Searching Polyhedra by Rotating Half-Planes

The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar one-way sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no well-defined notion in 3D of planar "clockwise rotation". Next follow two results: every polyhedron with r>0 reflex edges can be searched by at most r^2 suitably placed guards, whereas just r guards suffice if the polyhedron is orthogonal. (Minimizing the number of guards to search a given polyhedron is easily seen to be NP-hard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NP-hard, and that deciding if a given target area is searchable at all is strongly PSPACE-hard, even for orthogonal polyhedra. A number of peripheral results are proved en route to these central theorems, and several open problems remain for future work.Comment: 45 pages, 26 figure

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

Advanced Development of Space Structures in Domains of 3D Transformation

The emergence of the new architectural solutions and structural forms of Mengeringhausen, Tsuboi, Safdie, Foster, Calatrava and other creators of magnificent structures, may be taken as an initiation and explosion of inventiveness which has continued up to till present. Consequently, the topic of this paper is to show a part of broad range of structural systems which have not been sufficiently disclosed in Serbia and surroundings, in spite of their attractiveness in contemporary architecture, in terms of space transformations, materialization and technology. The basic properties of all analyzed space structures lies in their geometric shape (Archimedean and Platonic polyhedra, polyhedron structures, and bionic of structures as well), which applies regularity, symmetry, speed of mounting, as well as modularity of the original matrices. Solutions and analyses shown deal with multifunctional space matrices, which make its potential very important both in architectural design and in structural theory. The topic of this paper is to consider the development of matrix structure in context of architectural forms in future, emphasizing the importance of structural geometry and its possible applications

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Meshing Deforming Spacetime for Visualization and Analysis

We introduce a novel paradigm that simplifies the visualization and analysis of data that have a spatially/temporally varying frame of reference. The primary application driver is tokamak fusion plasma, where science variables (e.g., density and temperature) are interpolated in a complex magnetic field-line-following coordinate system. We also see a similar challenge in rotational fluid mechanics, cosmology, and Lagrangian ocean analysis; such physics implies a deforming spacetime and induces high complexity in volume rendering, isosurfacing, and feature tracking, among various visualization tasks. Without loss of generality, this paper proposes an algorithm to build a simplicial complex -- a tetrahedral mesh, for the deforming 3D spacetime derived from two 2D triangular meshes representing consecutive timesteps. Without introducing new nodes, the resulting mesh fills the gap between 2D meshes with tetrahedral cells while satisfying given constraints on how nodes connect between the two input meshes. In the algorithm we first divide the spacetime into smaller partitions to reduce complexity based on the input geometries and constraints. We then independently search for a feasible tessellation of each partition taking nonconvexity into consideration. We demonstrate multiple use cases for a simplified visualization analysis scheme with a synthetic case and fusion plasma applications