3-D Vertical Ray Shooting and 2-D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects

We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting--preprocess a set/C of n non-intersecting convex simply-shaped fiat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure--preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching--preprocess a set C of n convex fat polygons, so that the k objects intersecting a "not-too-large" query polygon can be reported efficiently, and (iv) bounded-size segment shooting--preprocess a set C as in (iii), so that the first object (if exists) hit by a "not-too-long" oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(As(n)log 3 n), where s is the maximum number of intersections between the boundaries of the (xy-projections) of any pair of objects, and As(n) is the maximum length of (n, s) Davenport-Schinzel sequences. The data structure for the fourth problem is of size O(As(n) log 2 n). The query time in the first problem is O(log 4 n), the query time in the second and third problems is O(log 3 n + k log 2 n), and the query time in the fourth problem is O(log 3 n). We also present a simple algorithm for computing a depth order for a set /C as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for/C, if exists, is a linear order of/C, such that, if KI, K2 C/C and K1 lies vertically above K2, then K1 precedes K2.) Unlike the algorithm of Agarwal et al. (1995) that might output a false order when a depth order does not exist, the new algorithm is able to determine whether such an order exists, and it is often more efficient in practical situations than the former algorithm. © 1997 Elsevier Science B.V.

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