Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multiresolution models of scalar fields, such as terrains, and static or time-varying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures and in other applications. In this state of the art report, we distinguish between approaches that focus on a specific dimension and those that apply to all dimensions. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplexes, called diamonds, as the modeling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension–independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain
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