Terrain Visibility With Multiple Viewpoints

Abstract

We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with m > 1 viewpoints (or guards) located on the terrain surface. A point on the terrain is considered visible if it has an unobstructed line of sight to at least one viewpoint. We study several natural and fundamental visibility structures: (1) the visibility map, which is a partition of the terrain into visible and invisible regions; (2) the colored visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and (3) the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting. To the best of our knowledge, the other structures have not been studied before.

Altitude terrain guarding and guarding uni-monotone polygons

“…INPUT : Terrain T , altitude line A, its leftmost point a, sets C, O, S of closing, opening, and soft opening points for all edges in T , all ordered from left to right. OUTPUT: An optimal guard set G. 1 E g = E(T ) // set of edges that still need to be guarded 2 i := 1 3 g 0 := a // the point on A before the first guard is a, g 0 is NOT a guard 4 while E g = ∅ // as long as there are still unseen edges 5 do Shoot a visibility ray from g i onto e // We shoot a ray from g i though all vertices to the right of it, and then check if one of them is the occluding vertex, we use the ray through this occluding vertex 17 Let the intersection point be r e // all points on e to the right of r e (incl. r e ) are seen 18 Identify the mark m e immediately to the right of r e on e…”

“…Moreover, as g i : p o e ≤ g i ≤ p c e , e is never completely deleted from E g in lines 10-12. Consequently, for some i we have p o e > g i and g i ≥ p s e (lines [14][15][16][17][18][19][20][21][22]. As p / ∈ V T (G), we have p ∈ e ⊂ e (e being the still unseen interval of e).…”

Algorithms for use of modeling methods in selection process of hydrotechnical construction project

“…Figure 3 in project variant 1, we take slope i=1:2.5, i=1:3 and make artificial horizontals parallel to the main body of the dam. The process of transferring the artificial horizontals is also found using the formula for transferring a parallel line by placing the distance between the given line from above, i.e., the slope scale lines to the main contour of the dam are λ=2.5 m, λ=3 m [11,13]. We draw a parallel straight line at a distance of 3m in Fig.…”

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“…They describe three different strategies for constructing the sparse network and present experimental results showing that their approach is significantly faster than others, and that the different strategies offer a trade-off between higher-quality paths and lower preprocessing times. Hurtado et al [2014] present a pioneering theoretical study about visibility in polyhedral terrains in the presence of multiple viewpoints. They analyze the complexity and describe algorithms to compute three visibility structures: the visibility map, which is a partition of the terrain into visible and invisible regions; the colored visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint.…”

“…They analyze the complexity and describe algorithms to compute three visibility structures: the visibility map, which is a partition of the terrain into visible and invisible regions; the colored visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint. For example, in Theorem 11 of Hurtado et al [2014], the authors prove that the colored visibility map of a polyhedral terrain may be computed in O(m(nα(n) + min(k c , n 2 )) log n + mk c ) time, where n is the number of vertices, m is the number of viewpoints, α(·) is the inverse Ackermann's function (for practical purposes, a constant), and k c is the size of the colored visibility map. The max size of k c = (m 2 n 2 ).…”