Several solid representation schemes exist today in solid modeling. As none of them has properties that are uniformly better than any other, the need for modeling systems working on two or more different schemes which perform every operation in the most suitable scheme, became evident. Then it is necessary to provide geometric modeling systems with the ability of converting from one scheme into another scheme. As most solid modellers use either, a boundary representation (BR) or a constructive solid geometry representation (CSG), conversion from one into the other appears to be of capital importance. Conversion from CSG into BR has been carefully studied but a few attention has been deserved to the conversion from BR to CSG. This paper presents an algorithm which generates CSG binary trees from BR representations over a restricted class of polyhedra. The tree is optimal in the output size. Unfortunately, the algorithm does not in general warrant a correct conversion and, consequently some open questions and ideas for further research are presented.

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.