Abstract:
Retrieving points that can be plotted in a predetermined area are achieved where the locations of the points are indexed in an index that includes regional data. The regional data defines a plurality of regions, and each region encompasses one or more of the points. In addition, the index includes linking data, which, for each region, identifies the point or points encompassed by that region. The method includes (i) reviewing the regional data in the index to identify regions that are wholly contained by the predetermined area; and (ii) reviewing the linking data to retrieve points encompassed by the identified regions.

Description:
This application is the US national phase of international application PCT/GB01/05479 filed 11 Dec. 2001 which designated the U.S. 
   BACKGROUND 
   1. Technical Field 
   The present invention relates to a method of retrieval, and is suitable for use with entities stored in a database, or equivalent storage. 
   2. Related Art 
   It can readily be seen that when there are vast numbers of entities in a database, identifying entities in accordance with a query in respect of data in the database within a reasonable period of time is a non-trivial exercise. To ease the retrieval process, data in a database is generally indexed in some way, and queries are then performed on the index. The way in which the entities are indexed can be expected to have a significant bearing on the quality and speed of retrieval, and as information is increasingly being stored in databases, there is significant interest in finding improved ways of indexing data. 
   It is known to index location data based on place names. It is also known to retrieve a set of geographic coordinates from place names, and build an index based on topological information extracted from the coordinates (e.g. “GIPSY”: developed at U.C. Berkeley in conjunction with a joint NSF/NASA/ARPA (Wilensky et al., 1994) initiative). Furthermore, it is known to build an index based on the geographical coordinates themselves: database vendors such as Oracle™ have developed systems for storing and indexing geometrical data—e.g. Oracle spatial data cartridge, which allows a spatial querying to be carried out using an extended (non-standard) form of SQL. Other vendors, like Maplnfo™, SpatialWare™, Innogistic™ and Informix™ have similar proprietary ways of dealing with spatial data. In particular, Innogistic™ have developed a product known as Cartology DSI, which stores geometrical vector data as blobs (binary large objects—which are not intrinsically recognisable by the underlying database). It also creates indexes outside of the database based on the well-known ‘quad tree’ idea. The index data is stored in binary-tree structures and is accessed by Distributed Component Object Model (DCOM) middleware services. 
   Both the Oracle™ and lnnogistic™ systems make use of the quad-tree method, in which an entire area of a layer is divided and subdivided into a series of four nested squares. The entire area is assigned to one of four squares designated  0 ,  1 ,  2 , and  3 . Each of these squares is subdivided into four smaller squares. The area of square  1  becomes  10 ,  11 ,  12 , and  13 . Each of these is further subdivided, meaning, for example, that the subdivisions of square  11  would be assigned index values of  110 ,  111 ,  112 , and  113 . As a result, any location in the map can be referred to by a single index number. The disadvantage with this quad-tree method is that processing time is wasted if there are no points within the subdivided squares; if indexing is performed over a large area, this wasted processing time is non-trivial and costly. 
   BRIEF SUMMARY 
   According to a first aspect of the present invention there is provided a method of retrieving points that are contained within a predetermined area. The method comprises
     (i) retrieving data identifying a region, where the region encompasses one or more points and is associated with linking data which, for each region, identifies the point or points encompassed by that region;   (ii) performing a process in respect of the region, the process comprising the steps of:   

   comparing extents of the region with extents of the predetermined area in order to establish whether the region overlaps with the predetermined area; 
   if there is overlap, retrieving data identifying sub-regions of the region and identifying any such sub-regions whose extents are wholly within the predetermined area;
     (iii) for each sub-region, repeating the process until all sub-regions thereof falling wholly within the predetermined area are identified, and   (iv) accessing linking data corresponding to the identified sub-regions so as to retrieve points encompassed by the sub-regions.   

