Abstract: A method for determining positional relationships among objects represented in a database. A plurality of tiles are defined. A distribution of objects with respect to the tiles is determined. The distribution of objects is compared with respect to the tiles to identify objects fulfilling a primary filter condition related to an interaction of the objects with respect to the tiles. Objects are identified that fulfill a secondary filter condition related to an interaction of the geometries of the objects by analyzing the distribution of objects that fulfill the primary filter condition with respect to the tiles. Objects are identified that fulfill the secondary filter condition by comparing geometries of objects that fulfill the primary filter condition that the analysis does not identify as fulfilling the secondary filter condition.

Abstract: A method for evaluating a spatial query comprises receiving a spatial query defining a query window including a void, identifying an interior circle for the query window, wherein the interior circle includes a void, and processing the spatial query by either (1) modifying the at least one interior circle to exclude the void, and using the modified interior circle to evaluate the spatial query by checking whether a data MBR is inside the modified interior circle and when it does, including the data item in the query result set, or (2) by checking whether a data MBR is inside the interior circle and when it does, checking whether the data MBR intersects the MBRs of any of the voids, and including the data item in the query result set when there is no intersection.

Abstract: Determining relationships among objects represented in a database includes defining a plurality of tiles in the approximation of the first geometry by dividing the approximation of the first geometry in a first direction a plurality of times and dividing the approximation of the first geometry in a second direction perpendicular to the first direction a plurality of times. A second geometry is analyzed to determine if it fulfills a first filter condition with respect to any of the tiles defined in the approximation of the first geometry. If the second geometry fulfills the first filter condition with respect to any of the tiles defined in the first geometry carrying out a mathematical comparison of the first geometry and the second geometry.

Abstract: Quicker and more efficient processing of spatial queries is provided when the query window has holes. A method for evaluating a spatial query comprises receiving a spatial query defining a query window including a void, identifying an interior circle for the query window, wherein the interior circle includes a void, and processing the spatial query efficiently in one of the following two alternate ways: (1) modifying the at least one interior circle to exclude the void, and using the modified interior circle to evaluate the spatial query by checking if a data mbr is inside the modified interior circle and if it does, including the data item in the query result set. Or (2) by checking for a data MBR is inside the interior circle and if so, checking if the data MBR intersects the MBRs of any of the voids, and including the data item in the query result set if there is no intersection.

Abstract: Determining relationships among objects represented in a database includes defining a plurality of tiles in the approximation of the first geometry by dividing the approximation of the first geometry in a first direction a plurality of times and dividing the approximation of the first geometry in a second direction perpendicular to the first direction a plurality of times. A second geometry is analyzed to determine if it fulfills a first filter condition with respect to any of the tiles defined in the approximation of the first geometry. If the second geometry fulfills the first filter condition with respect to any of the tiles defined in the first geometry carrying out a mathematical comparison of the first geometry and the second geometry.