7 research outputs found
Querying for the Largest Empty Geometric Object in a Desired Location
We study new types of geometric query problems defined as follows: given a
geometric set , preprocess it such that given a query point , the
location of the largest circle that does not contain any member of , but
contains can be reported efficiently. The geometric sets we consider for
are boundaries of convex and simple polygons, and point sets. While we
primarily focus on circles as the desired shape, we also briefly discuss empty
rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission
arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the
results have been improved in Section 5.Please note that the change in title
and abstract indicate that we have expanded the scope of the problems we
stud
Algorithms for computing diffuse reflection paths in polygons
Let s be a point source of light inside a polygon P of n vertices. A polygonal path from s to some point t inside P is called a diffuse reflection path if the turning points of the path lie on polygonal edges of P. We present three different algorithms for computing diffuse reflection paths from s to t inside P. For constructing such a path, the first algorithm uses a greedy method, the second algorithm uses a transformation of a minimum link path, and the third algorithm uses the edge-edge visibility graph of P. The first two algorithms are for polygons without holes, and they run in O(n + k log n) time, where k denotes the number of reflections in the path. The third algorithm is for both polygons with or without holes, and it runs in O(n 2) time. The number of reflections in the path produced by this algorithm can be at most 3 times that of an optimal diffuse reflection path. The problem of computing a diffuse reflection path between two points inside a polygon has not been considered in the past
Localized geometric query problems
A new class of geometric query problems are studied in this paper. We are required to preprocess a set of geometric objects P in the plane, so that for any arbitrary query point q, the largest circle that contains q but does not contain any member of P, can be reported efficiently. The geometric sets that we consider are point sets and boundaries of simple polygons