   The plurality of points is advantageously pre-stored as a list of points, in an order given by the predetermined relationship between the region and sub-regions. Furthermore the linking data includes a value indicating the position of a first of the corresponding encompassed points in the list of points. The accessing step then involves retrieving an identifier representative of the number of encompassed points and retrieving the position value associated with the identified region or sub-region. This enables retrieval of the number of encompassed points from a position in the list associated with the position value. 
   Conveniently the points correspond to data that can be expressed in two dimensions, for example location (longitude and latitude) data or range data. Range data can include business hours (e.g. opening and closing times), price margins (e.g. maximum and minimum prices), and/or medical data (e.g. maximum and minimum blood pressure). Thus a predetermined area could be a range of prices—such as a maximum house prices and a minimum house price. In accordance with the method described above, the extents of the predetermined area (i.e. price range) are compared with a region retrieved from the index. All regions that overlap with the price range are then successively retrieved until a region, which falls wholly within the specified price range, is identified. All points falling within this identified region thus represent goods being of a price that falls within the specified maximum and minimum price range. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further aspects, features and advantages of the present invention will be apparent from the following description of preferred embodiments of the invention, which refers to the accompanying drawings, in which 
       FIG. 1  is a schematic diagram illustrating aspects of a communications system used by the invention; 
       FIG. 2  is a schematic diagram showing an example of points to be indexed according to the invention; 
       FIG. 3  is a schematic diagram showing an expanded view of  FIG. 2 ; 
       FIGS. 4   a  &amp;  4   b  in combination comprise a flow diagram showing an embodiment of an indexing process according to the present invention when indexing the points shown in  FIG. 2 ; 
       FIG. 5  is a schematic diagram showing application of the process of  FIGS. 4   a  &amp;  4   b  to create a quad around the points shown in  FIG. 2 ; 
       FIG. 6  is a schematic diagram showing application of the process of  FIGS. 4   a  &amp;  4   b  to create a sub-quad of the quad created according to  FIG. 5 ; 
       FIG. 7  is a schematic diagram showing application of the process of  FIGS. 4   a  &amp;  4   b  to create a sub-quad of one of the sub-quads of  FIG. 6 ; 
       FIG. 8  is an expanded view of  FIG. 7 ; 
       FIG. 9  is an expanded view of  FIG. 8  showing application of the process of  FIGS. 4   a  &amp;  4   b  to create another of the sub-quads shown in  FIG. 7 ; 
       FIG. 10  is an expanded view of  FIG. 8  showing application of the process of  FIGS. 4   a  &amp;  4   b  to create another of the sub-quads shown in  FIG. 7 ; 
       FIG. 11  is a schematic diagram showing application of the process of  FIGS. 4   a  &amp;  4   b  to create another sub-quad of one of the sub-quads of  FIG. 6 ; 
       FIG. 12  is a schematic diagram illustrating a process of storing points according to the invention; 
       FIGS. 13   a  &amp;  13   b  in combination comprise a flow diagram showing an embodiment of a retrieving process according to the present invention when retrieving points in accordance with an area of interest; 
       FIG. 14  is a schematic diagram showing an example of an area of interest for which points are to be retrieved; 
       FIG. 15  is a schematic diagram showing application of the process of  FIGS. 13   a  &amp;  13   b  to one of the sub-quads of  FIG. 6 ; 
       FIGS. 16 and 17  are an enlarged view of  FIG. 15  and are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to a first of the sub-quads of  FIG. 7 ; 
       FIG. 18  is an enlarged view of  FIG. 15  and shows the area of interest and a second of the sub-quads of  FIG. 7 ; 
       FIG. 19  is a schematic diagram corresponding to  FIG. 18  showing application of the process of  FIGS. 13   a  &amp;  13   b  to the second of the sub-quads of  FIG. 7 ; 
       FIGS. 20   a  &amp;  20   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to a (first) sub-quad of the sub-quad shown in  FIG. 18 ; 
       FIGS. 21   a  &amp;  21   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to another (a second) sub-quad of the sub-quad shown in  FIG. 18 ; 
       FIGS. 22   a  &amp;  22   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to another sub-quad (a third) of the sub-quad shown in  FIG. 18 ; 
       FIGS. 23   a  &amp;  23   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to another sub-quad (a fourth) of the sub-quad shown in  FIG. 18 ; 
       FIG. 24  is an enlarged view of  FIG. 15  and shows the area of interest and a third of the sub-quads of  FIG. 7 ; 
       FIG. 25  is a schematic diagram corresponding to  FIG. 24  showing application of the process of  FIGS. 13   a  &amp;  13   b  to the third of the sub-quads of  FIG. 7 ; 
       FIGS. 26   a  &amp;  26   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to a (first) sub-quad of the sub-quad shown in  FIG. 24 ; 
       FIGS. 27   a  &amp;  27   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to another (a second) sub-quad of the sub-quad shown in  FIG. 24 ; 
       FIGS. 28   a  &amp;  28   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to another sub-quad (a third) of the sub-quad shown in  FIG. 24 ; 
       FIGS. 29   a  &amp;  29   b  are schematic diagrams showing application of the process of  FIGS. 13   a  &amp;  13   b  to another sub-quad (a fourth) of the sub-quad shown in  FIG. 24 ; 
       FIG. 30  is an enlarged view of  FIG. 15  and shows the area of interest and a fourth of the sub-quads of  FIG. 7 ; 
       FIG. 31  is a schematic diagram corresponding to  FIG. 30  showing application of the process of  FIGS. 13   a  &amp;  13   b  to the fourth of the sub-quads of  FIG. 7 ; 
       FIG. 32  is a flow diagram showing further steps involved in the retrieving process of  FIGS. 13   a  &amp;  13   b ; and 
     Each of  FIGS. 33   a - 33   e  is a schematic illustration of a region of interest for which points are to be retrieved according to the invention. 
   

   DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
   Overview 
   Database servers DB 1 , DB 2 , such as those shown in  FIG. 1 , typically store information for retrieval by users. At the physical level, the communications environment within which the database servers DB 1 , DB 2  are located includes at least one user interface, commonly provided by a computer terminal or workstation T 3 . 
   Embodiments of the invention can be executed on the workstation T 3 , which is connected to database servers DB 1 , DB 2 . Although the database servers DB 1 , DB 2  are shown on the same LAN N 1  as the terminal T 3 , it is understood that the database servers DB 1 , DB 2  could be connected to different networks, which in turn are connected to LAN N 1  via one or more switches and/or routers (not shown). Embodiments receive data as input, for instance as a file, and build an index to the data, as is described in more detail below. The built index is then saved in one of the databases DB 1 , DB 2 , and the indexed data is also saved, in an order given by the structure of the built index, to one of the databases DB 1 , DB 2 . The built index can be saved on the same, or a different, database as the database on which the data is stored. 
   Embodiments of the present invention are concerned with indexing entities that are defined by 2-dimensions. Obvious examples of entities that can be indexed according to embodiments include the location of objects, such as petrol stations, cash points etc., as the position of objects is commonly defined in terms of latitude and longitude. Many other entities can be represented by 2-dimensions—e.g. acceleration of a motorbike as a function of time and speed, conductivity of a material as a function of material properties and temperature, deformation of an object as a function of material properties and force applied to the object etc. Furthermore, transformations can be applied to n-dimensional parameters to reduce them to 2-dimensional parameters, which can be displayed in a 2-dimensional space. 
   A further example of entities that can be indexed using embodiments of the present invention include range information, e.g. temporal information, price information, and medical condition information. 
   An example of temporal range information is opening and closing times of business and leisure establishments—these times can be expressed in two dimensions, with, for example, the closing and opening times respectively on the ordinate and abscissa axes. Similarly, delivery times (earliest and latest) can be expressed in two dimensions. 
   An example of price information includes prices of goods, so that, for example, maximum and minimum prices of goods can be respectively expressed on one of two dimensions, and so indexed using embodiments of the invention. Price information also includes trading information, as used to buy and sell stocks, shares, bonds etc. 
   An example of medical condition information includes statistics relating to measurable conditions such as body temperature and blood pressure, and conditions that can be translated into numerical representations, such as cancer sites. 
   In the following description, entities are generally referred to as “points” in order to disassociate the context of the entities (e.g. petrol station, cash point etc.) from the mechanics of the embodiment. 
   In overview, an embodiment of a method of indexing points is described with reference to  FIG. 2 .  FIG. 2  shows an area R 1  within which a plurality of points  200  is located. Each of the points is defined in x,y co-ordinate space. Essentially the area R 1  comprising points to be indexed is examined and split into areas R 3 , R 4  containing points and area R 2  not containing points (the areas could be split into any shape, such as a rectangle, triangle, or strips;  FIG. 2  shows the areas split into strips for the purposes of describing the inventive concept of this invention). The embodiment then examines the distribution of points in areas R 3  and R 4 , identifying on a smaller scale than was considered for region R 1 , areas in R 3  and R 4  that comprise points. Referring to  FIG. 3 , R 4  essentially becomes R 11  and the distribution of points within R 11  is examined. By concentrating on the distribution of points in this way, areas that do not contain any points, Region R 2  in  FIG. 2  and region R 12  in  FIG. 3  are implicitly discarded. The process is continually repeated, effectively “burrowing down” through a series of areas of diminishing size, until the size of an area is such that it collapses to the size of a single point. As the embodiment “burrows down”, each area is linked to the area above it, such that each point is linked by a series of areas. An index to these points comprises the series of areas, and these areas and points are used to create an index (described in more detail below) that is saved in database DB 1 . The relationship between the points and areas enables points to be identified by identification of an area in the index. 
   One of the advantages of creating an index as described above is that the query search domain is confined to areas that are known to contain points—i.e. queries will only be carried out on the areas saved in the database DB 1 , and as these areas by definition include points, the search domain is relatively compact. Referring back to  FIG. 2 , the process of identifying points in respect of a query is faster according to the embodiment described above, than if the index comprised information relating to the whole of area R 1 . 
   In a particular form of an embodiment, presented below with reference to  FIGS. 4-11 , points are 2-dimensional co-ordinates in x, y space. If the entities to be indexed are acceleration values, defined by a corresponding set of time and speed values, the time and speed values map directly onto an x,y co-ordinate space, so that (t 1 , v 1 ), (t 2 , v 2 ) . . . (tn, vn) are co-ordinates of points corresponding to the acceleration values. Similarly, if the entities to be indexed are location values, defined by a corresponding set of latitude and longitude values, the latitude and longitude values map directly onto an x, y co-ordinate space, such that (lat_ 1 , long_ 1 ) . . . (lat_n, long_n) are co-ordinates of points corresponding to location values. It is assumed that the points have been stored (e.g. written to a file), so that embodiments of the invention read the points in from a file. In alternative embodiments a user may input the points when the index to those points is about to be built. 
   Furthermore, in the embodiment presented below, the areas are squares, referred to as “quads” and “sub-quads” in the description below, and each quad is successively split into four sub-quads. Each sub-quad is examined for the existence of points. Those with no points are discarded, which is an equivalent process to discarding the area R 2  described with reference to  FIGS. 2 and 3  above, and each new sub-quad containing points is “shrunk wrapped” around the smallest area that contains points in that sub-quad. (The embodiment analyses the areas in accordance with squares, but many other shapes could be used to “shrink-wrap” around the points). Each sub-quad is then divided again into four sub-quads, empty sub-quads are again discarded and each remaining sub-quad “shrunk-wrapped” about its smallest area containing points. Eventually, each remaining sub-quad will have been “shrunk-wrapped” onto a single point and its co-ordinates will be those of the point concerned. Once the quad and all the sub-quads, including both the intervening sub quads which haven&#39;t been discarded and the sub-quads coinciding with single points, have been identified, an index to the points, comprising the quad and sub quads relevant to each one, is created. This is described in detail after the discussion of  FIGS. 5-10 . 
   In the following, steps S 4 . 1  through to S 4 . 9  are shown in sequence in the flow chart of  FIGS. 4   a  and  4   b  and illustrated by the operations with the same reference numerals as shown in  FIGS. 5-11 . 
   
     FIG. 5 
   
   
       
       Step S 4 . 1  Read in x, y co-ordinates of all points to be indexed (as stated above, the points may be read from either a storage location, such as a file, or directly from a user); 
       Step S 4 . 2  Draw up a bounding box for all points, identifying, and “shrink-wrapping” around, the co-ordinates of the outermost points (the bounding box is given by the difference between the co-ordinates of the outermost points in both the x and y dimensions: dx and dy respectively). This bounding box is the outermost quad  501 ; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save extents of quad  501 —i.e. dx, dy—and the co-ordinates of the points in it; 
       Step S 4 . 5  Check whether the outermost quad  501  has positive size (i.e. are dx, dy of quad  501  equal to zero?) In the example shown in  FIG. 5 , there is an outermost quad  501 , because the quad has multiple points in it, dx, dy are non-zero, and thus quad  501  has positive size; 
       Step S 4 . 6  Split the outermost quad  501  into four sub-quads,  503   a ,  503   b ,  503   c ,  503   d;    
       Step S 4 . 6 . 1  Tag each point with its relevant sub-quad and record the number of points in each sub-quad; 
       Step S 4 . 7  Starting with sub-quad  0  ( 503   a ) check whether there are any points in the sub-quad  503   a.  
 
As there are points, input the points within this sub-quad  0  ( 503   a ) to Step S 4 . 1  and run through steps Step S 4 . 1  onwards for sub-quad  0  ( 503   a ), as described below with reference to  FIG. 6 .
 
