The doubly connected edge list (DCEL), also known as half-edge data structure, is a

data structure In computer science, a data structure is a data organization, management, and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the rel ...

to represent an

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...

of a

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...

in the plane, and

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

s in 3D. This data structure provides efficient manipulation of the topological information associated with the objects in question (vertices, edges, faces). It is used in many

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

s of computational geometry to handle polygonal subdivisions of the plane, commonly called

planar straight-line graph In computational geometry and geometric graph theory, a planar straight-line graph, in short ''PSLG'', (or ''straight-line plane graph'', or ''plane straight-line graph'') is a term used for an embedding of a planar graph in the plane such that ...

s (PSLG). For example, a

Voronoi diagram In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...

is commonly represented by a DCEL inside a bounding box. This data structure was originally suggested by Muller and Preparata for representations of 3D

convex polyhedra A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wor ...

. Later, a somewhat different data structure was suggested, but the name "DCEL" was retained. For simplicity, only

connected graph In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgra ...

s are considered, however the DCEL structure may be extended to handle disconnected graphs as well by introducing dummy edges between disconnected components.

Data structure

DCEL is more than just a

doubly linked list In computer science, a doubly linked list is a linked data structure that consists of a set of sequentially linked records called nodes. Each node contains three fields: two link fields (references to the previous and to the next node in the se ...

of edges. In the general case, a DCEL contains a record for each edge,

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

and

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

of the subdivision. Each record may contain additional information, for example, a face may contain the name of the area. Each edge usually bounds two faces and it is, therefore, convenient to regard each edge as two "half-edges" (which are represented by the two edges with opposite directions, between two vertices, in the picture on the right). Each half-edge is "associated" with a single face and thus has a pointer to that face. ''All'' half-edges associated with a face are clockwise or counter-clockwise. For example, in the picture on the right, all half-edges associated with the middle face (i.e. the "internal" half-edges) are counter-clockwise. A half-edge has a pointer to the next half-edge and previous half-edge of the same face. To reach the other face, we can go to the twin of the half-edge and then traverse the other face. Each half-edge also has a pointer to its origin vertex (the destination vertex can be obtained by querying the origin of its twin, or of the next half-edge). Each vertex contains the coordinates of the vertex and also stores a pointer to an arbitrary edge that has the vertex as its origin. Each face stores a pointer to some half-edge of its outer boundary (if the face is unbounded then pointer is null). It also has a list of half-edges, one for each hole that may be incident within the face. If the vertices or faces do not hold any interesting information, there is no need to store them, thus saving space and reducing the data structure's complexity.

References

{{Reflist Graph data structures Geometric graph theory Geometric data structures

Data structure

See also

References