Abstract: Embeddings of spherical triangles onto a planar surface permit locations on a sphere to be represented as cells on the planar surface. Embeddings define paths based on one or more sets of great circles which can in turn be based on rotations of an icosahedron. Distances between locations as well as locations can be determined as integer values unlike conventional latitude/longitude-based systems that require floating point arithmetic. Some locations correspond to cells on different paths defined by one or more sets of great circles. Distance between two locations can be estimated as a minimum of distances associated with the cell locations on the different paths. Methods for processing data defined with respect to an origin point in three-dimensional space include establishing a set of concentric spherical shells with the origin point as their origin and establishing a discrete global grid on the concentric spherical shells.

Abstract: Embeddings of spherical triangles onto a planar surface permit locations on a sphere to be represented as cells on the planar surface. Embeddings can define paths based on one or more sets of great circles on the sphere which can in turn be based on rotations of an icosahedron about various axes. Distances between locations as well as locations themselves can be determined as integer values unlike conventional latitude/longitude based systems that require floating point arithmetic. Some locations correspond to cells on different paths defined by one or more sets of great circles. Distance between two locations can be estimated as a minimum of distances associated with the cell locations on the different paths. Methods for processing data defined with respect to an origin point in three-dimensional space include establishing a set of concentric spherical shells with the origin point as their origin and establishing a discrete global grid on each of the concentric spherical shells.

Abstract: Embeddings of spherical triangles onto a planar surface permit locations on a sphere to be represented as cells on the planar surface. Embeddings can define paths based on one or more sets of great circles on the sphere which can in turn be based on rotations of an icosahedron about various axes. Distances between locations as well as locations themselves can be determined as integer values unlike conventional latitude/longitude based systems that require floating point arithmetic. Some locations correspond to cells on different paths defined by one or more sets of great circles. Distance between two locations can be estimated as a minimum of distances associated with the cell locations on the different paths. Methods for processing data defined with respect to an origin point in three-dimensional space include establishing a set of concentric spherical shells with the origin point as their origin and establishing a discrete global grid on each of the concentric spherical shells.

Abstract: Spatial location systems can be based on multi-resolution grids such as mixed aperture combinations of hexagonal cells. A particular finest resolution can be selected based on a suitable combination of intermediate grid apertures such as arbitrary combinations of aperture 3, aperture 4, and aperture 7 hexagons. Location identifiers can be uniquely assigned by generating child cells from a parent cell so that the generated child cells do not overlap child cells of other parent cells. One or more child blocks can be used at any resolution to avoid such overlap. For aperture three and aperture four hexagonal cells, blocks of four and three child cells, respectively, can be used to provide unique location identifiers.

Abstract: A method for assigning path address-form location codes to objects represented using aperture 3 hexagon discrete global grid systems in both vector systems and bucket and raster systems in which hexagons in a first resolution are given a linear code and hexagons in subsequent finer resolutions have identifiers added to the linear code, the method iteratively applying the assigning step to further finer resolutions to a maximum resolution. In vector systems each hexagon has seven hexagons in a finer resolution and in raster and bucket systems each hexagon is assigned to be an open or closed generator class, an open generator creating a closed generator in a finer resolution, and a closed generator generating six open generator hexagons and a seventh closed generator hexagon.

Abstract: A method for assigning path address-form location codes to objects represented using aperture 3 hexagon discrete global grid systems in both vector systems and bucket and raster systems in which hexagons in a first resolution are given a linear code and hexagons in subsequent finer resolutions have identifiers added to the linear code, the method iteratively applying the assigning step to further finer resolutions to a maximum resolution. In vector systems each hexagon has seven hexagons in a finer resolution and in raster and bucket systems each hexagon is assigned to be an open or closed generator class, an open generator creating a closed generator in a finer resolution, and a closed generator generating six open generator hexagons and a seventh closed generator hexagon.