... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.

Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a range-searching problem. A typical range-searching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.

An output-sensitive visibility algorithm is one whose runtime is proportional to the number of visible graphic primitives in a scene model---not to the total number of primitives, which can be much greater. The known practical output-sensitive visibility algorithms are suitable only for static scenes, because they include a heavy preprocessing stage that constructs a spatial data structure which relies on the model objects' positions. Any changes to the scene geometry might cause significant modifications to this data structure. We show how these algorithms may be adapted to dynamic scenes. Two main ideas are used: first, update the spatial data structure to reflect the dynamic objects' current positions; make this update efficient by restricting it to a small part of the data structure. Second, use temporal bounding volumes (TBVs) to avoid having to consider every dynamic object in each frame. The combination of these techniques yields efficient, output-sensitive visibility algorithms for scenes with multiple dynamic objects. The performance of our methods is shown to be significantly better than previous output-sensitive algorithms, intended for static scenes. TBVs can be

The repetitive hidden-surface-removal problem can be rephrased as the problem of finding the most compact representation of all views of a polyhedral scene that allows efficient on-line retrieval of a single view. We assume that a polyhedral scene in 3-space is given in advance and is preprocessed off-line into a data structure. Afterwards, the data structure is accessed repeatedly with view-points given on-line and the portions of the polyhedra visible from each view-point are produced on-line. This mode of operation is close to that of real interactive display systems. The main difficulty is to preprocess the scene without knowing the query view-points. In this paper we present a novel approach to this problem. Let n be the total number of edges, vertices and faces of the polyhedral objects and let k be the number of vertices and edges of the image. The main result of this paper is that, using an off-line data structure of size m with n 1+ffl m n 2+ffl , it is possibl...

This report discusses some trends and achievements in computational geometry during the past five years, with emphasis on problems related to computer graphics. Furthermore, a direction of research in computational geometry is discussed, which could help in bringing the fields of computational geometry and computer graphics closer together.

We define a new model of complexity, called object complexity, for measuring the performance of hidden-surface removal algorithms. This model is more appropriate for predicting the performance of these algorithms on current graphics rendering systems than the standard measure of scene complexity used in computational geometry. We also

We propose a hybrid image-space/object-space solution to the classical hidden surface removal problem: Given n disjoint triangles in IR 3 and p sample points (\pixels") in the xy-plane, determine the rst triangle directly behind each pixel. Our algorithm constructs the sampled visibility map of the triangles with respect to the pixels, which is the subset of the trapezoids in a trapezoidal decomposition of the analytic visibility map that contain at least one pixel. The sampled visibility map adapts to local changes in image complexity, and its complexity is bounded both by the number of pixels and by the complexity of the analytic visibility map. Our algorithm runs in time O(n 1+" + n 2=3+" t 2=3 + p), where t is the output size. This is nearly optimal in the worst case and compares favorably with the best output-sensitive algorithms for both ray casting and analytic hidden surface removal. In the special case where the pixels form a regular grid, a sweepline variant of our a...