Computing Realistic Terrains from Imprecise Elevations

Abstract

In the imprecise 2.5D terrain model, each vertex of a triangulated terrain has precise x- and y-coordinates, but the elevation (z-coordinate) is an imprecise value only known to lie within some interval. The goal is to choose elevation values from the intervals so that the resulting precise terrain is "realistic" as captured by some objective function.

We consider four objectives: #1 minimizing local extrema; #2 optimizing coplanar features; #3 minimizing surface area; #4 minimizing maximum steepness.

We also consider the problems down a dimension in 1.5D, where a terrain is a poly-line with precise x-coordinates and imprecise y-coordinate elevations. In 1.5D we reduce problems #1, #3, and #4 to a shortest path problem, and show that problem #2 can be 2-approximated via a minimum link path.

In 2.5D, problem #1 was proved NP-hard by Gray et al.~[Computational Geometry, 2012]. We give a polynomial time algorithm
for a triangulation of a polygon. We prove that problem #2 is strongly NP-complete, but give a constant-factor approximation when the triangles form a path and lie in a strip. We show that problems #3 and #4 can be solved efficiently via Second Order Cone Programming.