Minimizing Slope Change in Imprecise 1.5D terrains

A 1.5D terrain is an x-monotone polyline with n vertices. An imprecise 1.5D terrain is a 1.5D terrain with a y-interval at each vertex, rather than a fixed y-coordinate. A realization of an imprecise terrain is a sequence of n y-coordinates—one for each interval— such that each y-coordinate is within its corresponding interval. For certain applications in terrain analysis, it is important to be able to find a realization of an imprecise terrain that is smooth. In this paper we model smoothness by considering the change in slope between consecutive edges of the terrain. The goal is to find a realization of the terrain where the maximum slope change is minimized. We present an exact algorithm that runs in O(n) time.