The field of solid modeling deals with design and representation of physical objects. The two major representation schemata used in solid modeling are constructive solid geometry (CSG) and boundary representations (B-rep). Both these representations have different inherent strengths and weaknesses and for most applications both are desired. Currently, most solid models are able to support solids composed of polyhedral models and quadric surfaces (like spheres, cylinders etc.) and their Boolean combinations. At the same time, the field of geometric modeling has been developed to model classes of piecewise surfaces based on particular conditions of shape and smoothness. Such models are also referred to as sculptured models . Over the years there has been considerable effort put into integrating geometric and solid modeling. In particular, there is considerable interest in building complete solid representations from spline surfaces and their Boolean combinations. However, the major bottle necks are in performing robust, efficient and accurate Boolean operations on the sculptured models. The main difficulty is in evaluating and representing the intersection of parametric surface patches and it has hindered the development of solid modelers that incorporate parametric surface models. The topology of a surface patch becomes quite complicated when Boolean operations are performed and finding a convenient representation for these topologies has been a major challenge. As a result, most of the current solid modelers use polyhedral approximations to these surfaces and apply existing algorithms to design and manipulate these polyhedral objects. Not only does this approach lead to data proliferation, the resulting algorithms are inefficient and inaccurate.
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