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Environmental Monitoring and Assessment

, Volume 64, Issue 1, pp 93–103 | Cite as

Global Grids from Recursive Diamond Subdivisions of the Surface of an Octahedron or Icosahedron

  • Denis White
Article

Abstract

In recent years a number of methods have been developed for subdividing the surface of the earth to meet the needs of applications in dynamic modeling, survey sampling, and information storage and display. One set of methods uses the surfaces of Platonic solids, or regular polyhedra, as approximations to the surface of the earth. Diamond partitions are similar to recursive subdivisions of the triangular faces of either the octahedron or icosahedron. This method views the surface as either four (octahedron) or ten (icosahedron) tessellated diamonds, where each diamond is composed of two adjacent triangular faces of the figure. The method allows for a recursive partition on each diamond, creating nested sub-dimaonds, that is implementable as a quadtree, including the provision for a Peano or Morton type coding system for addressing the hierarchical pattern of diamonds and their neighborhoods, and for linearizing storage. Furthermore, diamond partitions, in an aperture-4 hierarchy, provide direct access through the addressing system to the aperture-4 hierarchy of hexagons developed on the figure. Diamond partitions provide a nested hierarchy of grid cells for applications that require nesting and diamond cells have radial symmetry for those that require this property. Finally, diamond partitions can be cross-referenced with hierarchical triangle partitions used in other methods.

global grids geometric models sampling grids hierarchical spatial sampling 
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References

  1. Arlinghaus, S.L.: 1993, ‘Central place fractals: theoretical geography in an urban setting’ Fractals in GeographyLam, N.S.N. and DeCola, L. (eds.), Prentice Hall, Englewood Cliffs, NJ, pp. 213–227.Google Scholar
  2. Baumgardner, J.R. and Frederickson, P.O.: 1985, ‘Icosahedral discretization of the two-sphere’ SIAM Journal on Numerical Analysis22(6), 1107–1115.Google Scholar
  3. Bell, S.B.M., Diaz, B.M., Holroyd, F. and Jackson, M.J.: 1983, ‘Spatially referenced methods of processing raster and vector data’ Image and Vision Computing 1(4)211–220.Google Scholar
  4. Commission for Environmental Cooperation, 1997: Ecological Regions of North America: Toward a Common PerspectiveMontreal, Canada, 71 pp.Google Scholar
  5. Dutton, GH.: 1998, A hierarchical coordinate system for geoprocessing and cartographyLecture Notes in Earth Science 79, Springer-Verlag, Berlin, 231 pp.Google Scholar
  6. Fekete, G: 1990, ‘Rendering and managing spherical data with sphere quadtrees’ Proceedings of Visualization ‘90IEEE Computer Society, Los Alamitos, CA, pp. 176–186.Google Scholar
  7. Gibson, L. and Lucas, D.: 1982, ‘Vectorization of raster images using hierarchical methods’ Computer Graphics and Image Processing20, 82–89.Google Scholar
  8. Griinbaum, B. and Shephard, G.C.: 1977, ‘The eighty-one types of isohedral tilings in the plane’ Mathematical Proceedings of the Cambridge Philosophical Society82(2), 177–196.Google Scholar
  9. Heikes, R and Randall, D.A.: 1995, ‘Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: basic design and results oftest’ Monthly Weather Review123, 1862–1880.Google Scholar
  10. Huang, H. C. and Cressie, N.: 1997, ‘Multiscale spatial modeling’ Proceedings of the Section on Statistics and the EnvironmentAmerican Statistical Association, Alexandria, VA, pp. 49–54.Google Scholar
  11. Kimerling, A.J., Sahr, K., White, D. and Song, L.: in press, ‘Comparing geometrical properties of discrete global grids’ Cartography and Geographic Information Systems. Google Scholar
  12. Langran, G and Chrisman, N.R.: 1988, ‘A framework for temporal geographic information’ Cartographica25(3), 1–14.Google Scholar
  13. Lee, M. and Samet, H.: 1998, ‘Traversing the triangle elements of an icosahedral spherical representation in constant time’ Proceedings 8th International Symposium on Spatial Data HandlingInternational Geographical Union, Burnaby, BC, Canada, pp. 22–33.Google Scholar
  14. Mark, D.M.: 1990, ‘Neighbor-based properties of some orderings of two-dimensional space’ Geographical Analysis22(2), 145–157.Google Scholar
  15. Murray, B.G., Jr.: 1967, ‘Dispersal in vertebrates’ Ecology 48(6), 975–978.Google Scholar
  16. Olsen, A.R., Stevens, D.L., Jr. and White, D.: 1998, ‘Application of global grids in environmental sampling’ Proceedings of the 30th Symposium on the Interface, Computing Science and Statistics30, 279–284.Google Scholar
  17. Saalfeld, A.: 1998, ‘Sorting spatial data for sampling and other geographic applications’ GeoInformatica2, 37–57.Google Scholar
  18. Sahr, K. and White, D.: 1998, ‘Discrete global grid systems’ Proceedings of the 30th Symposium on the Interface, Computing Science and Statistics30, 269–278.Google Scholar
  19. Samet, H.: 1984, ‘The quadtree and related hierarchical data structures’ Computing Surveys16(2), 188–260.Google Scholar
  20. Stevens, D.L., Jr.: 1997, ‘Variable density grid-based sampling designs for continuous spatial populations’ Environmetrics8, 167–195.Google Scholar
  21. Thubum, J.: 1997, ‘A PV based shallow-water model on a hexagonal-icosahedral grid’ Monthly Weather Review125(9), 2328–2347.Google Scholar
  22. White, D., Kimerling, A.J. and Overton, W.S.: 1992, ‘Cartographic and geometric components of a global sampling design for environmental monitoring’ Cartography and Geographic Information Systems 19(1)5-22.Google Scholar
  23. White, D., Kimerling, A.J., Sahr, K. and Song, L.: 1998, ‘Comparing area and shape distortion on polyhedral-based recursive partitions of the sphere’ International Journal of Geographical Information Science12(8), 805–827.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Denis White
    • 1
  1. 1.National Health and Environmental Effects Research Laboratory, Western Ecology DivisionU.S. Environmental Protection AgencyCorvallisUSA

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