@MISC{Eppstein92speed-upsin, author = {David Eppstein}, title = {Speed-ups in Constructive Solid Geometry}, year = {1992} }

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Abstract

We convert constructive solid geometry input to explicit representations of polygons, polyhedra, or more generally d-dimensional polyhedra, in time O(n d ), improving a previous O(n d log n) bound. We then show that any Boolean formula can be preprocessed in time O(n log n/ log log n) so that the value of the formula can be maintained, as variables are changed one by one, in time O(log n/ log log n) per change; this speeds up certain output-sensitive algorithms for constructive solid geometry. 1 Introduction Computational geometry typically deals with explicit representations of geometric input (polygons, polyhedra, etc) as cell complexes of facets. However another representation is possible: a shape may be described implicitly,in terms of the method of its construction. Such a representation is the Constructive Solid Geometry (CSG) formula, in which a shape is built up from primitive shapes such as halfspaces and spheres, by means of Boolean combinations such as set unio...