Abstract:
A system and method for providing a perspective-corrected representation of a multi-dimensional cluster is presented. Clusters are displayed based on independent spatial orientations within multiple dimensions. For each such spatial orientation, the clusters are located at an independent distance from a common origin. A relationship between a pair of the clusters is picked and analyzed. A criteria for perspective correction is sought. The criteria is applied to the pair of clusters to determine whether the clusters need be reoriented. The clusters are reoriented in the display by a perspective-corrected distance based on the relationship between the pair of clusters.

Description:
CROSS-REFERENCE TO RELATED APPLICATION  
       [0001]     This patent application is a continuation of commonly-assigned U.S. patent application, Ser. No. 11/110,452, filed Apr. 19, 2005, pending; which is a continuation of U.S. Pat. No. 6,888,548, issued May 3, 2005, the priority date of which is claimed and the disclosure of which is incorporated by reference. 
     
    
     FIELD OF THE INVENTION  
       [0002]     The present invention relates in general to data visualization and, in particular, to a system and method for generating a visualized data representation preserving independent variable geometric relationships.  
       BACKGROUND OF THE INVENTION  
       [0003]     Computer-based data visualization involves the generation and presentation of idealized data on a physical output device, such as a cathode ray tube (CRT), liquid crystal diode (LCD) display, printer and the like. Computer systems visualize data through the use of graphical user interfaces (GUIs) which allow intuitive user interaction and high quality presentation of synthesized information.  
         [0004]     The importance of effective data visualization has grown in step with advances in computational resources. Faster processors and larger memory sizes have enabled the application of complex visualization techniques to operate in multi-dimensional concept space. As well, the interconnectivity provided by networks, including intranetworks and internetworks, such as the Internet, enable the communication of large volumes of information to a wide-ranging audience. Effective data visualization techniques are needed to interpret information and model content interpretation.  
         [0005]     The use of a visualization language can enhance the effectiveness of data visualization by communicating words, images and shapes as a single, integrated unit. Visualization languages help bridge the gap between the natural perception of a physical environment and the artificial modeling of information within the constraints of a computer system. As raw information cannot always be digested as written words, data visualization attempts to complement and, in some instances, supplant the written word for a more intuitive visual presentation drawing on natural cognitive skills.  
         [0006]     Effective data visualization is constrained by the physical limits of computer display systems. Two-dimensional and three-dimensional information can be readily displayed. However, n-dimensional information in excess of three dimensions must be artificially compressed. Careful use of color, shape and temporal attributes can simulate multiple dimensions, but comprehension and usability become difficult as additional layers of modeling are artificially grafted into the finite bounds of display capabilities.  
         [0007]     Thus, mapping multi-dimensional information into a two- or three-dimensional space presents a problem. Physical displays are practically limited to three dimensions. Compressing multi-dimensional information into three dimensions can mislead, for instance, the viewer through an erroneous interpretation of spatial relationships between individual display objects. Other factors further complicate the interpretation and perception of visualized data, based on the Gestalt principles of proximity, similarity, closed region, connectedness, good continuation, and closure, such as described in R. E. Horn, “Visual Language: Global Communication for the 21 st  Century,” Ch. 3, MacroVU Press (1998), the disclosure of which is incorporated by reference.  
         [0008]     In particular, the misperception of visualized data can cause a misinterpretation of, for instance, dependent variables as independent and independent variables as dependent. This type of problem occurs, for example, when visualizing clustered data, which presents discrete groupings of data which are misperceived as being overlaid or overlapping due to the spatial limitations of a three-dimensional space.  
         [0009]     Consider, for example, a group of clusters, each cluster visualized in the form of a circle defining a center and a fixed radius. Each cluster is located some distance from a common origin along a vector measured at a fixed angle from a common axis through the common origin. The radii and distances are independent variables relative to the other clusters and the radius is an independent variable relative to the common origin. In this example, each cluster represents a grouping of points corresponding to objects sharing a common set of traits. The radius of the cluster reflects the relative number of objects contained in the grouping. Clusters located along the same vector are similar in theme as are those clusters located on vectors having a small cosine rotation from each other. Thus, the angle relative to a common axis&#39; distance from a common origin is an independent variable with a correlation between the distance and angle reflecting relative similarity of theme. Each radius is an independent variable representative of volume. When displayed the overlaying or overlapping of clusters could mislead the viewer into perceiving data dependencies where there are none.  