 FIG. 6 
 
       Step S 4 . 1  Read in x, y co-ordinates of points corresponding to sub-quad  503   a;    
       Step S 4 . 2  Draw up bounding box for all points in  503   a , creating “shrink-wrapped” sub-quad  0   601 . This illustrates the principle described above—the embodiment identifies an area within sub-quad  0  where there are no points, and this area is then discarded; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save extents of sub-quad  0  ( 601 )—i.e. dx, dy—and the co-ordinates of points in it; 
       Step S 4 . 5  Check whether sub-quad  601  has positive size? (i.e. are dx, dy of quad  601  equal to zero?) As can be seen in  FIG. 6 , quad  601  has multiple points in it, dx, dy for quad  601  are non-zero, and thus sub-quad  601  has positive size; 
       Step S 4 . 6  Split sub-quad  0   601  into 4 sub-quads:  0 , 0  ( 603   a ),  0 , 1  ( 603   b ),  0 , 2  ( 603   c ),  0 , 3  ( 603   d ) 
       Step S 4 . 6 . 1  Tag each point with its relevant sub-quad and record the number of points in each sub-quad; 
       Step S 4 . 7  Starting with sub-quad  0 , 0  ( 603   a ) Check whether there are any points in the sub-quad  0 , 0  ( 603   a ): As there are points, input the points within this sub-quad  0 , 0  ( 603   a ) to Step S 4 . 1  and run through steps Step S 4 . 1  onwards for sub-quad  0 , 0  ( 603   a ), as described below with reference to  FIG. 7 .
 
 FIG. 7 
 
       Step S 4 . 1  Read in x, y co-ordinates of points corresponding to sub-quad  603   a;    
       Step S 4 . 2  Draw up bounding box for all points in sub-quad  0 , 0  ( 603   a ), creating “shrink-wrapped” sub-quad  0 , 0   701 . As for sub-quad  0 , the area without points within sub-quad  603   a  is ignored; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save extents of sub-quad  0 ,  0  ( 701 )—i.e. dx, dy—and the co-ordinates of points in it; 
       Step S 4 . 5  Check whether sub-quad  701  has positive size? (i.e. are dx, dy of quad  701  equal to zero?) As can be seen in  FIG. 7 , quad  701  has multiple points in it, dx, dy for quad  701  are non-zero, and this sub-quad  701  has positive size; 
       Step S 4 . 6  Split sub-quad  0 , 0   701  into 4 sub-quads:  0 , 0 , 0  ( 703   a ),  0 , 0 , 1  ( 703   b ),  0 , 0 , 2  ( 703   c ),  0 , 0 , 3  ( 703   d ) 
       Step S 4 . 6 . 1  Tag each point with its relevant sub-quad and record the number of points in each sub-quad; 
       Step S 4 . 7  Starting with sub-quad  0 , 0 , 0  ( 703   a ) check whether there are any points in the sub-quad  0 , 0 , 0  ( 703   a ): there is one point in sub-quad  0 , 0 , 0  ( 703   a ) so input the points within this sub-quad  0 , 0 , 0  to Step S 4 . 1  and run through steps Step S 4 . 1  onwards for sub-quad  0 , 0 , 0  ( 703   a ), as described below with further reference to  FIG. 7 .
 
Also  FIG. 7 
 
       Step S 4 . 1  Read in x, y co-ordinates of points corresponding to sub-quad  703   a;    
       Step S 4 . 2  Draw up bounding box for all points in sub-quad  0 , 0 , 0  ( 703   a ), creating “shrink-wrapped” sub-quad  0 , 0 , 0 : here around a single point; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save the extents of the “shrink-wrapped” sub-quad  0 , 0 , 0 , which is now down to a single point such that dx, dy=0, and save the co-ordinates of the point; 
       Step S 4 . 5  Check whether sub-quad the point has positive size? (i.e. are dx, dy of the point equal to zero?) In fact dx and dy are both zero because the sub-quad  703   a  collapsed into a single point. So onto Step S 4 . 8 ; 
       Step S 4 . 8  Increment the sub-quad counter i at this level ( 0 , 0 ,i), input the points (Step S 4 . 7 ) within sub-quad  0 , 0 , 1  ( 703   b ) to Step S 4 . 1  and run through steps Step S 4 . 1  onwards for sub-quad  0 , 0 , 1  ( 703   b ), as described below with reference to  FIG. 8 .
 
 FIG. 8 
 
       Step S 4 . 1  Read in x, y co-ordinates of points corresponding to sub-quad  703   b;    
       Step S 4 . 2  Draw up bounding box for all points in sub-quad  0 , 0 , 1  ( 703   b ) creating “shrink-wrapped” sub-quad  0 , 0 , 1   801 . As for sub-quad  0 , the area without points within sub-quad  703   b  is ignored; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save extents of sub-quad  801 —i.e. dx, dy—and the co-ordinates of points in it; 
       Step S 4 . 5  Check whether sub-quad  801  has positive size? (i.e. are dx, dy of quad  703   a  equal to zero?). Sub-quad  801  has multiple points in it, dx, dy for quad  801  are non-zero, and thus sub-quad  801  does have positive size; 
       Step S 4 . 6  Split sub-quad  0 , 0 , 1   801  into 4 sub-quads:  0 , 0 , 1 , 0  ( 803   a ),  0 , 0 , 1 , 1  ( 803   b ),  0 , 0 , 1 , 2  ( 803   c ),  0 , 0 , 1 , 3  ( 803   d ) 
       Step S 4 . 6 . 1  Tag each point with its relevant sub-quad and record the number of points in each sub-quad; 
       Step S 4 . 7  Starting with sub-quad  0 , 0 , 1 , 0  ( 803   a ) Check whether there are any points in the sub-quad  0 , 0 , 1 , 0  ( 803   a ): There are no points in sub-quad  0 , 0 , 1 , 0  ( 803   a ); 
       Step S 4 . 8  Increment the sub-quad counter i at this level ( 0 , 0 , 1 ,i) and input the points (Step S 4 . 7 ) within sub-quad  0 , 0 , 1 , 1  ( 803   b ) to Step S 4 . 1  and run through steps Step S 4 . 1  onwards for sub-quad  0 , 0 , 1 , 1  ( 803   b ), as described below with reference to  FIG. 9 .
 
