Abstract: Distances between geographic real-world entities modeled as geometric shapes are measured. At least two shapes are positioned in a coordinate space based on their respective sets of points. A non-empty quadrant set from a plurality of non-empty quadrant sets formed of non-empty quadrants in the coordinate space is identified such that the non-empty quadrants of the identified quadrant set associated with each of the at least two shapes satisfy a predetermined condition and a distance between portions of the at least two shapes in the non-empty quadrants of the identified non-empty quadrant set is an optimal value for the requested distance. The distance between the portions of the shapes in the identified quadrant set is determined to be the requested distance.

Abstract: Distances between geographic real-world entities modeled as geometric shapes are measured. At least two shapes are positioned in a coordinate space based on their respective sets of points. A non-empty quadrant set from a plurality of non-empty quadrant sets formed of non-empty quadrants in the coordinate space is identified such that the non-empty quadrants of the identified quadrant set associated with each of the at least two shapes satisfy a predetermined condition and a distance between portions of the at least two shapes in the non-empty quadrants of the identified non-empty quadrant set is an optimal value for the requested distance. The distance between the portions of the shapes in the identified quadrant set is determined to be the requested distance.

Abstract: According to an example, an index for three-dimensional geographic data stores information regarding a set of spherical polygons. The index includes position information of the vertices and edges of the set of spherical polygons in a coordinate system. At least a subset of the edges of the set of spherical polygons are represented as arcs within the index. Geo-positioning information can be determined based on the index.