         [0010]     Therefore, there is a need for an approach to presenting arbitrarily dimensioned data in a finite-dimensioned display space while preserving independent data relationships. Preferably, such an approach would maintain size and placement relationships relative to a common identified reference point.  
         [0011]     There is a further need for an approach to reorienting data clusters to properly visualize independent and dependent variables while preserving cluster radii and relative angles from a common axis drawn through a common origin.  
       SUMMARY OF THE INVENTION  
       [0012]     The present invention provides a system and method for reorienting a data representation containing clusters while preserving independent variable geometric relationships. Each cluster is located along a vector defined at an angle θ from a common axis x. Each cluster has a radius r. The distance (magnitude) of the center c i  of each cluster from a common origin and the radius r are independent variables relative to other clusters and the radius r of each cluster is an independent variable relative to the common origin. The clusters are selected in order of relative distance from the common origin and optionally checked for an overlap of bounding regions. Clusters having no overlapping regions are skipped. If the pair-wise span s ij  between the centers c i  and c j  of the clusters is less than the sum of the radii r i  and r j , a new distance d i  for the cluster is determined by setting the pair-wise span s ij  equal to the sum of the radii r i  and r j  and solving the resulting quadratic equation for distance d i . The operations are repeated for each pairing of clusters.  
         [0013]     An embodiment is a system and method for reorienting a cluster rendering within a display. A plurality of clusters that have been visually rendered on a two-dimensional display are obtained to model independent spatial orientations within multiple dimensions. The spatial orientations of the clusters in each such dimension are evaluated by identifying two-dimensional relationships for pairs of the clusters. A distance for at least one such cluster is determined from a common origin within the display that visually preserves the spatial orientation of the at least one cluster as a function of the relationships of that cluster.  
         [0014]     A further embodiment is a system and method for providing a perspective-corrected representation of a multi-dimensional cluster. Clusters are displayed based on independent spatial orientations within multiple dimensions. For each such spatial orientation, the clusters are located at an independent distance from a common origin. A relationship between a pair of the clusters is picked and analyzed. A criteria for perspective correction is sought. The criteria is applied to the pair of clusters to determine whether the clusters need be reoriented. The clusters are reoriented in the display by a perspective-corrected distance based on the relationship between the pair of clusters.  
         [0015]     Still other embodiments of the present invention will become readily apparent to those skilled in the art from the following detailed description, wherein is described embodiments of the invention by way of illustrating the best mode contemplated for carrying out the invention. As will be realized, the invention is capable of other and different embodiments and its several details are capable of modifications in various obvious respects, all without departing from the spirit and the scope of the present invention. Accordingly, the drawings and detailed description are to be regarded as illustrative in nature and not as restrictive. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0016]      FIG. 1  is a block diagram showing a system for generating a visualized data representation preserving independent variable geometric relationships, in accordance with the present invention.  
         [0017]      FIG. 2  is a data representation diagram showing, by way of example, a view of overlapping clusters generated by the cluster display system of  FIG. 1 .  
         [0018]      FIG. 3  is a graph showing, by way of example, the polar coordinates of the overlapping clusters of  FIG. 2 .  
         [0019]      FIG. 4  is a data representation diagram showing, by way of example, the pair-wise spans between the centers of the clusters of  FIG. 2 .  
         [0020]      FIG. 5  is a data representation diagram showing, by way of example, an exploded view of the clusters of  FIG. 2 .  
         [0021]      FIG. 6  is a data representation diagram showing, by way of example, a minimized view of the clusters of  FIG. 2 .  
         [0022]      FIG. 7  is a graph, showing, by way of example, the polar coordinates of the minimized clusters of  FIG. 5 .  
         [0023]      FIG. 8  is a data representation diagram showing, by way of example, the pair-wise spans between the centers of the clusters of  FIG. 2 .  
         [0024]      FIG. 9  is a flow diagram showing a method for generating a visualized data representation preserving independent variable geometric relationships, in accordance with the present invention.  