 FIG. 9 
 
       Step S 4 . 1  Read in x, y co-ordinates of points corresponding to sub-quad  803   b;    
       Step S 4 . 2  Draw up bounding box for all points in sub-quad  0 , 0 , 1 , 1  ( 803   b ), creating “shrink-wrapped” sub-quad  0 , 0 , 1 , 1 , which is a single point; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save extents of the single point—i.e. dx, dy—and the co-ordinates of the point; 
       Step S 4 . 5  Check whether the point has positive size? (i.e. are dx, dy of the point equal to zero?) In fact dx and dy are both zero because the sub-quad  803   b  collapsed into a single point. So onto Step S 4 . 8 ; 
       Step S 4 . 8  Increment the sub-quad counter i at this level ( 0 , 0 , 1 ,i) and input the points within sub-quad  0 , 0 , 1 , 2  ( 803   b ) to Step S 4 . 1  and run through steps Step S 4 . 1  onwards for sub-quad  0 , 0 , 1 , 2  ( 803   b ), as described with further reference to  FIG. 9 
 
Also  FIG. 9 
 
       Step S 4 . 1  Read in x, y co-ordinates of points corresponding to sub-quad  803   c;    
       Step S 4 . 2  Draw up bounding box for all points in sub-quad  0 , 0 , 1 , 2  ( 803   c ), creating “shrink-wrapped” sub-quad  0 , 0 , 1 , 2 , which is a single point; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save extents of the single point—i.e. dx, dy—and the co-ordinates of the point; 
       Step S 4 . 5  Check whether the point has positive size? (i.e. are dx, dy of the point equal to zero?) In fact dx and dy are both zero because the sub-quad  803   c  collapsed into a single point. So onto Step S 4 . 8 ; 
       Step S 4 . 8  Increment the sub-quad counter i at this level ( 0 , 0 , 1 ,i). There are no points (Step S 4 . 7 ) within sub-quad  0 , 0 , 1 , 3  ( 803   d ), so back to Step S 4 . 8 : increment the sub-quad counter i at this level ( 0 , 0 , 1 ,i): but i&gt;3 so 
       Step S 4 . 9  Return to sub-quad level  0 , 0 ,i and increment the sub-quad counter from 1 to 2, and thus consider sub-quad  0 , 0 , 2  ( 703   c ): There is a point in sub-quad  0 , 0 , 2  ( 703   c ) so input the points (Step S 4 . 7 ) within sub-quad  0 , 0 , 2  ( 703   c ) to Step S 4 . 1  and run through steps Step S 4 . 1  onwards for sub-quad  0 , 0 , 2  ( 703   c ), as described with reference to  FIG. 10 .
 
 FIG. 10 
 
       Step S 4 . 1  Read in x, y co-ordinates of points corresponding to sub-quad  703   c;    
       Step S 4 . 2  Draw up bounding box for all points in sub-quad  0 , 0 , 2  ( 703   c ), creating “shrink-wrapped” sub-quad  0 , 0 , 2 , which is a single point; 
       Step S 4 . 3  &amp; Step S 4 . 4  Save extents of the single point—i.e. dx, dy—and the co-ordinates of the point; 
       Step S 4 . 5  Check whether the point has positive size? (i.e. are dx, dy of the point equal to zero?) In fact dx and dy are both zero because the sub-quad  703   c  collapsed into a single point. So onto Step S 4 . 8 ; 
       Step S 4 . 8  Increment the sub-quad counter i at this level ( 0 , 0 ,i). There are no points (Step S 4 . 7 ) within sub-quad  0 , 0 , 3  ( 703   d ), so back to Step S 4 . 8 : increment the sub-quad counter i at this level ( 0 , 0 ,i): but i&gt;3 so 
       Step S 4 . 9  Return to sub-quad level  0 ,i and increment the sub-quad counter i from 0 to 1, and thus consider sub-quad  0 , 1  ( 603   b ), as described with reference to  FIG. 11 .
 
 FIG. 11 
 
The process described in  FIGS. 6-10  is repeated, but for sub-quad  0 , 1 , and once all of the points within sub-quad  0 ,n have been assigned to sub-quads, the process moves onto sub-quad  1 .
 
     
  
   As described earlier, building an index to these points is then a matter of saving the sub-quad information. This can be engineered in many ways, but preferably the index comprises sub-quad information saved at steps S  4 . 3  and S  4 . 4 , namely the extents of sub-quad (in x, y co-ordinates) and co-ordinates of points falling therein, and a link to the 4 sub-quadrants within the sub-quad. Thus the index essentially comprises a hierarchy of sub-quad structures where the hierarchy is given by the relationship between each successive sub-quad and its 4 sub-quads. The sub-quads are written to the index in accordance with the sub-quad hierarchy, from the largest sub-quad (here  501  on  FIG. 5 ), down to the smallest sub-quad. 
   In addition to saving the sub-quad structures in the database DB 1 , the points are written to the database (either the same database or a different database), e.g. in a file, in an order given by the inverse of the sub-quad hierarchy. Thus in this case, points in the sub-quads at the bottom of the hierarchy are written to the file first. As the points are written to a file, a running tally of the total number of points is maintained, such that as each point is written to the file, a counter representing: current number of points encountered so far+1 is written to the respective sub-quad structure. The tally works from the smallest sub-quad up, and, for each sub-quad, essentially indicates the position of the first of all points in that sub-quad in terms of all of the points being indexed (Pos  sub     —     quad ) e.g. Referring to  FIG. 12 , 
   
     
       
             
             
             
             
           
             
             
             
             
           
         
             
                 
             
             
                 
                 
               Number of first 
                 
             
             
                 
                 
               point written to 
                 
             
             
                 
               Points in sub- 
               sub-quad 
               Highlighted point in 
             
             
               Sub-quad 
               quad (N) 
               (Pos  sub     —     quad ) 
               points file 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               0, 0, 1, 1 
               n (N = 1) 
               1 
                 n , m, p, o . . . 
             
