# Patent

## Title
System and method for presenting independent variable geometric relationships

## Abstract

A pair of convex clusters are selected. Each convex cluster has a center of mass located at an original fixed distance from a common origin, and is oriented along a vector formed at a fixed angle from a common polar axis. A span is measured between the centers of mass of each convex cluster. A segment is measured from the center of mass of each convex shape to a point closest to the other convex shape along the span. A new fixed distance from the common origin for the center of mass for one of the convex clusters located along the vector for that convex cluster is evaluated if the span is less than the sum of the segments of the convex clusters. The pair of convex clusters are displayed rendered using at least the new fixed distance for the center of mass of the one convex cluster.

## Background
CROSS-REFERENCE TO RELATED APPLICATION
This patent application is a continuation of commonly-assigned U.S. Ser. No. 09/944,475, filed Aug. 31, 2001, now 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
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
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.
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.
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.
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.
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 21st Century,” Ch. 3, MacroVU Press (1998), the disclosure of which is incorporated by reference.
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.
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' 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.
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.
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
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 ci 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 sij between the centers ci and cj of the clusters is less than the sum of the radii ri and rj, a new distance di for the cluster is determined by setting the pair-wise span sij equal to the sum of the radii ri and r and solving the resulting quadratic equation for distance di. The operations are repeated for each pairing of clusters.
An embodiment of the present invention is a system and method for providing a perspective-corrected representation of a multi-dimensional cluster rendering. A span between centers for a pair of clusters is measured. Each center has an independent radius and is located in concept space at an independent distance along a vector drawn from a common origin and formed at a dependent angle from a common axis drawn through the common origin. A perspective-corrected independent distance from the common origin for one such cluster is determined if the span does not substantially equal the sum of the independent radii of the clusters. The perspective-corrected independent distance substantially equals a root of a quadratic equation of the vectors formed by the independent distances of the clusters and angle formed therebetween.
A further embodiment is a system and method for generating a visualized data representation preserving independent variable geometric relationships. A pair of convex clusters is selected. Each convex cluster is rendered on a display. Each convex cluster has a center of mass located at an original fixed distance from a common origin, and is oriented along a vector formed at a fixed angle from a common polar axis. A span is measured between the centers of mass of each convex cluster. For each convex cluster, a segment is measured from the center of mass of each convex shape to a point closest to the other convex shape along the span. A new fixed distance from the common origin for the center of mass for one of the convex clusters located along the vector for that convex cluster is evaluated if the span is less than the sum of the segments of the convex clusters. The pair of convex clusters is displayed rendered using at least the new fixed distance for the center of mass of the one convex cluster.
A further embodiment is a system and method for providing a perspective-corrected representation of a multi-dimensional convex shape rendering. A plurality of shapes is rendered. Each shape defines a convex volume representing a data grouping located within a multi-dimensional concept space and includes a center of mass logically located within the convex volume. A span is measured between the centers of mass for a pair of the convex shapes. Each convex shape is located at an independent distance along a vector drawn from a common origin and is formed at an independent angle from a common axis drawn through the common origin. A perspective-corrected independent distance from the common origin for one such convex shape is determined if the span does not substantially equal the sum of the distances of center of mass to point closest to the other convex shape of each convex shape. The perspective-corrected independent distance substantially equals a root of a quadratic equation formed by the independent distances of the convex shapes and angle formed there between.
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
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.
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.
FIG. 3 is a graph showing, by way of example, the polar coordinates of the overlapping clusters of FIG. 2.
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.
FIG. 5 is a data representation diagram showing, by way of example, an exploded view of the clusters of FIG. 2.
FIG. 6 is a data representation diagram showing, by way of example, a minimized view of the clusters of FIG. 2.
FIG. 7 is a graph, showing, by way of example, the polar coordinates of the minimized clusters of FIG. 5.
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.
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.
FIG. 10 is a routine for reorienting clusters for use in the method of FIG. 9.
FIG. 11 is a flow diagram showing a routine for calculating a new distance for use in the routine of FIG. 10.
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.
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.
FIG. 14 is a routine for checking for overlapping clusters for use in the routine of FIG. 10.
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
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.
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. Pat. No. 6,778,995, issued Aug. 17, 2004, the disclosure of which is incorporated by reference.
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.
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.
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.
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.
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. Pat. No. 6,978,274, issued Dec. 20, 2005 the disclosure of which is incorporated by reference.
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.
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.
FIG. 4 is a data representation diagram 50 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 are separated respectively by pair-wise spans s 60–65. Each span s 60–65 is also dependent on the independent variables radii r 41–44 (shown in FIG. 2) and angles θ 52–55.
For each cluster 33–36, the radii r is an independent variable. The distances d 56–59 and angles θ 52–55 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.
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 72–75, 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).
Although the “exploded” data visualization view 71 preserves the relative pair-wise spans s 60–65 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.
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. 4), the radii r 41–44 and angles θ 52–55 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′.
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 and angles θ 102–105 relative to the shared axis x 101 (shown in FIG. 2) are preserved. The new distances d′ 106–109 also approximate the proportionate pair-wise spans s′ 111–115 between the centers c 37–40.
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.
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.
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.
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.
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.
Thus, during the outer iterative loop (blocks 132–146), an initial Clusteri is selected (block 133) and the radius ri, center ci, angle θi, and distance di for the selected Clusteri are obtained (block 134). Next, during the inner iterative loop (blocks 135–145), another Clusterj (block 136) is selected and the radius rj, center cj, angle θj, and distance dj are obtained (block 137).
In a further embodiment, bounding regions are determined for Clusteri and Clusterj and the bounding regions are checked for overlap (block 138), as further described below with reference to FIG. 14.
Next, the distance di of the cluster being compared, Clusteri, is multiplied by the coefficient k (block 139) to establish an initial new distance d′i for Clusteri. A new center ci is determined (block 140). The span sij between the two clusters, Clusteri and Clusterj, is set to equal the absolute distance between center ci plus center cj. If the pair-wise span sij is less than the sum of radius ri and radius rj for Clusteri and Clusterj, respectively (block 143), a new distance di for Clusteri is calculated (block 144), as further described below with reference to FIG. 11. Processing of each additional Clusteri continues (block 145) until all additional clusters have been processed (blocks 135–145). Similarly, processing of each Clusterj (block 146) continues until all clusters have been processed (blocks 132–146), after which the routine returns.
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 ci of a selected clusteri from a common origin. In the described embodiment, the new distance d′i is determined by solving the quadratic equation formed by the distances di and dj and adjacent angle.
Thus, the sum of the radii (ri+rj)2 is set to equal the square of the distance dj plus the square of the distance di minus the product of the 2 times the distance dj times the distance di times cos θ (block 171), as expressed by equation (1):

