1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (T ) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size O(n 2 ) and a witness set of size O(n 3 ) have been presented, where n is the number of vertices on T . We show that there exists an even smaller FDS of cardinality O(k) and a witness set of cardinality O(n), where k is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that k ≤ n for n ≥ 2 and since it is possible to construct terrains such that n = 2 k , the existence of an FDS with cardinality O(k) and a witness set of cardinality of O(n) leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem.

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Locating a minimum number of guards on a terrain such that every point on the terrain is guarded by at least one of the guards is known as the Terrain Guarding Problem (TGP). In this paper, a realistic example of the terrain guarding problem is studied, involving the surveillance of a rugged geographical terrain by means of thermal cameras. A number of issues related to TGP are addressed with integer-programming models proposed to solve the problem. Also, a sensitivity analysis is carried out in which five fictitious terrains are created to see the effect of the resolution of the terrain, and of terrain characteristics, on coverage optimization and the required number of guards. Finally, a new problem, which is called the Blocking Path Problem (BPP), is introduced. BPP is about guarding a path on the terrain with a minimum number of guards such that the path blocks all possible infiltration routes. A discussion is provided about the relation of BPP to the Network Interdiction Problem (NIP), which has been studied extensively by the operations research community, and to the k-Barrier Coverage Problem, which has been studied under the Sensor Deployment Problem. BPP is solved via an integer-programming formulation based on a network paradigm.

We model the projection of a triangle onto another triangle when viewed from a given viewpoint in 3D space. The motivation arises from the need to calculate the viewshed of a viewpoint on a triangulated terrain. A triangulated terrain (TIN) is a representation of a real terrain, where the surface of the TIN is composed of triangles. Calculating the viewshed involves finding the invisible region on a triangle caused by the terrain surface. To this end, some studies either projected the vertices of the horizon of the terrain or projected the vertices of a triangle directly onto the supporting plane of the triangle of interest, and then connected the projections to find the invisible region on the target triangle. Such a projection involves sending a ray from the viewpoint that passes through the vertex of the horizon or the vertex of the triangle, and finding out where this ray hits on the supporting plane of the target triangle. These studies assumed that such a ray hits the supporting plane of the triangle in front of the viewpoint. Our key contribution is to show, by a counter example, that the ray may hit the plane behind the viewpoint. Taking into account this fact, we show that the projection of a triangle onto another triangle is characterized by a system of nonlinear equations, which are linearized to obtain a polyhedron. Our approach can be extended to projecting objects of general shapes.