             
               0, 0, 1, 2 
               m (N = 1) 
               2 
               n,  m , p, o . . . 
             
             
               0, 0, 0 
               p (N = 1) 
               3 
               n, m,  p , o . . . 
             
             
               0, 0, 1 
               n, m (N = 2) 
               1 
                 n , m, p, o . . . 
             
             
               0, 0, 2 
               o (N = 1) 
               4 
               n, m, p, o . . . 
             
             
               0, 0 
               n, m, p, o (N = 4) 
               1 
                 n , m, p, o . . . 
             
             
               0, 1, c, d 
               some points 
               5 
               n, m, p, o,  Next  . . . 
             
             
                 
             
           
        
       
     
   
   Thus points file for sub-quad  501  (the outermost quad, see  FIG. 5 ) reads n,m,p,o . . . (starting from the smallest sub-quad  0 , 0 , 1 , 1 ). As both the number of points, (N) and the number in the running tally of points (Pos  sub     —     quad ) corresponding to the first point in a sub-quad, are written to the sub-quad, then once a sub-quad of interest has been identified, the points that lie within the identified sub-quad can be extracted by moving to Pos  sub     —     quad  in the points file and extracting N points from that position. This is demonstrated in an embodiment demonstrating retrieval of points. 
   Retrieval of Information 
   The second invention relates to a method of retrieving entities by means of an index of elements related to the entities, when the relationship between each element and other elements in the index, and the relationship between elements in the index and the entities being indexed, is known. The method is readily applicable to an index created in accordance with the method of indexing presented above, because the index comprises a hierarchy of sub-quad structures, and the hierarchy is well defined (by quad-&gt;sub-quad relationships). However, it should be borne in mind that the method is equally applicable to any type of index that satisfies these conditions. 
   In the following description, an embodiment of the retrieval process is described, where an index to points is queried with a query specifying a “region of interest”. It is assumed that:
     the elements in the index are a plurality of areas within a predetermined area,   the entities being indexed are points defined by two dimensions;   there is a predetermined relationship between the areas; and   there is a predetermined relationship between the areas and the points.
 
Essentially the embodiment identifies which of the areas
   a) overlaps with the region of interest, and   b) contains points within those areas that overlap with the region of interest.
 
The predetermined relationship between areas and the points is then used to extract points falling within the identified areas.
   

   A region of interest refers to a region within the two dimensional representation of the entity. Thus for the temporal range described above (closing and opening times of business and leisure establishments), a region of interest would be a time period of interest—such as “shops open between 10:00 and 13:00 hours”. The region of interest would then be defined by a region (preferably a square) bounded within prespecified points. Referring to  FIG. 33   a - e , the region of interest could be any one of:
     ( 0 , 10 ), ( 13 , 24 )≡establishments that are open sometime between 10:00 &amp; 13:00 ( FIG. 33   a )   ( 0 , 13 ), ( 10 , 24 )≡establishments that are open continuously between 10:00 &amp; 13:00 ( FIG. 33   b )   ( 0 , 10 ), ( 10 , 13 )≡establishments that are open before 10:00, but close before 13:00 ( FIG. 33   c )   ( 10 , 13 ), ( 13 , 24 ) establishments that open after 10:00 but close after 13:00 ( FIG. 33   d )   ( 10 , 10 ), ( 13 , 13 ) establishments that open after 10:00 &amp; close before 13:00 ( FIG. 33   e )   

   Similarly, for the pricing range described above, a region of interest would be range of prices, so that, for a retrieval requirement of “all items that fall somewhere in the range of £50.00 and £80.00”, the region of interest could be defined by a region bounded within the points ( 0 , 50 ) and ( 80 , 80 ). 
   For the location information described above, a region of interest would be a range of positions, such as “all items located between a first position ((lat, long) 1 ) and a second position ((lat, long) 2 )”. 
   A flow diagram showing steps of a method of identifying areas that overlap with the region of interest, when the region of interest relates to location information, is shown in  FIGS. 13   a  and  13   b . The steps are then illustrated, for the region of interest shown in  FIG. 14 , in  FIGS. 14-31 . In this embodiment, specific examples of areas within a predetermined region are referred to as quads and sub-quads. The method steps shown in  FIG. 13  are described below with reference to each of  FIGS. 14-31 . 
   
     FIG. 14 
   
   
       
       S 13 . 1 . 1  Read in x, y co-ordinates of a region of interest (x 1 ,y 1 ) (x 2 ,y 2 ). As an example, if a user wants to locate garages in a certain area, these might be indexed as latitude/longitude pairs defining points in a two-dimensional space and the geographical area that the user is interested in can be expressed as a “region of interest” in the two-dimensional space; 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of quad for the outermost quad and set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of outermost quad; 
       S 13 . 2  Assess whether the region of interest requires cropping: if x 1 &lt;X 1 _Q set x 1 =X 1 _Q; if y 1 &lt;Y 1 _Q set y 1 =Y 1 _Q; if x 2 &gt;X 2 _Q set x 2 =X 2 _Q; if y 2 &gt;Y 2 _Q set y 2 =Y 2 _Q. For the example region of interest shown in  FIG. 14 , none of these conditions are satisfied; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case neither conditions are satisfied;
 
Steps S 13 . 2  and S 13 . 3  are only one example of conditions that can be applied to establish whether the sub quad retrieved at S 13 . 1 . 2  overlaps with the region of interest, and whether, if there is overlap, there are any points within the overlapping region; it is envisaged that alternative methods could be applied to establish this.
 
