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 eecient and rather simple solutions to several problems including (i) vertical ray shooting | preprocess a set K of n non-intersecting convex simply-shaped at objects in 3-space, whose xy-projections are fat, for eecient 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 eeciently, (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 eeciently, and (iv) bounded-size segment shooting | preprocess a set C as in (iii), so that the rst object (if exists) hit by a `not-too-long' oriented query segment can be found eeciently. For the rst three problems we construct data structures of size O(s(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 s(n) is the maximum length of (n; s) Davenport-Schinzel sequences. The data structure for the fourth problem is of size O(s(n) log 2 n). The query time in the rst 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 K as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for K, if exists, is a linear order of K, such that, if K1; K2 2 K and K1 lies vertically above K2, then K1 precedes K2.) Lancer de rayons verticaux 3-D et inclusion de points 2-D, requ^ etes de domaines, et lancer d'arcs dans un ensemble d'objets convexes et epais R esum e : Nous pr esentons une nouvelle structure de donn ees pour un ensemble de n objets epais de complexit e born ee dans le plan. Nous l'utilisons pour obtenir des solutions simples et eecaces pour plusieurs probl emes, parmi lesquels (i) lancer de rayons verticaux | un ensemble K de n objets convexes de complexit e born ee et plats en 3-D dont …