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. We show that the visibility map of a point for a realistic terrain with <i>n</i> triangles has complexity Θ(<i>n</i>√<i>n</i>). We also prove that the shortest path between two points <i>p</i> and <i>q</i> on a realistic terrain passes through Θ(√<i>n</i>) triangles, and that the bisector between <i>p</i> and <i>q</i> has complexity <i>O</i>(<i>n √n</i>). We use these results to show that the shortest path map for any point on a realistic terrain has complexity Θ(<i>n√n</i>), and that the Voronoi diagram for any set of <i>m</i> points on a realistic terrain has complexity Ω(<i>n + m√n</i>) and <i>O</i>(<i>(n+m)√n</i>). Our results immediately imply more efficient algorithms for computing the various structures on realistic terrains.

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