Vicinities for Spatial Data Processing: a Statistical Approach to Algorithm Design

Abstract Spatial data processing is often the core function in many information system applications. Algorithm design for these applications generally aims at being worst case optimal for processing efficiency. We propose a different approach applying the notion of vicinity. We partition the object space into grid cells of size adapted to the statistical dimensions of the input data objects for processing, and consider only those data objects sharing the same common grid cells. We describe the processing steps of the algorithm in our approach and analyze the performance. We also experimented with different data patterns in our implementation. We believe that our approach can be efficient and practicable for the computation of geometric intersection and spatial interference detection. These are essentially the core functions in geographic information systems, computer graphics and computer aided design systems as well. We also briefly discuss our understanding of how the grid cell size may affect the performance with regard to varying patterns of the input data objects

Exact Algorithms For Circles On The Sphere

We describe exact representations and algorithms for geometric operations on general circles and circular arcs on the sphere, using integer homogeneous coordinates. The algorithms include testing a point against a circle, computing the intersection of two circles, and ordering three arcs out of the same point. These tools support robust and efficient operations on maps overs the sphere, such as point location and map overlay, and provide a reliable framework for robotics, geographic information systems, and other geometric applications.113267290Aho, A., Johnson, D.S., Karp, R.M., Kosaraju, S.R., McGeoch, C.C., Papadimitriou, C.H., Pevzner, P., (1996) "Theory of Computing: Goals and Directions,", , ManuscriptAndrade, M.V.A., (1999) Representação e Manipulação Exatas de Mapas Esféricos, , Ph. D. Thesis, Institute of Computing, University of Campinas, (In Portuguese)Brönnimann, H., Emiris, I., Pan, V., Pion, S., Computing exact geometric predicates using modular arithmetic with single precision (1997) Proc. 13th Ann. ACM Sympos. on Comput. Geom., pp. 174-182Brönnimann, H., Yvinec, M., Efficient exact evaluation of signs of determinants (1997) Proc. 13th Ann. ACM Sympos. on Comput. Geom., pp. 166-173(1996) Applications Challenges to Computational Geometry, , Technical Report TR-521-96, Princeton UniversityCox, D., Little, J., O'Shea, D., (1992) Ideals, Varieties, and Algorithms, , Springer-VerlagEdelsbrunner, H., Guibas, L., Topologically Sweeping an Arrangement (1989) J. Comput. System Sci., 38, pp. 165-194Finke, U., Hinrichs, K., Overlaying Simply Connected Planar Subdivisions in Linear Time (1995) Proc. 11th Ann. ACM Symp. on Comput. Geom., pp. 119-126Fortune, S., Van Wyk, C.J., Efficient exact arithmetic for computational geometry (1993) Proc. 9th Ann. ACM Sympos. Comput. Geom., pp. 163-172Goodrich, M.T., Guibas, L.J., Hershberger, J., Tannenbaum, P.J., Snap Rounding Line Segments Efficiently in Two and Three Dimensions (1997) Proc. 13th Ann. ACM Symp. on Comput. Geom., pp. 284-293Granlund, T., The GNU Multiple Precision Arithmetic Library, , http://www.gnu.org/manual/gmp/gmp.html, Free Software FoundationGreene, D.H., Yao, F.F., Finite-resolution computational geometry (1986) Proc. 27th Ann. IEEE Sympos. Found. Comput. Sci., pp. 143-152Guibas, L., Marimont, D., Rounding arrangements dynamically (1995) Proc. 11th Ann. ACM Sympos. Comput. Geom., pp. 190-199Guibas, L., Stolfi, J., Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams (1985) ACM Transactions on Graphics, 4 (2), pp. 74-123Halperin, D., Shelton, C.R., A perturbation scheme for spherical arrangements with application to molecular modeling (1998) Computational Geometry: Theory and Applications, 10 (4), pp. 273-288Hoffmann, C.M., The problems of accuracy and robustness in geometric computation (1989) IEEE Computer, 22 (3), pp. 31-42Maguire, D.J., Goodchild, M.F., Rhind, D., (1991) Geographical Information Systems - Principles and Applications, , John Wiley & SonsNelson, G., (1991) Systems Programming with Modula-3, , Prentice HallStolfi, J., (1991) Oriented Protective Geometry - A Framework for Geometric Computations, , Academic PressWu, P.Y.F., Franklin, W.R., A Logic Programming Approach to Cartographic Map Overlay (1990) Canadian Computational Intelligence Journal, 6 (2), pp. 61-70Yap, C.K., Towards Exact Geometric Computation (1993) Proc. 5th Canad. Conf. Comput. Geom, pp. 405-419Yap, C.K., Dubé, T., The exact computation paradigm (1995) Computing in Euclidean Geometry, Volume 1 of Lecture Notes Series on Computing, pp. 452-492. , D.-Z Du and F. K. Hwang, editors, World Scientific Press, Singapore, 2nd. editio