Quadtree representation of two-dimensional objects is performed with a tree that describes the recursive subdivision of the more complex parts of a picture until the desired resolution is reached. At the end, all the leaves of the tree are square cells that lie completely inside or outside the object. There are two great disadvantages in the use of quadtrees as a representation scheme for objects in geometric modeling system: The amount of memory required for polygonal objects is too great, and it is difficult to recompute the boundary representation of the object after some Boolean operations have been performed. In the present paper a new class of quadtrees, in which nodes may contain zero or one edge, is introduced. By using these quadtrees, storage requirements are reduced and it is possible to obtain the exact backward conversion to boundary representation. Algorithms for the generation of the quadtree, Boolean operations, and recomputation of the boundary representation are presented, and their complexities in time and space are discussed. Three-dimensional algorithms working on octrees are also presented. Their use in the geometric modeling of three-dimensional polyhedral objects is discussed.

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This paper presents a new approach for generating coarse-level approximations of topologically complex models. Dramatic\ud topology reduction is achieved by converting a 3D model to and from a volumetric representation. Our approach produces valid,\ud error-bounded models and supports the creation of approximations that do not interpenetrate the original model, either being completely contained in the input solid or bounding it. Several simple to implement versions of our approach are presented and discussed. We show that these methods perform significantly better than other surface-based approaches when simplifying topologically-rich models such as scene parts and complex mechanical assemblies.Postprint (published version

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Set membership classification and, specifically, the evaluation of a CSG tree, are problems of a certain complexity. Several techniques to speed up these processes have been proposed. This include Active Zones, Geometric Bounds and the Extended Convex Differences Tree.Boxes are the most commonly studied geometric bounds, although other bounds such as spheres, convex hulls and prisms have also been proposed.On the other hand, there is an extended bibliography dealing with convex polyhedra and solving problems for this class of polyhedra. Orthogonal polyhedra are also a class of polyhedra and several problems have been solved for them.In this work we propose orthogonal polyhedra as geometric bounds in the CSG model. CSG primitives are approximated by orthogonal polyhedra, and the orthogonal bound of the object is obtained by applying the corresponding boolean algebra. A specific model for orthogonal polyhedra is presented that facilitates a simple and robust boolean operations algorithm between orthogonal polyhedra. This algorithm has linear complexity (is based on a merging process) and avoids floating-point computation.

In this paper we present algorithms to extract the boundary representation of orthogonal polygons and polyhedra, either manifold or pseudomanifold. The algorithms we develop reconstruct not only the polygons of the boundaries but also the hole-face inclusion relationship. Our algorithms have a simple input so they can be used to convert many different kinds of models to B-Rep. In the 2D case, the input is the set of vertices, and in the 3D case, some small additional information must be supplied for every vertex. All proposed algorithms run in O(n log n) time and use O(n) space, where n is the number of vertices of the input. Moreover, we explain how to use our proposal to extract the boundary from the well-known voxel and octree models as well as from three vertex-based models found in the related literature: the neighbourhood, the EVM, and the weighted vertex list models.