Abstract

We study the complexity of the visibility map of terrains whose triangles are fat, not too steep and have roughly the same size. It is known that the complexity of the visibility map of such a terrain with n triangles is θ(n2) in the worst case. We prove that if the elevations of the vertices of the terrain are subject to uniform noise which is proportional to the edge lengths, then the worst-case expected (smoothed) complexity is only θ(n). This provides an explanation why visibility maps of superlinear complexity are unlikely to be encountered in practice.

Highlights

  • We study the smoothed complexity of the visibility map of T for perspective views, that is, the map as it appears in the projection on a viewing plane hview for a given viewing point pview

  • We prove a tight upper bound for the smoothed complexity of the visibility map of such a terrain that contains at most k bad triangles

  • We proved that the smoothed complexity of the visibility map of not-too-steep terrains with fat triangles of similar size is O(n)

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Summary

Introduction

We assume that the fatness φ, steepness θ, and scale factor σ of the unperturbed terrain are constants that are independent of the number of triangles n, with φ > 0, and θ < π/2, and σ 1 These assumptions are used in other papers [1, 10], the steepness assumption is not needed for the specific result on visibility maps by Moet et al In her thesis, Moet [9] experimentally investigates terrain models of various mountainous regions in the US. Our main result is that the smoothed complexity of any visibility map of a terrain satisfying the abovementioned assumptions is only Θ(n) This result can be generalized to certain nonuniform noise distributions. To avoid technical details regarding what happens if one looks at the boundary of the terrain from the side, we focus on the case of perspective views with the view point being located above the terrain

Visibility maps resulting from perspective projection
Terrains that almost satisfy the assumptions
Concluding remarks
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