(ri+rj)2=di2+dj2−2·didjcos θ  (1)

The distance di can be calculated by solving a quadratic equation (5) (block 172), derived from equation (1) as follows:





  
    
      
        
          
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In the described embodiment, the ‘±’ operation is simplified to a ‘+’ operation, as the distance di is always increased.

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 di divided by the old distance di (block 173), as expressed by equation (6):




  
    
      
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The routine then returns.

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):




  
    
      
        
          
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where di and di−1, are the distances from the common origin and ri and ri−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 di of the further cluster need not be adjusted.

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 185a–b from the common origin 180. Similarly, a bounding region 188 for cluster 182 is formed by taking a pair of tangent vectors 186a–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.
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 195a–b from the common origin 190. Similarly, a bounding region 198 for cluster 192 is formed by taking a pair of tangent vectors 196a–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.
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).
Thus, the bounding region of a first Clusteri is determined (block 201) and the bounding region of a second Clusterj is determined (block 202). If the respective bounding regions do not overlap (block 203), the second Clusterj is skipped (block 204) and not reoriented. The routine then returns.
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 cm 217–220 and is oriented around a common origin 212. The center of mass as cm 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.
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.
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 p air 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.
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.

## Claims

1. Computer-readable medium encoded with stored rendering data for output as a perspective-corrected visualization of data clusters, comprising:
a plurality of stored clusters to individually represent thematically related objects, each stored cluster visually defining a bounded area within a two-dimensional visualization with a center of mass located within the bounded area;
a concept space to span the two-dimensional visualization, which outputs the stored clusters positioned with each center of mass at a distance and angle of rotation determined from an origin common to the concept space; and
a perspective-corrected distance determined as a function of spacing between the centers of mass and the distances and angles of rotation for each pair of stored clusters, wherein the distance of at least one of the pair of stored clusters is conditionally replaced with the perspective-corrected distance;
a stored quadratic equation to provide the function by which the perspective-corrected distance can be determined, wherein, for the at least one pair of clusters di and dj, the perspective-corrected distance di for the cluster i is calculated according to the quadratic equation comprising:





  
    d
    i
  
  =
  
    
      
        (
        
          
            2
            ·
            
              d
              j
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          θ
        
        )
      
      ±
      
        
          
            
              (
              
                
                  2
                  ·
                  
                    d
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                θ
              
              )
            
            2
          
          -
          
            4
            ·
            
              (
              
                
                  d
                  j
                  2
                
                -
                
                  
                    [
                    
                      
                        r
                        i
                      
                      +
                      
                        r
                        j
                      
                    
                    ]
                  
                  2
                
              
              )
            
          
        
      
    
    2
  




where dj represents the distance for the cluster j, and ri and rj are radii for the clusters i and j, respectively; and the two-dimensional visualization is output on a display.