         [0025]      FIG. 10  is a routine for reorienting clusters for use in the method of  FIG. 9 .  
         [0026]      FIG. 11  is a flow diagram showing a routine for calculating a new distance for use in the routine of  FIG. 10 .  
         [0027]      FIG. 12  is a graph showing, by way of example, a pair of clusters with overlapping bounding regions generated by the cluster display system of  FIG. 1 .  
         [0028]      FIG. 13  is a graph showing, by way of example, a pair of clusters with non-overlapping bounding regions generated by the cluster display system of  FIG. 1 .  
         [0029]      FIG. 14  is a routine for checking for overlapping clusters for use in the routine of  FIG. 10 .  
         [0030]      FIG. 15  is a data representation diagram showing, by way of example, a view of overlapping, non-circular clusters generated by the clustered display system of  FIG. 1 . 
     
    
     DETAILED DESCRIPTION  
       [0031]      FIG. 1  is a block diagram  10  showing a system for generating a visualized data representation preserving independent variable geometric relationships, in accordance with the present invention. The system consists of a cluster display system  11 , such as implemented on a general-purpose programmed digital computer. The cluster display system  11  is coupled to input devices, including a keyboard  12  and a pointing device  13 , such as a mouse, and display  14 , including a CRT, LCD display, and the like. As well, a printer (not shown) could function as an alternate display device. The cluster display system  11  includes a processor, memory and persistent storage, such as provided by a storage device  16 , within which are stored clusters  17  representing visualized multi-dimensional data. The cluster display system  11  can be interconnected to other computer systems, including clients and servers, over a network  15 , such as an intranetwork or internetwork, including the Internet, or various combinations and topologies thereof.  
         [0032]     Each cluster  17  represents a grouping of one or more points in a virtualized concept space, as further described below beginning with reference to  FIG. 2 . Preferably, the clusters  17  are stored as structured data sorted into an ordered list in ascending (preferred) or descending order. In the described embodiment, each cluster represents individual concepts and themes categorized based on, for example, Euclidean distances calculated between each pair of concepts and themes and defined within a pre-specified range of variance, such as described in common-assigned U.S. patent application Ser. No. 09/943,918, entitled “System And Method For Efficiently Generating Cluster Groupings In A Multi-Dimensional Concept Space,” filed Aug. 31, 2001, pending, the disclosure of which is incorporated by reference.  
         [0033]     The cluster display system  11  includes four modules: sort  18 , reorient  19 , display and visualize  20 , and, optionally, overlap check  21 . The sort module  18  sorts a raw list of clusters  17  into either ascending (preferred) or descending order based on the relative distance of the center of each cluster from a common origin. The reorient module  19 , as further described below with reference to  FIG. 10 , reorients the data representation display of the clusters  17  to preserve the orientation of independent variable relationships. The reorient module  19  logically includes a comparison submodule for measuring and comparing pair-wise spans between the radii of clusters  17 , a distance determining submodule for calculating a perspective-corrected distance from a common origin for select clusters  17 , and a coefficient submodule taking a ratio of perspective-corrected distances to original distances. The display and visualize module  20  performs the actual display of the clusters  17  via the display  14  responsive to commands from the input devices, including keyboard  12  and pointing device  13 . Finally, the overlap check module  21 , as further described below with reference to  FIG. 12 , is optional and, as a further embodiment, provides an optimization whereby clusters  17  having overlapping bounding regions are skipped and not reoriented.  
         [0034]     The individual computer systems, including cluster display system  11 , are general purpose, programmed digital computing devices consisting of a central processing unit (CPU), random access memory (RAM), non-volatile secondary storage, such as a hard drive or CD ROM drive, network interfaces, and peripheral devices, including user interfacing means, such as a keyboard and display. Program code, including software programs, and data are loaded into the RAM for execution and processing by the CPU and results are generated for display, output, transmittal, or storage.  
         [0035]     Each module is a computer program, procedure or module written as source code in a conventional programming language, such as the C++ programming language, and is presented for execution by the CPU as object or byte code, as is known in the art. The various implementations of the source code and object and byte codes can be held on a computer-readable storage medium or embodied on a transmission medium in a carrier wave. The cluster display system  11  operates in accordance with a sequence of process steps, as further described below with reference to  FIG. 9 .  
