Point location in zones of k-flats in arrangements

Abstract

Let A(H) be the arrangement of a set H of n hyperplanes in d-space. A k-flat is a k-dimensional affine subspace of d-space. The zone of a k-flat f with respect to H is the set of all faces in A(H) that intersect f. In this paper we study some problems on zones of k-flats. Our most important result is a data structure for point location in the zone of a k-flat. This structure uses O(n([d/2]+epsilon) + n(k+epsilon)) preprocessing time and space and has a query time of O(log(2) n). We also show how to test efficiently whether two flats are visible from each other with respect to a set of hyperplanes. Then point location in m faces in arrangements is studied. Our data structure for this problem has size O(n([d/2)]+epsilon) m([d/2]/d)) and the query time is O(log(2) n).