2. Stored rendering data according to claim 1, further comprising:
stored multidimensional data to provide the thematically related objects, wherein each thematically related object corresponds to a different dimension within the concept space.

3. An output device providing rendering data for output as a perspective-corrected visualization of data clusters, comprising:
a two-dimensional output area to render data for physical presentation in a display; and
an output buffer to store the rendering data pending output, comprising:
a plurality of clusters to individually represent thematically related objects, each cluster visually defining a bounded area within a two-dimensional visualization with a center of mass located within the bounded area; and
a concept space to span the two-dimensional visualization, which outputs the clusters positioned with each center of mass at a distance and angle of rotation determined from an origin common to the concept space; and
a perspective-corrected distance determined as a function of spacing between the centers of mass and the distances and angles of rotation for each pair of clusters, wherein the distance of at least one of the pair of clusters is conditionally replaced with the perspective-corrected distance,
a stored quadratic equation to provide the function by which the perspective-corrected distance can be determined, wherein, for the at least one pair of clusters di and dj, the perspective-corrected distance di for the cluster i is calculated according to the quadratic equation comprising:






  
    d
    i
  
  =
  
    
      
        (
        
          
            2
            ·
            
              d
              j
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          θ
        
        )
      
      ±
      
        
          
            
              (
              
                
                  2
                  ·
                  
                    d
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                θ
              
              )
            
            2
          
          -
          
            4
            ·
            
              (
              
                
                  d
                  j
                  2
                
                -
                
                  
                    [
                    
                      
                        r
                        i
                      
                      +
                      
                        r
                        j
                      
                    
                    ]
                  
                  2
                
              
              )
            
          
        
      
    
    2
  




where dk represents the distance for the cluster j, and ri and rj are radii for the clusters i and j, respectively; and wherein clusters are visually rendered on the display of the two-dimensional output.

4. An output device according to claim 3, further comprising:
multidimensional data further stored in the output buffer to provide the thematically related objects, wherein each thematically related object corresponds to a different dimension within the concept space.

5. An output device according to claim 3, wherein the display of the two-dimensional output is selected from the group comprising a CRT display, LCD display, and printer.

6. A system for rendering data for output as a perspective-corrected visualization of data clusters, comprising:
a memory, comprising:
a plurality of stored clusters to individually represent thematically related objects, each stored cluster visually defining a bounded area within a two-dimensional visualization with a center of mass located within the bounded area;
a concept space to span the two-dimensional visualization, which outputs the stored clusters positioned with each center of mass at a distance and angle of rotation determined from an origin common to the concept space; and

a processor, comprising:
a reorientation module to determine a perspective-corrected distance as a function of spacing between the centers of mass and the distances and angles of rotation for each pair of stored clusters; and
a visualization module to render the stored clusters for output on the two-dimensional output, wherein the distance of at least one of the pair of stored clusters is conditionally replaced with the perspective-corrected distance,
a stored quadratic equation to provide the function by which the perspective-corrected distance can be determined, wherein, for the at least one pair of clusters di and dj, the perspective-corrected distance di for the cluster i is calculated according to the quadratic equation comprising:






  
    d
    i
  
  =
  
    
      
        (
        
          
            2
            ·
            
              d
              j
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          θ
        
        )
      
      ±
      
        
          
            
              (
              
                
                  2
                  ·
                  
                    d
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                θ
              
              )
            
            2
          
          -
          
            4
            ·
            
              (
              
                
                  d
                  j
                  2
                
                -
                
                  
                    [
                    
                      
                        r
                        i
                      
                      +
                      
                        r
                        j
                      
                    
                    ]
                  
                  2
                
              
              )
            
          
        
      
    
    2
  




where dj represents the distance for the cluster j, and ri and rj are radii for the clusters i and j, respectively; and
displaying the two-dimensional visualization.

7. A system according to claim 6, further comprising:
a coefficient submodule to determine a coefficient to modify the perspective-corrected distance as applied to the at least one stored cluster for which the distance is replaced proportionately to the at least one pair of stored clusters.

8. A system according to claim 7, wherein the coefficient k is calculated according to the equation comprising:




  k
  =
  
    
      d
      
        i
        ⁡
        
          (
          new
          )
        
      
    
    
      d
      
        i
        ⁡
        
          (
          old
          )
        
      
    
  




where di(new) represents the perspective-corrected distance and di(old) represents the distance for the at least one cluster.

9. A system according to claim 6, wherein the memory further comprises stored thematically related objects to provide multidimensional data, wherein each thematically related object corresponds to a different dimension within the concept space.