         [0036]      FIG. 2  is a data representation diagram  30  showing, by way of example, a view  31  of overlapping clusters  33 - 36  generated by the cluster display system  11  of  FIG. 1 . Each cluster  33 - 36  has a center c  37 - 40  and radius r  41 - 44 , respectively, and is oriented around a common origin  32 . The center c of each cluster  33 - 36  is located at a fixed distance (magnitude) d  45 - 48  from the common origin  32 . Cluster  34  overlays cluster  33  and clusters  33 ,  35  and  36  overlap.  
         [0037]     Each cluster  33 - 36  represents multi-dimensional data modeled in a three-dimensional display space. The data could be visualized data for a virtual semantic concept space, including semantic content extracted from a collection of documents represented by weighted clusters of concepts, such as described in commonly-assigned U.S. patent application Ser. No. 09/944,474, entitled “System And Method For Dynamically Evaluating Latent Concepts In Unstructured Documents,” filed Aug. 31, 2001, pending, the disclosure of which is incorporated by reference.  
         [0038]      FIG. 3  is a graph  50  showing, by way of example, the polar coordinates of the overlapping clusters  33 - 36  of  FIG. 2 . Each cluster  33 - 36  is oriented at a fixed angle θ  52 - 55  along a common axis x  51  drawn through the common origin  32 . The angles θ  52 - 55  and radii r  41 - 44  (shown in  FIG. 2 ) of each cluster  33 - 36 , respectively, are independent variables. The distances d  56 - 59  represent dependent variables.  
         [0039]     Referring back to  FIG. 2 , the radius r  41 - 44  (shown in  FIG. 2 ) of each cluster  33 - 36  signifies the number of documents attracted to the cluster. The distance d  56 - 59  increases as the similarity of concepts represented by each cluster  33 - 36  decreases. However, based on appearance alone, a viewer can be misled into interpreting cluster  34  as being dependent on cluster  33  due to the overlay of data representations. Similarly, a viewer could be misled to interpret dependent relationships between clusters  33 ,  35  and  36  due to the overlap between these clusters.  
         [0040]      FIG. 4  is a data representation diagram  60  showing, by way of example, the pair-wise spans between the centers of the clusters of  FIG. 2 . Centers c  37 - 40  of the clusters  33 - 36  (shown in  FIG. 2 ) are separated respectively by pair-wise spans s  61 - 66 . Each span s  61 - 66  is also dependent on the independent variables radii r  41 - 44  (shown in  FIG. 2 ) and angles θ  52 - 55 .  
         [0041]     For each cluster  33 - 36  (shown in  FIG. 2 ), the radii r is an independent variable. The distances d  56 - 59  (shown in  FIG. 3 ) and angles θ  52 - 55  (shown in  FIG. 3 ) are also independent variables. However, the distances d  56 - 59  and angles θ  52 - 55  are correlated, but there is no correlation between different distances d  56 - 59 . As well, the relative angles θ  52 - 55  are correlated relative to the common axis x, but are not correlated relative to other angles θ  52 - 55 . However, the distances d  56 - 59  cause the clusters  33 - 36  to appear to either overlay or overlap and these visual artifacts erroneously imply dependencies between the neighboring clusters based on distances d  56 - 59 .  
         [0042]      FIG. 5  is a data representation diagram  70  showing, by way of example, an exploded view  71  of the clusters  33 - 36  of  FIG. 2 . To preserve the relationships between the dependent variables distance d and span s, the individual distances d  56 - 59  (shown in  FIG. 3 ) are multiplied by a fixed coefficient to provide a proportionate extension e  71 - 75 , respectively, to each of the distances d  56 - 59 . The resulting data visualization view  71  “explodes” clusters  33 - 36  while preserving the independent relationships of the radii r  41 - 44  (shown in  FIG. 2 ) and angles θ  52 - 55  (shown in  FIG. 3 ).  
         [0043]     Although the “exploded” data visualization view  71  preserves the relative pair-wise spans s  61 - 66  between each of the clusters  33 - 36 , multiplying each distance d  56 - 59  by the same coefficient can result in a potentially distributed data representation requiring a large display space.  
