Graphics in flatland revisited

Abstract

Let S be a set of n non-intersecting convex objects in the plane. A view of S from a given viewpoint p is the circular sequence of objets intersected by a semi-infinite line rotating continuously around the viewpoint p. We show how to preprocess the set S in time O(n ² · α(n)) where α(n) is a pseudo-inverse of Ackermann's function, so that the view from a given point in the plane can be computed in time O(n · 2^α(n)). Then we show how, with the same preprocessing time, the view can be maintained in O(log n) time per change while advancing the viewpoint along a given connected curve lying in the complement of the convex hull of S. Both algorithms use O(n ²) storage. In the process we show how to reduce by a log n factor the preprocessing time of the ray-shooting problem and we investigate a preprocessing version of the edge visibility problem. This work also serves to demonstrate that the geometric duality between the plane and the set of oriented straight lines of the plane is a “good” framework in dealing with objects more general then the classical points and segments which we usually meet in the context of computational geometry.