Comparing area and shape distortion on polyhedral-based recursive partitions of the sphere

Abstract

Regular grid sampling structures in the plane are a common spatial framework for many studies. Constructing grids with desirable properties such as equality of area and shape is more di cult on a sphere. We studied the distortion characteristics of recursive partitions of the surface of the globe starting with the octahedron and icosahedron polyhedral models. We used ® ve di erent methods for mapping from the polyhedral model to the surface of the sphere: the Gnomonic projection, Fuller's Dymaxion projection, Snyder's equal area polyhedral projection, direct spherical subdivision, and a recursive polyhedral projection. We increased partition density using both a 4-fold and a 9-fold ratio at each level of recursive subdivision by subdividing to the 8th level with the 4-fold density ratio (65 536 cells per polyhedral face) and to the ® fth level with the 9-fold density ratio (59 049 cells per polyhedral face). We measured the area and perimeter of each cell at each level of recursion for each method on each model using each density ratio. From these basic measurements we calculated the range and standard deviation of the area measurement, and the mean, range, and standard deviation of a compactness measurement de® ned as the ratio of (the ratio of the perimeter to the area of the cell ) to (the ratio of the perimeter to the area of a spherical circle with the same area). We looked at these basic measurements and their statistics using graphs of variation with recursion level, sums of squares analyses of variation, histograms of the distributions, maps of the spatial variation, and correlograms. The Snyder projection performed best in area distortion and the Gnomonic projection performed best in compactness distortion. The Fuller projection and the Sphere method had moderate distortion in both area and compactness relative to the worst methods. There was little di erence in distortion performance between partitions using the 4-fold density ratio and those using the 9-fold density ratio. Partitions based on the icosahedron performed better for all statistics than those based on the octahedron.

Continuous indexing of hierarchical subdivisions of the globe

“…Another partition often discussed is that of an equilateral triangle into nine equivalent equilateral subtriangles (White et al 1998 ). Though we do not address this explicitly, our methods also apply.…”

The Geometrical Distortion of Discrete Global Grid in Multi-resolution Expression

“…Many methods of subdivision and projection models were created such as QTM [2] 、 spherical quadtree [3]、ISEA model [4]、polyhedral equal area projection [5] and so on. In the field of error, the research work focused on the geometrical distortion of partitions in discrete global grid, such as Denis White [6]、GoodChild [7]、Kimerling [8]、Clark [9] and so on. In these error research work, the geometrical distortion of partitions in the same subdivision level were often studied.…”

Parallels plane projection and its geometric features

“…The global grid system consists of a set of approximately congruent and regular polygons that subdivide the earth's spherical surface into a hierarchical structure. This method has many potential applications in large-scale spatial data management and earth surface processing simulations [6,7] . Equivalent map projections (the Lambert and Snyder projection, for example) are widely employed in the global discrete grid system for modeling demands [3] .…”

“…Any map projection, however, cannot avoid length, angular and shape distortion completely because the earth ellipsoid is a non-developable surface. We can control projection distortion errors to meet the needs of certain studies and applications by choosing different projection methods such as azimuthal, cylindrical and conic projections [1] .Recently, the global discrete grid system has been proposed as a representation approach to spatial objects [2][3][4][5][6] . The global grid system consists of a set of approximately congruent and regular polygons that subdivide the earth's spherical surface into a hierarchical structure.…”

“…Recently, the global discrete grid system has been proposed as a representation approach to spatial objects [2][3][4][5][6] . The global grid system consists of a set of approximately congruent and regular polygons that subdivide the earth's spherical surface into a hierarchical structure.…”