         [0044]      FIG. 6  is a data representation diagram  80  showing, by way of example, a minimized view  81  of the clusters  33 - 36  of  FIG. 2 . As in the exploded view  71  (shown in  FIG. 5 ), the radii r  41 - 44  (shown in  FIG. 2 ) and angles θ  52 - 55  (shown in  FIG. 3 ) of each cluster  33 - 36  are preserved as independent variables. The distances d  56 - 59  are independent variables, but are adjusted to correct to visualization. The “minimized” data representation view  81  multiplies distances d  45  and  48  (shown in  FIG. 2 ) by a variable coefficient k. Distances d  46  and  47  remain unchanged, as the clusters  34  and  35 , respectively, need not be reoriented. Accordingly, the distances d  45  and  48  are increased by extensions e′  82  and  83 , respectively, to new distances d′.  
         [0045]      FIG. 7  is a graph  90  showing, by way of example, the polar coordinates of the minimized clusters  33 - 36  of  FIG. 5 . Although the clusters  33 - 36  have been shifted to distances d′  106 - 109  from the common origin  32 , the radii r  41 - 44  (shown in  FIG. 2 ) and angles θ  102 - 105  relative to the shared axis x  101  are preserved. The new distances d′  106 - 109  also approximate the proportionate pair-wise spans s′  110 - 115  between the centers c  37 - 40 .  
         [0046]      FIG. 8  is a data representation diagram  110  showing, by way of example, the pair-wise spans between the centers of the clusters of  FIG. 2 . Centers c  37 - 40  (shown in  FIG. 2 ) of the clusters  33 - 36  are separated respectively by pair-wise spans s  111 - 116 . Each span s  111 - 116  is dependent on the independent variables radii r  41 - 44  and the angles θ  52 - 55  (shown in  FIG. 3 ). The length of each pair-wise span s  111 - 116  is proportionately increased relative to the increase in distance d  56 - 69  of the centers c  37 - 40  of the clusters  33 - 36  from the origin  32 .  
         [0047]      FIG. 9  is a flow diagram showing a method  120  for generating a visualized data representation preserving independent variable geometric relationships, in accordance with the present invention. As a preliminary step, the origin  32  (shown in  FIG. 2 ) and x-axis  51  (shown in  FIG. 3 ) are selected (block  121 ). Although described herein with reference to polar coordinates, any other coordinate system could also be used, including Cartesian, Logarithmic, and others, as would be recognized by one skilled in the art.  
         [0048]     Next, the clusters  17  (shown in  FIG. 1 ) are sorted in order of relative distance d from the origin  32  (block  122 ). Preferably, the clusters  17  are ordered in ascending order, although descending order could also be used. The clusters  17  are reoriented (block  123 ), as further described below with reference to  FIG. 10 . Finally, the reoriented clusters  17  are displayed (block  124 ), after which the routine terminates.  
         [0049]      FIG. 10  is a flow diagram showing a routine  130  for reorienting clusters  17  for use in the method  120  of  FIG. 9 . The purpose of this routine is to generate a minimized data representation, such as described above with reference to  FIG. 5 , preserving the orientation of the independent variables for radii r and angles θ relative to a common x-axis.  
         [0050]     Initially, a coefficient k is set to equal 1 (block  131 ). During cluster reorientation, the relative distances d of the centers c of each cluster  17  from the origin  32  is multiplied by the coefficient k The clusters  17  are then processed in a pair of iterative loops as follows. During each iteration of an outer processing loop (blocks  132 - 146 ), beginning with the innermost cluster, each cluster  17 , except for the first cluster, is selected and processed. During each iteration of the inner processing loop (blocks  135 - 145 ), each remaining cluster  17  is selected and reoriented, if necessary.  
         [0051]     Thus, during the outer iterative loop (blocks  132 - 146 ), an initial Cluster i  is selected (block  133 ) and the radius r i , center c i , angle θ i , and distance d i  for the selected Cluster i  are obtained (block  134 ). Next, during the inner iterative loop (blocks  135 - 145 ), another Cluster j  (block  136 ) is selected and the radius r j , center c j , angle θ j , and distance d j  are obtained (block  137 ).  
