On realistic terrains

We study worst-case complexities of visibility and distance structures on terrains under realistic assumptions on edge length ratios and the angles of the triangles, and a more general low-density assumption. We show that the visibility map of a point for a realistic terrain with n triangles has complexity @Q(nn). We also prove that the shortest path between two points p and q on a realistic terrain passes through @Q(n) triangles, and that the bisector of p and q has complexity O(nn). We use these results to show that the shortest path map for any point on a realistic terrain has complexity @Q(nn), and that the Voronoi diagram for any set of m points on a realistic terrain has complexity @W(n+mn) and O((n+m)n). Our results immediately imply more efficient algorithms for computing the various structures on realistic terrains.

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