mirror of https://github.com/CGAL/cgal
63 lines
3.2 KiB
TeX
63 lines
3.2 KiB
TeX
A halfedge data structure \cite{hds} is an edge-centered data
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structure capable of maintaining incidence informations of vertices,
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edges and faces, for planar maps, polyhedron, or other orientable,
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two-dimensional surfaces embedded in arbitrary dimension. Each edge is
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decomposed into two halfedges with opposite orientations. Two types
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of halfedge can be chosen for the data structure: facet-edge pair or
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vertex-edge pair. One incident face and one incident vertex are
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stored in each halfedge. For each face and each vertex, one incident
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halfedge is stored (see Fig.\ref{fig:halfedge}). The halfedge is
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designated as the connectivity primitive. Connectivity manipulations
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are based on the reconfiguration and the mesh traversal is equivalent
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to the adjacency walking of the halfedges. Halfedge data structures
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have been very successful for the design of algorithms on meshes for
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several reasons:
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\begin{itemize}
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\item halfedges contain a constant number of adjacencies.
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\item halfedges encode the mesh orientation.
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\item navigation around vertices or facets is easy.
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\item corner attributes, such as crease normals, is allowed
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to be associated with halfedges.
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\end{itemize}
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\begin{figure}[htb]
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\centering{\includegraphics[width=5.0cm]{figs/halfedge}}
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\caption{One halfedge and its incident primitives.}
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\label{fig:halfedge}
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\end{figure}
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Halfedge data structure is a combinatorial data structure. The
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geometric interpretation is instantiated with a defined kernel. A
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geometry kernel defines the dimensional primitives such as points and
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vectors and the geometry metrics such as distances and angles. A
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polyhedron is a constrained combinatorial structure that can only
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represent a 2-manifold. A 2-manifold is a surface where every point is
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embedded by a disk (interior points) or a half disk (boundary
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points). Polyhedrons are usually interpreted as the shell of a solid
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bounded by plane polygons in the 3D Euclidean space.
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%% In the \poly, the \hds\ is interfaced and indirectly manipulated
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%% by the \poly. Since \hds\ is the internal representation of the
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%% connectivity structure, primitives travesal of the \poly\ is
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%% supported by the \hds . As a container, the \hds\ provides
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%% iterators of the primitives, i.e. vertices, halfedges and faces.
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%% This primitive iterators visit the primitives on the storage
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%% order of the internal list or vector. As a connected graph,
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%% the \hds\ provides circulators around vertices or faces.
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%% The circulator visits the halfedges ajacent to the vertex
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%% in CW order and the face in CCW order. Details of using iterators
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%% and circulators can be found at \ref{sec:??}.
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%% Halfedges in a \hds\ also provide a set of low-level
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%% traversal operators. These operators include the prev(), next(),
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%% opposite(), vertex(), and face(). The prev(), next() and opposite()
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%% return the handle of the previous, next and opposite
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%% halfedge respectively. The vertex() returns the handle of the
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%% incidence vertex and the face() returns the handle of the
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%% incidence face. The next() and opposite() are mandatory and others
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%% are optionally support. Though in this tutorial, a compelete
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%% support is assumed. For how to define a partially support \hds ,
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%% readers should refer to \cite{cgalmanul}.
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