         [0052]     In a further embodiment, bounding regions are determined for Cluster i  and Cluster j  and the bounding regions are checked for overlap (block  138 ), as further described below with reference to  FIG. 14 .  
         [0053]     Next, the distance d i  of the cluster being compared, Cluster i , is multiplied by the coefficient k (block  139 ) to establish an initial new distance d′ i  for Cluster i . A new center c i  is determined (block  140 ). The span s ij  between the two clusters, Cluster i  and Cluster j , is set to equal the absolute distance between center c i  plus center c j . If the pair-wise span s ij  is less than the sum of radius r i  and radius r j  for Cluster i  and Cluster j , respectively (block  143 ), a new distance d i  for Cluster i  is calculated (block  144 ), as further described below with reference to  FIG. 11 . Processing of each additional Cluster i  continues (block  145 ) until all additional clusters have been processed (blocks  135 - 145 ). Similarly, processing of each Cluster j  (block  146 ) continues until all clusters have been processed (blocks  132 - 146 ), after which the routine returns.  
         [0054]      FIG. 11  is a flow diagram showing a routine  170  for calculating a new distance for use in the routine  130  of  FIG. 10 . The purpose of this routine is to determine a new distance d′ i  for the center c i  of a selected cluster i  from a common origin. In the described embodiment, the new distance d′ i  is determined by solving the quadratic equation formed by the distances d i  and d j  and adjacent angle.  
         [0055]     Thus, the sum of the radii (r i +r j ) 2  is set to equal the square of the distance d j  plus the square of the distance d i  minus the product of the 2 times the distance d j  times the distance d i  times cos θ (block  171 ), as expressed by equation (1): 
 
( r   i   +r   j ) 2   =d   i   2   +d   j   2 −2 ·d   i   d   j  cos θ  (1) 
 
 The distance d i  can be calculated by solving a quadratic equation (5) (block  172 ), derived from equation (1) as follows:  
                 1   ·     d   i   2       +       (       2   ·     d   j       ⁢   cos   ⁢           ⁢   θ     )     ⁢     d   i         =     (       d   j   2     -       [       r   i     +     r   j       ]     2       )             (   2   )                   1   ·     d   i   2       +       (       2   ·     d   j       ⁢   cos   ⁢           ⁢   θ     )     ⁢     d   i       -     (       d   j   2     -       [       r   i     +     r   j       ]     2       )       =   0           (   3   )                 d   i     =         (       2   ·     d   j       ⁢   cos   ⁢           ⁢   θ     )     ±           (       2   ·     d   j       ⁢   cos   ⁢           ⁢   θ     )     2     -     4   ·   1   ·     (       d   j   2     -       [       r   i     +     r   j       ]     2       )               2   ·   1               (   4   )                 d   i     =         (       2   ·     d   j       ⁢   cos   ⁢           ⁢   θ     )     ±           (       2   ·     d   j       ⁢   cos   ⁢           ⁢   θ     )     2     -     4   ·     (       d   j   2     -       [       r   i     +     r   j       ]     2       )             2             (   5   )             
 
 In the described embodiment, the ‘±’ operation is simplified to a ‘+’ operation, as the distance d i  is always increased. 
 
         [0056]     Finally, the coefficient k, used for determining the relative distances d from the centers c of each cluster  17  (block  139  in  FIG. 10 ), is determined by taking the product of the new distance d i  divided by the old distance d i  (block  173 ), as expressed by equation (6):  
             k   =       d     i   new         d     i   old                 (   6   )             
 
 The routine then returns. 
 
         [0057]     In a further embodiment, the coefficient k is set to equal 1 if there is no overlap between any clusters, as expressed by equation (7):  
                 if   ⁢           ⁢         d     i   -   1       +     r     i   -   1             d   i     -     r   i           &gt;   1     ,       then   ⁢           ⁢   k     =   1             (   7   )             
 
 where d i  and d i-1  are the distances from the common origin and r i  and r i-1  are the radii of clusters i and i-1, respectively. If the ratio of the sum of the distance plus the radius of the further cluster i-1 over the difference of the distance less the radius of the closer cluster i is greater than 1, the two clusters do not overlap and the distance d i  of the further cluster need not be adjusted. 
