Computing the set of all distant horizons of a terrain

We study the problem of computing the set of all distant horizons of a terrain, represented as either: the set of all edges that appear in the set of all distant horizons; the connected sets in the union of all points that appear in the set of all distant horizons (the set of edge fragments); or a search structure to efficiently calculate the edge fragments or edges on a distant horizon from a particular viewing direction. We describe a randomized algorithm that can be used to solve all three forms of the problem with an expected run time of O(n2+ ) for any > 0 where n is the number of edges in the piecewise linear terrain. We show that solving either of the first two versions of the problem is 3SUM hard, and we also construct a terrain with a single local maxima and a quadratic number of edge fragments in the set of all distant horizons, showing that our solution to the second version of the problem is essentially optimal.

• Type: Conference
• Year: 2004
• Where: CCCG
• Authors: William S. Evans, Daniel Archambault, David G. Kirkpatrick
• Keywords: CCCG 2004, CCCG 2007, Distant Horizons, Edge Fragments, Particular Viewing Direction