the opposite record's origin. This mesh record is distinct from a dangling half-edge of a graph.
The compact implementation answer isThere is deliberate redundancy: the same adjacency can be reached from both an edge and a face, a stored previous edge is recoverable by walking the loop, and a face normal vector can be computed from its ordered noncollinear vertices of a polyhedron. These copies make neighbourhood queries and local edits cheap, and cached plane equations speed geometric tests. They impose consistency obligations: an edit must update reciprocal references, loops, orientations and affected geometric caches together. A half-edge data structure should share the undirected edge's geometry or keep its paired records synchronized.
Vertex of a polyhedron 2026-10-31
A corner of a polyhedron, stored with its spatial coordinates and incidences with boundary edges and faces. In a closed manifold boundary representation, the incident faces form a single cyclic fan. The vertices of a polyhedron and edges define a graph, but geometric embedding and valid face incidence require additional information.