 
         [0058]      FIG. 12  is a graph showing, by way of example, a pair of clusters  181 - 182  with overlapping bounding regions generated by the cluster display system  11  of  FIG. 1 . The pair of clusters  181 - 182  are respectively located at distances d  183 - 184  from a common origin  180 . A bounding region  187  for cluster  181  is formed by taking a pair of tangent vectors  185   a - b  from the common origin  180 . Similarly, a bounding region  188  for cluster  182  is formed by taking a pair of tangent vectors  186   a - b  from the common origin  180 . The intersection  189  of the bounding regions  187 - 188  indicates that the clusters  181 - 182  might either overlap or overlay and reorientation may be required.  
         [0059]      FIG. 13  is a graph showing, by way of example, a pair of clusters  191 - 192  with non-overlapping bounding regions generated by the cluster display system  11  of  FIG. 1 . The pair of clusters  191 - 192  are respectively located at distances d  193 - 194  from a common origin  190 . A bounding region  197  for cluster  191  is formed by taking a pair of tangent vectors  195   a - b  from the common origin  190 . Similarly, a bounding region  198  for cluster  192  is formed by taking a pair of tangent vectors  196   a - b  from the common origin  190 . As the bounding regions  197 - 198  do not intersect, the clusters  191 - 192  are non-overlapping and non-overlaid and therefore need not be reoriented.  
         [0060]      FIG. 14  is a flow diagram showing a routine  200  for checking for overlap of bounding regions for use in the routine  130  of  FIG. 10 . As described herein, the terms overlap and overlay are simply referred to as “overlapping.” The purpose of this routine is to identify clusters  17  (shown in  FIG. 1 ) that need not be reoriented due to the non-overlap of their respective bounding regions. The routine  200  is implemented as an overlap submodule in the reorient module  19  (shown in  FIG. 1 ).  
         [0061]     Thus, the bounding region of a first Cluster i  is determined (block  201 ) and the bounding region of a second Cluster j  is determined (block  202 ). If the respective bounding regions do not overlap (block  203 ), the second Cluster j  is skipped (block  204 ) and not reoriented. The routine then returns.  
         [0062]      FIG. 15  is a data representation diagram  210  showing, by way of example, a view  211  of overlapping non-circular cluster  213 - 216  generated by the clustered display system  11  of  FIG. 1 . Each cluster  213 - 216  has a center of mass c m    217 - 220  and is oriented around a common origin  212 . The center of mass as c m  of each cluster  213 - 216  is located at a fixed distance d  221 - 224  from the common origin  212 . Cluster  218  overlays cluster  213  and clusters  213 ,  215  and  216  overlap.  
         [0063]     As described above, with reference to  FIG. 2 , each cluster  213 - 216  represents multi-dimensional data modeled in a three-dimension display space. Furthermore, each of the clusters  213 - 216  is non-circular and defines a convex volume representing a data grouping located within the multi-dimensional concept space. The center of mass Cm at  217 - 220  for each cluster  213 - 216 , is logically located within the convex volume. The segment measured between the point closest to each other cluster along a span drawn between each pair of clusters is calculable by dimensional geometric equations, as would be recognized by one skilled in the art. By way of example, the clusters  213 - 216  represent non-circular shapes that are convex and respectively comprise a square, triangle, octagon, and oval, although any other form of convex shape could also be used either singly or in combination therewith, as would be recognized by one skilled in the art.  
         [0064]     Where each cluster  213 - 216  is not in the shape of a circle, a segment is measured in lieu of the radius. Each segment is measured from the center of mass  217 - 220  to a point along a span drawn between the centers of mass for each pair of clusters  213 - 216 . The point is the point closest to each other cluster along the edge of each cluster. Each cluster  213 - 216  is reoriented along the vector such that the edges of each cluster  213 - 216  do not overlap.  
         [0065]     While the invention has been particularly shown and described as referenced to the embodiments thereof, those skilled in the art will understand that the foregoing and other changes in form and detail may be made therein without departing from the spirit and scope of the invention.