10. A system according to claim 6, further comprising:
an overlap submodule to replace the distance of the at least one stored cluster when the bounded areas of the at least one pair of stored clusters overlap within the two-dimensional visualization.

11. Computer-readable medium encoded with a structured record to store rendering data for output as a perspective-corrected visualization of data clusters, comprising:
structured data collectively organized as a database record, comprising:
a plurality of stored clusters to individually represent thematically related objects, each stored cluster visually defining a bounded area within a two-dimensional visualization with a center of mass located within the bounded area;
a concept space to span the two-dimensional visualization, which outputs the stored clusters positioned with each center of mass at a distance and angle of rotation determined from an origin common to the concept space; and
a perspective-corrected distance determined as a function of spacing between the centers of mass and the distances and angles of rotation for each pair of stored clusters,
a stored quadratic equation to provide the function by which the perspective-corrected distance can be determined, wherein, for the at least one pair of clusters di and dj, the perspective-corrected distance di for the cluster i is calculated according to the quadratic equation comprising:






  
    d
    i
  
  =
  
    
      
        (
        
          
            2
            ·
            
              d
              j
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          θ
        
        )
      
      ±
      
        
          
            
              (
              
                
                  2
                  ·
                  
                    d
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                θ
              
              )
            
            2
          
          -
          
            4
            ·
            
              (
              
                
                  d
                  j
                  2
                
                -
                
                  
                    [
                    
                      
                        r
                        i
                      
                      +
                      
                        r
                        j
                      
                    
                    ]
                  
                  2
                
              
              )
            
          
        
      
    
    2
  




where dj represents the distance for the cluster j, and ri and rj are radii for the clusters i and j, respectively; and
wherein the distance of at least one of the pair of stored clusters is conditionally replaced with the perspective-corrected distance and the two-dimensional visualization is output on a display.

12. A structured record according to claim 11, wherein the perspective-corrected distance is modified by a coefficient as applied to the at least one stored cluster for which the distance is replaced proportionately to the at least one pair of stored clusters.

13. A structured record according to claim 11, wherein the distance of the at least one stored cluster is replaced when the bounded areas of the at least one pair of stored clusters overlap within the two-dimensional visualization.

14. A method for rendering data for output as a perspective-corrected visualization of data clusters, comprising:
individually representing thematically related objects as a plurality of clusters, each cluster visually defining a bounded area within a two-dimensional visualization with a center of mass located within the bounded area;
defining a concept space to span the two-dimensional visualization;
outputting the clusters positioned with each center of mass at a distance and angle of rotation determined from an origin common to the concept space;
determining a perspective-corrected distance as a function of spacing between the centers of mass and the distances and angles of rotation for each pair of clusters;
defining a quadratic equation as the function by which the perspective-corrected distance can be determined, wherein, for the at least one pair of clusters di and dj, the perspective-corrected distance di for the cluster i is calculated according to the quadratic equation comprising:





  
    d
    i
  
  =
  
    
      
        (
        
          
            2
            ·
            
              d
              j
            
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          θ
        
        )
      
      ±
      
        
          
            
              (
              
                
                  2
                  ·
                  
                    d
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                θ
              
              )
            
            2
          
          -
          
            4
            ·
            
              (
              
                
                  d
                  j
                  2
                
                -
                
                  
                    [
                    
                      
                        r
                        i
                      
                      +
                      
                        r
                        j
                      
                    
                    ]
                  
                  2
                
              
              )
            
          
        
      
    
    2
  



 where dj represents the distance for the cluster j, and ri and rj are radii for the clusters i and j, respectively:
conditionally replacing the distance of at least one of the pair of clusters with the perspective-corrected distance; and
displaying the two-dimensional visualization.

15. A method according to claim 14, further comprising:
providing multidimensional data as the thematically related objects, wherein each thematically related object corresponds to a different dimension within the concept space.

16. A method according to claim 14, further comprising:
replacing the distance of the at least one cluster when the bounded areas of the at least one pair of clusters overlap within the two-dimensional visualization.

17. A method according to claim 14, further comprising:
a coefficient submodule to determine a coefficient to modify the perspective-corrected distance as applied to the at least one cluster for which the distance is replaced proportionately to the at least one pair of clusters.

18. A method according to claim 17, wherein the coefficient k is calculated according to the equation comprising:




  k
  =
  
    
      d
      
        i
        ⁡
        
          (
          new
          )
        
      
    
    
      d
      
        i
        ⁡
        
          (
          old
          )
        
      
    
  




where di(new) represents the perspective-corrected distance and di(old) represents the distance for the at least one cluster.

19. A computer-readable storage medium holding code for performing the method according to claim 14.