This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometric objects. We have demonstrated this by developing new algorithms for two kinds of geometric approximation problems that have been motivated by a single driving problem--- the efficient computation and display of smooth solvent-accessible molecular surfaces. The applications of these detail hierarchies are in biochemistry and computer graphics. The smooth solvent-accessible surface of a molecule is useful in studying the structure and interactions of proteins, in particular for attacking the protein-substrate docking problem. We have developed a parallel linear-time algorithm for computing molecular surfaces. Molecular surfaces are equivalent to the weighted ff-hulls. Thus our work is potentially useful in the application areas of ff-hulls which include astronomy and surface modeling, besides biochemistry.
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