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Halfedge Data Structures

Introduction

A halfedge data structure is an edge-centered data structure capable of maintaining incidenceinformations of vertices, edges and facets, for example for planar maps or polyhedrons. Each edge isdecomposed into two halfedges with opposite orientations. One incident facet and one incident vertexare stored in each halfedge. For each facet and each vertex one incident halfedge is stored. Reducedvariants of the halfedge data structure can omit some of these informations, for example the halfedgepointers in vertices or the storage of vertices at all.

The data structure provided here is known as the FE-structure [Wei85], as halfedges [Män88, BFH95] oras the doubly connected edge list (DCEL) [dBvKOS97], although the original reference for theDCEL [MP78] describes a different data structure. We continue naming it halfedge data structure(HDS). This data structure can also be seen as one of the variants of the quad-edge datastructure [GS85], reduced to support only orientable surfaces. In general, the quad-edge data structurehas a richer modeling space and can represent non-orientable 2-manifolds. The dual and the primalgraph are symmetrically represented; the same algorithms can be applied to the dual graph as well as tothe primal graph. This implies a lack of static type checking to distinguish between facets and vertices.Even though halfedges are similar to the winged edge data structure [Bau75] or the original reference forthe DCEL [MP78], they are not equivalent, since the traversal algorithms are forced to search for certainincidences where the information is directly available in the halfedge data structure. An overview andcomparison of these different data structures together with a thorough description of the designimplemented here can be found in [Ket98].

Design Overview

The design strictly separates topology and geometry. Vertices, halfedges and facets carry both kinds ofinformation. The Halfedge_data_structure acts as a container class and stores all items and manages

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Requirements for a Halfedge

Definition

A halfedge must store a pointer to the next halfedge and a pointer to its opposite halfedge. It optionallystores pointers to its incident vertex and incident facet. Type tags indicate whether these memberfunctions are supported. Figure depicts the relationship between a halfedge and its incident halfedges,vertices, and facets. The is_border() predicate is mandatory, but may return always false. A halfedge isan oriented edge between two vertices. It is always paired with its counterpart that has the oppositedirection. They are mutually linked with the opposite() member function.

Types

Halfedge::Vertex corresponding vertex type.

Halfedge::Halfedge self.

Halfedge::Facet corresponding facet type.

Operations

Halfedge* h.opposite () the opposite halfedge. const Halfedge* h.opposite () const Halfedge* h.next () the next halfedge around the facet. const Halfedge* h.next () const Halfedge* h.prev () the previous halfedge around the facet. const Halfedge* h.prev () const Vertex* h.vertex () the incident vertex. const Vertex* h.vertex () const Facet* h.facet () the incident facet. If h is a border halfedge the

result might be NULL or a unique facetrepresenting this hole or a unique facetrepresenting all holes.

const Facet* h.facet () const bool h.is_border () const

is true if h is a border halfedge. void h.set_next ( Halfedge* next) void h.set_prev ( Halfedge* prev) void h.set_vertex ( Vertex* v) void h.set_facet ( Facet* f)

if f == NULL h becomes a border edge.

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