       S 13 . 4  Assess whether the region of interest overlaps exactly with the quad. No: 
       S 13 . 5  Retrieve a sub-quad ( 503   a ) of the present quad ( 501 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIG. 15 
 
 FIG. 15 
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0   601 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  601 ; 
       S 13 . 2  Assess whether the region of interest requires cropping: if x 1 &lt;X 1 _Q set x 1 =X 1 _Q; if y 1 &lt;Y 1 _Q set y 1 =Y 1 _Q; if x 2 &gt;X 2 _Q set x 2 =X 2 _Q; if y 2 &gt;Y 2 _Q set y 2 =Y 2 _Q. In this case none of these conditions are satisfied; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case neither conditions are satisfied; 
       S 13 . 4  Assess whether the region of interest overlaps exactly with the sub-quad. No: 
       S 13 . 5  Retrieve a sub-quad of the present sub-quad in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 16 and 17 
 
 FIGS. 16 &amp; 17 
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 0   701 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  701 ; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 2 &gt;X 2 _Q so set x 2 =X 2 _Q and y 2 &gt;Y 2 _Q so set y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case ( FIG. 17 ) both conditions are satisfied, which means that there are no points in sub-quad  0 , 0  ( 701 ) that fall within the region of interest; 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 ,i), and retrieve sub-quad ( 0 , 1 ) in accordance with S 13 . 1 . 2 , as described with reference to  FIGS. 18 and 19 .
 
 FIGS. 18 &amp; 19 
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 1 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 1 ; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 1 &lt;X 1 _Q so set x 1 =X 1  Q and y 2 &gt;Y 2 _Q so set y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case neither conditions are satisfied; 
       S 13 . 4  Assess whether the region of interest (now cropped) overlaps exactly with the sub-quad. No: 
       S 13 . 5  Retrieve a sub-quad  0 , 1 , 0  of the present sub-quad  0 , 1  in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 20   a  and  20   b.  
 
 FIGS. 20   a  &amp;  20   b  
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 1 , 0 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 1 , 0 ; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 2 &gt;X 2 _Q so set x 2 =X 2 _Q and y 2 &gt;Y 2 _Q so set y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case ( FIG. 20   b ) both conditions are satisfied, which means that there are no points in sub-quad  0 , 1 , 0  that fall within the region of interest; 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 , 1 ,i), and (S 13 . 6 . 1 ) retrieve sub-quad ( 0 , 1 , 1 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 21   a  and  21   b.  
 
 FIGS. 21   a  &amp;  21   b  
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 1 , 1 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 1 , 1 —i.e. a single point; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 1 &lt;X 1 _Q so set x 1 =X 1 _Q and y 2 &gt;Y 2 _Q so set y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case ( FIG. 21   b ) both conditions are satisfied, which means that there are no points in sub-quad  0 , 1 , 1  that fall within the region of interest; 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 , 1 ,i), and (S 13 . 6 . 1 ) retrieve sub-quad ( 0 , 1 , 2 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 22   a  and  22   b.  
 
 FIGS. 22   a  &amp;  22   b  
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 1 , 2 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 1 , 2 —i.e. a single point; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 1 &lt;X 1 _Q so set x 1 =X 1 _Q, y 1 &lt;Y 1 _Q, x 2 &gt;X 2 _Q and y 2 &gt;Y 2 _Q so set x 1 =X 1 _Q, y 1 =Y 1 _Q, x 2 =X 2 _Q, y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . No 
       S 13 . 4  Assess whether the region of interest (now cropped) overlaps exactly with the sub-quad: YES 
       S 13 . 4 . 2  Record the sub-quad and number of points within the sub-quad (here  1 ); 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 , 1 ,i), and (S 13 . 6 . 1 ) retrieve sub-quad ( 0 , 1 , 3 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 23   a  and  23   b.  
 
 FIGS. 23   a  &amp;  23   b  
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 1 , 3 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 1 , 3 —i.e. a single point; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 1 &lt;X 1 _Q so set x 1 =X 1 _Q, y 1 &lt;Y 1 _Q, x 2 &gt;X 2 _Q and y 2 &gt;Y 2 _Q so set x 1 =X 1 _Q, y 1 =Y 1 _Q, x 2 =X 2 _Q, y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . No 
       S 13 . 4  Assess whether the region of interest (now cropped) overlaps exactly with the sub-quad: YES 
       S 13 . 4 . 2  Record the sub-quad and number of points within the sub-quad (here  1 ); 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 , 1 ,i) . . . i&gt;3 so (S 13 . 6 . 2 ) retrieve sub-quad ( 0 , 2 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 24 and 25 .
 
 FIGS. 24 &amp; 25 
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 2 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 2 ; 
       S 13 . 2  Assess whether the region of interest requires cropping: y 1 &lt;Y 1 _Q so set y 1 =Y 1 _Q and x 2 &gt;X 2 _Q so set x 2 =X 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case neither conditions are satisfied; 
       S 13 . 4  Assess whether the region of interest (now cropped) overlaps exactly with the sub-quad. No: 
       S 13 . 5  Retrieve a sub-quad  0 , 2 , 0  of the present sub-quad  0 , 2  in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 26   a  and  26   b.  
 
 FIGS. 26   a  &amp;  26   b  
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 2 , 0 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 2 , 0 ; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 2 &gt;X 2 _Q so set x 2 =X 2 _Q and y 2 &gt;Y 2 _Q so set y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case ( FIG. 26   b ) both conditions are satisfied, which means that there are no points in sub-quad  0 , 2 , 0  that fall within the region of interest; 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 , 2 ,i), and (S 13 . 6 . 1 ) retrieve sub-quad ( 0 , 2 , 1 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 27   a  and  27   b.  
 
 FIGS. 27   a  &amp;  27   b  
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 2 , 1 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 2 , 1 —i.e. a single point; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 1 &lt;X 1 _Q so set x 1 =X 1 _Q, y 1 &lt;Y 1 _Q, x 2 &gt;X 2 _Q and y 2 &gt;Y 2 _Q so set x 1 =X 1 _Q, y 1 =Y 1 _Q, x 2 =X 2 _Q, y 2 =Y 2 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . No 
       S 13 . 4  Assess whether the region of interest (now cropped) overlaps exactly with the sub-quad: YES 
       S 13 . 4 . 2  Record the sub-quad and number of points within the sub-quad (here  1 ); 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 , 1 ,i) . . . i&lt;3 so (S 13 . 6 . 1 ) retrieve sub-quad ( 0 , 2 , 2 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 28   a  and  28   b.  
 
