Hidden surface removal for axis-parallel polyhedra

An efficient, output-sensitive method for computing the visibility map of a set of axis-parallel polyhedra (i.e. polyhedra with their faces and edges parallel to the coordinate axes) as seen from a given viewpoint is introduced. For nonintersecting polyhedra with n edges in total, the algorithm runs in time O((n+k)log n), where k is the complexity of the visibility map. The method can handle cyclic overlap of the polyhedra and perspective views without any problem. For c-oriented polyhedra (with faces and edges in c orientations, for some constant c) the method can be extended to run in the same time bound. The method can be extended even further to deal with intersecting polyhedra with only a slight increase in the time bound

An intersection-sensitive algorithm for snap rounding

Snap rounding is a method for converting arbitrary-precision arrangements of segments into fixed-precision representation. We present an algorithm for snap rounding with running time O((n+I)logn), where I is the number of intersections between the input segments. In the worst case, our algorithm is an order of magnitude more efficient than the best previously known algorithms. We also propose a variant of the traditional snap-rounding scheme. The new method has all the desirable properties of traditional snap rounding and, in addition, guarantees that the rounded arrangement does not have degree-2 vertices in the interior of edges. This simplified rounded arrangement can also be computed in O((n+I)logn) time

On R-trees with low stabbing number

The R-tree is a well-known bounding-volume hierarchy that is suitable for storing geometric data on secondary memory. Unfortunately, no good analysis of its query time exists. We describe a new algorithm to construct an R-tree for a set of planar objects that has provably good query complexity for point location queries and range queries with ranges of small width. For certain important special cases, our bounds are optimal. We also show how to update the structure dynamically, and we generalize our results to higher-dimensional spaces