We prove that b c vertex guards are always sufficient and sometimes necessary to guard the surface of an n-vertex polyhedral terrain. We also show that b guards are sometimes necessary to guard the surface of an n-vertex polyhedral terrain.

We describe several fast programs to compute viewsheds and weighted visibility indices for observation points in a raster terrain. These programs explore various tradeoffs between speed and accuracy. We have analyzed many cells of data; there is no strong correlation between a point's elevation and its weighted visibility index. However, the, very few, high visibility points tend to characterize features of the terrain. This work can form a basis for automatically locating observers jointly to cover a terrain region of interest. Keywords: visibility, viewshed, line-of-sight. 1 Introduction Visibility research, calculating lines-of-sight and viewsheds, on terrain databases, is an established, and important, field in GIS. Nevertheless, progress is still possible, since both more powerful Unix workstation hardware, with tens of megabytes of memory and software environments, and much larger amounts of input data are available. The research reported here would not have been feasible on ...

Petersen's theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3-regular graphs. We have developed an O(n log^4 n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

Geometric aspects of the visibility problem in the siting of air defense missile batteries were studied in this project. It theoretically analyzed the problem, produced several new and efficient algorithms, implemented them, and tested them on many cells of data. The theoretical analysis studied the terrain characteristics, formally defined the Hawk siting problem, formalized the optimal placement of observers, and considered the minimum elevation of an airplane flying over varying terrain. Three algorithms for finding the viewshed around a particular observer were studied. R3 is slow but accurate. R2 is much faster, yet almost as accurate. Xdraw computes an approximate viewshed with error bounds. Four visibility index algorithms were studied. VL runs a fixed number of lines of sight out from every possible observer. WeightF weights the points by distance. Weight approximates WeightF, and DEM/LOS skips points along each line of sight for increased speed. The visibility indices of 20 DT...

Several applications of analytical cartography are presented. They include terrain visibility (including visibility indices, viewsheds, and intervisibility), map overlay (including solving roundoff errors with C++ class libraries and computing polygon areas from incomplete information) , mobility, and interpolation and approximation of curves and of terrain (including curves and surfaces in CAD/CAM, smoothing terrains with overdetermined systems of equations, and drainage patterns). General themes become apparent, such as simplicity, robustness, and the tradeoff between different data types. Finally several future applications are discussed, such as the lossy compression of correlated layers, and just good enough computation when high precision is not justified. # to appear, Cartography and Geographic Information Systems, 2000 # http://www.ecse.rpi.edu/Homepages/wrf/research/gisapps/gisapps.pdf 1 Franklin Applications of Analytical Cartography Contents 1 Introduction 3 ...

We present efficient polynomial time algorithms that place bn=2c vertex guards which cover the surface of an n-vertex polyhedral terrain, and similarly, bn=3c edge guards which cover the surface of an n-vertex polyhedral terrain. The time complexity of both algorithms, dominated by the cost of finding a maximum matching in a graph, is O(n ).

We present an optimal \Theta (n)-time algorithm for the selection of a subset of the vertices of an n-vertex plane graph G so that each of the faces of G is covered by (i.e. incident with) one or more of the selected vertices. At most bn=2c vertices are selected, matching the worst-case requirement. Analogous results for edge-covers are developed for two different notions of "coverage". In particular,our linear-time algorithm selects at most n \Gamma 2 edges to strongly cover G, at most bn=3c diagonals to cover G, and in the case where G has no quadrilateral faces, at most bn=3c edges to cover G. All these bounds are optimal in the worst-case. Most of our results flow from the study of a relaxation of thefamiliar notion of a 2-coloring of a plane graph which we call a face-respecting 2-coloring that permits

In this paper we address the problem of computing visibility information on digital terrain models in parallel. We propose a parallel algorithm for computing the visible region of an observation point located on the terrain. The algorithm is based on a sequential triangle-sorting visibility approach proposed in [De Floriani et al. 1989]. Static and dynamic parallelization strategies, both in terms of partitioning criteria and scheduling policies, are discussed. The different parallelization strategies are implemented on an MIMD multicomputer and evaluated through experimental results.

Geometric aspects of the visibility problem in the siting of air defense missile batteries were studied in this project. It theoretically analyzed the problem, produced several new and efficient algorithms, implemented them, and tested them on many cells of data. The theoretical analysis studied the terrain characteristics, formally defined the Hawk siting problem, formalized the optimal placement of observers, and considered the minimum elevation of an airplane flying over varying terrain. Three algorithms for finding the viewshed around a particular observer were studied. R3 is slow but accurate. R2 is much faster, yet almost as accurate. Xdraw computes an approximate viewshed with error bounds. Four visibility index algorithms were studied. VL runs a fixed number of lines of sight out from every possible observer. WeightF weights the points by distance. Weight approximates WeightF, and DEM/LOS skips points along each line of sight for increased speed. The visibility indices of 20 ...