 FIGS. 28   a  &amp;  28   b  
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 2 , 2 : set (X 1 _Q, Y 1 _Q) (X 2 _Q, Y 2 _Q) to size of “shrink-wrapped” sub-quad  0 , 2 , 2 ; 
       S 13 . 2  Assess whether the region of interest requires cropping: x 2 &gt;X 2 _Q so set x 2 =X 2 _Q and y 1 &lt;Y 1 _Q so set y 1 =Y 1 _Q; 
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . In this case ( FIG. 28   b ) both conditions are satisfied, which means that there are no points in sub-quad  0 , 2 , 2  that fall within the region of interest; 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 , 2 ,i), and (S 13 . 6 . 1 ) retrieve sub-quad ( 0 , 2 , 3 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIG. 29 .
 
 FIG. 29 
 
       S 13 . 1 . 2  Retrieve co-ordinates of points and size of sub-quad from “shrink-wrapped” sub-quad  0 , 2 , 3 : there is no sub-quad corresponding to  0 , 2 , 3  because there are no points in the area corresponding to this sub-quad, so jump to S 13 . 6   
       S 13 . 6 . 2  Increment sub-quad counter i at this level ( 0 , 2 ,i): i&gt; 3 , so (S 13 . 6 . 2 ) retrieve sub-quad ( 0 , 3 ) in accordance with S 13 . 1 . 2 , as is described with reference to  FIGS. 30 and 31 .
 
 FIGS. 30 &amp; 31 
       S 13 . 2  Assess whether the region of interest requires cropping: x 1 &lt;X 1 _Q so set x 1 =X 1 _Q, y 1 &lt;Y 1 _Q, x 2 &gt;X 2 _Q and y 2 &gt;Y 2 _Q so set x 1 =X 1 _Q, y 1 =Y 1 _Q x 2 =X 2 _Q, y 2 =Y 2 _Q;   
     
       S 13 . 3  Assess whether x 1 &gt;x 2  or y 1 &gt;y 2 . No 
       S 13 . 4  Assess whether the region of interest (now cropped) overlaps exactly with the sub-quad: YES 
       S 13 . 4 . 2  Record the sub-quad and number of points within the sub-quad (here  2 ); 
       S 13 . 6  Increment sub-quad counter i at this level ( 0 ,i) . . . i&gt;3 so (S 13 . 6 . 2 ) retrieve sub-quad  1  in accordance with S 13 . 1 . 2 .
 
As can be seen from  FIG. 14 , the region of interest falls solely within sub-quad  0  and the sub-quads therein, such that when process described in  FIGS. 13   a  and  13   b  is applied to sub-quads  1 ,  2  and  3 , the conditions applied at steps S 13 . 2  and S 13 . 3  will cause the process to terminate within a few steps.
 
     
  
   At the end of the process of identifying sub-quads overlapping with the region of interest, the sub-quads, and the number of points within those sub-quads, that were recorded at steps S 13 . 4 . 2  are returned—for this example:
     sub-quad  0 , 1 , 2  number of points: 1 point;   sub-quad  0 , 1 , 3  number of points: 1 point;   sub-quad  0 , 2 , 1  number of points: 1 point;   sub-quad  0 , 3  number of points: 2 points.   

   Once the sub-quads have been identified, the actual points are retrieved. In this embodiment, and as described above, the points are stored in a flat file. Furthermore each sub-quad structure stores a number indicating the position, relative to the total number of points being indexed (Referring for example to  FIG. 4 , all of the points within quad  501 ), of the first point within a respective sub-quad. The process for actual retrieval of points is shown in  FIG. 32 : For each quad that was recorded at step S 13 . 4 . 2 :
     S 32 . 1  For that sub-quad retrieve number of points falling within sub-quad (N);   S 32 . 2  Retrieve position of the first point from the corresponding sub-quad structure (Pos  sub     —     quad );   S 32 . 3  Move a file pointer to a position in the points file given by Pos  sub     —     quad ;   S 32 . 4  From this position, extract N points from the file.
 
Implementation
   

   The processes described in  FIGS. 4   a  and  4   b ,  FIGS. 13   a  and  13   b  and  FIG. 32  are implemented in software, and run on one of, or distributed between, the terminals T 3 , T 4 . Terminals T 3 , T 4  are thus representative of one or a plurality of computers, and are preferably server computers. 
   Points to be indexed can be input to terminals T 3 , T 4  via a file or similar, the index created as described above can be stored in the database DB 1 , and the points file can also be stored in the database DB 1 . An area of interest can be input in the form of a database query, entered via a client terminal (not shown) and communicated over the network N 1  in a known manner. 
   Preferably the processes described above are implemented in the C programming language, and use recursive programming methods to “burrow down” to sub-quads within sub-quads. It is understood that such a method is inessential to the invention. 
   Additional Details and Modifications 
   As stated above, the invention can be used to index and retrieve data that is expressed in 2 dimensions. The invention can also be used to index and retrieve data of more than 2 dimensions, providing the data (n-dimensional data, where n&gt;2) can be transformed into 2-dimensions. In such cases the transformed, 2-dimensional, data can be indexed and retrieved according to the invention. For example, objects defined in 3-dimensional space can be transformed into 2-dimensions using a package such as NCAR Graphics, which is a Unix based graphics package that offers a wide range of capabilities for the display and manipulation of numerical data, and has been developed by the University Corporation for Atmospheric Research. (See http://www.dkrz.de/ngdoc/ng4.0.1 for information relating to NCAR graphics and http://ngwww.ucar.edu/ngdoc/ng/fund/chp16-21/threed.html for information relating to the 3 to 2 dimensional transformation aspects). 
   Other variations could be made. For instance, a simple one would be to use division of quads and sub-quads into different numbers of areas in each iteration, such as eight or ten instead of four.