Systems and methods for location representation using a discrete global grid system
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. Target locations are assigned in the three-dimensional space using a corresponding index on a spherical shell.
1. A computer-implemented method for processing data defined with respect to a spherical surface, comprising:
with a processor,
assigning a target cell on the spherical surface to an associated spherical data value;
establishing a grid that covers at least a portion of the spherical surface
embedding at least a portion of the grid on a planar surface; and
determining a cell in the embedded grid corresponding to the assigned target cell on the spherical surface, wherein at least a portion of the grid established on the spherical surface is embedded on a planar surface based on selected great circle paths and a cell orientation.
2. The computer-implemented method of claim 1 , wherein the assigned target cell is a first target cell, further comprising:
determining a cell in the embedded grid corresponding a second target cell on the spherical surface; and
estimating a distance from the first target cell to a second target cell in the embedded grid.
3. The computer-implemented method of claim 1 , wherein the assigned target cell is a first target cell, further comprising:
determining a cell in the embedded grid corresponding a second target cell on the spherical surface;
estimating a plurality of distances from the first target cell to a second target cell in the embedded grid; and
selecting a shortest distance among the plurality of distances and displaying an associated path in the embedding.
4. The computer-implemented method of claim 1 , further comprising establishing an array of cells that includes cells at a single grid resolution, or a hierarchical array of cells.
5. The computer-implemented method of claim 4 , wherein the processor is coupled to establish the hierarchical grid based on an embedding of one or more spherical triangles onto a plane.
6. The computer-implemented method of claim 4 , wherein the array of cells is based on at least one set of great circle paths.
7. The computer-implemented method of claim 6 , wherein the great circle paths are associated with Class I, Class II, or Class III great circles, or combinations thereof.
8. The computer-implemented method of claim 7 , wherein the array of cells is a hierarchical array of cells that includes cells associated with a first resolution and a second resolution.
9. The computer-implemented method of claim 1 , wherein the target cell is situated in a selected least common denominator triangle.
10. The computer-implemented method of claim 1 , wherein the target cell is assigned integer coordinates, and a distance to a second cell is determined as an integer value based on the integer coordinates.
11. A navigation system, comprising:
a position receiver situated to detect a plurality of location signals;
a processor coupled to the position receiver to:
establish a location based on the detected location signal;
assign the established location to a cell in an array of cells, wherein the array of cells includes cells situated on a path defined by an embedding of at least a portion of an icosahedral surface corresponding to an associated portion of a spherical surface and the array of cells is based on at least one set of great circles; and
estimate a distance from the established location to a destination location based on the cell associated with the established location and a destination cell associated with the destination.
12. The navigation system of claim 11 , wherein the array of cells defines either cells at a single grid resolution, or a hierarchical grid of cells.
13. The navigation system of claim 12 , wherein the distance is determined based on coordinates associated with the cells associated with the location and the destination.
14. The navigation system of claim 11 , wherein the position receiver is a GPS receiver.
15. The navigation system of claim 11 , wherein the processor is coupled to establish the hierarchical grid based on an embedding of one or more spherical triangles onto a plane.
16. The navigation system of claim 11 , wherein the destination location is associated with a plurality of paths, and the estimated distance is a minimum distance of the distances defined by the plurality of paths.
17. The navigation system of claim 11 , wherein the great circles are Class I great circles, Class I and Class II great circles, Class I and Class III great circles, or combinations thereof.
18. The navigation system of claim 11 , wherein the location is established as a longitude and a latitude.
19. The navigation system of claim 11 , wherein the array of cells is a hierarchical array of cells that includes cells associated with a first resolution and a second resolution.
20. A navigation system, comprising:
a position receiver situated to detect a plurality of location signals;
a processor coupled to the position receiver to:
establish a location based on the detected location signal;
assign the established location to a cell in an array of cells, wherein the array of cells includes cells situated on a path defined by an embedding of at least a portion of an icosahedral surface corresponding to an associated portion of a spherical surface; and
estimate a distance from the established location to a destination location based on the cell associated with the established location and a destination cell associated with the destination, wherein the cell associated with the established location is situated in a selected least common denominator triangle.
21. A method, comprising, in a navigation system that includes
a processor:
receiving a first location;
identifying at least one cell in a cellular grid associated with the first location, wherein the cell is on a path defined by embedding at least a portion of a spherical surface onto a plane based on a selected set of great circle paths on the spherical surface and a selected ordering of cells with respect to the selected set of great circle paths; and
identifying at least one cell in the cellular grid associated with a second location; and
determining a distance between the first location and the second location based on the associated identified cells, wherein the cells are hexagonal, a plurality of cells are identified as corresponding to the second location, and the distance between the first location and the second location is determined as a minimum of the distances between the cell associated with the first location and each of the cells associated with the second location, and wherein the cellular grid is defined based on a selection of one or more of Type I, Type II, and/or Type III Great Circles on the spherical surface, or combinations thereof, and the cells are oriented as Class I, Class II, Class I/III, or Class II/III cells.
22. The method of claim 21 , wherein the cells are hexagonal.
23. The method of claim 21 , wherein the cells associated with the first location and the second location are assigned integer coordinates, and the distance is determined as an integer value based on the integer coordinates.
24. The method of claim 21 , wherein the cells have a Class I orientation, and the cellular grid is defined based on Type I Great Circle paths.
25. The method of claim 21 , wherein the cells have a Class II orientation, and the cellular grid is defined based on both Type I and Type II Great Circle paths.
26. The method of claim 21 , wherein the cells have a Class I/III or Class II/III orientation, and the cellular grid is defined based on both Type I and combined Type III Great Circle paths.
27. A database system, comprising:
a processor configured to store location based data based on an embedding of a spherical grid into a planar grid, wherein each data element is assigned to a cell based on a selection of one or more great circle paths on the spherical surface and an orientation of cells with respect to the one or more great circles.
28. The database system of claim 27 , wherein a location associated with each data item is stored as one or more integer values.
29. The database system of claim 27 , wherein the selection of great circle paths includes selection of one or more Type I, Type II, and/or Type III great circle paths, or combinations thereof, and the cells are oriented as Class I, Class II, Class I/III, or Class II/III cells.