Computer-Implemented System and Method For Correcting A Rendering Of Clusters

Abstract

A computer-implemented system and method for correcting a rendering of clusters is provided. A pair of clusters is selected within a representation. A span between centers of the clusters is determined. Radii for each of the clusters in the pair is identified. The radii are summed and at least one of the clusters in the pair is moved within the representation when the span exceeds the sum.

Description

FIELD

The present invention relates in general to data visualization and, in particular, to a system and method for correcting a rendering of clusters.

BACKGROUND

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 21^st 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

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.

An embodiment provides a computer-implemented system and method for correcting a rendering of clusters. A pair of clusters is selected within a representation. A span between centers of the clusters is determined. Radii for each of the clusters in the pair is identified. The radii are summed and at least one of the clusters in the pair is moved within the representation when the span exceeds the sum.

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, to Gallivan, 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, to Gallivan, 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 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.

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.

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).

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.

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′.

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.

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 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).

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.

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.

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.

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_i²+d_j²−2·d_id_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:

$\begin{array}{cc}1·d_i^2+\left(2·d_j\cos \theta \right)·d_i=\left(d_j^2-{\left[r_i+r_j\right]}^2\right)& \left(2\right)\\ 1·d_i^2+\left(2·d_j\cos \theta \right)·d_i-\left(d_j^2-{\left[r_i+r_j\right]}^2\right)=0& \left(3\right)\\ d_i=\frac{\left(2·d_j\cos \theta \right)\pm \sqrt{{\left(2·d_j\cos \theta \right)}^2-4·1·\left(d_j^2-{\left[r_i+r_j\right]}^2\right)}}{2·1}& \left(4\right)\\ d_i=\frac{\left(2·d_j\cos \theta \right)\pm \sqrt{{\left(2·d_j\cos \theta \right)}^2-4·\left(d_j^2-{\left[r_i+r_j\right]}^2\right)}}{2}& \left(5\right)\end{array}$

In the described embodiment, the ‘±’ operation is simplified to a ‘+’ operation, as the distance d_i 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 d_i divided by the old distance d_i (block 173), as expressed by equation (6):

$\begin{array}{cc}k=\frac{d_{i_{\mathrm{new}}}}{d_{i_{\mathrm{old}}}}& \left(6\right)\end{array}$

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):

$\begin{array}{cc}\mathrm{if}\frac{d_{i-1}+r_{i-1}}{d_i-r_i}>1,\mathrm{then}k=1& \left(7\right)\end{array}$

where d_i and d_i-1 are the distances from the common origin r_i and r_i-1 and 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.

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 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.

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.

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 c_m 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 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.

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. A computer-implemented system for correcting a rendering of clusters, comprising: a selection module to select a pair of clusters within a representation;a determination module to determine a span between centers of the clusters;an identification module to identify radii for each of the clusters in the pair and summing the radii;an orientation module to move at least one of the clusters in the pair within the representation when the span exceeds the sum; anda processor to execute the modules.

2. A system according to claim 1, wherein each of the clusters is oriented within the representation at a distance from a common origin.

3. A system according to claim 2, further comprising: a representation module to further orient each cluster in the representation at a fixed angle along a common axis drawn through the common origin.

4. A system according to claim 2, wherein the orientation module increases the distance of the at least one cluster by multiplying the distance of that cluster by a variable coefficient.

5. A system according to claim 2, further comprising: a coefficient module to identify a fixed coefficient;a distance determination module to determine a further distance for each of the clusters in the pair by multiplying the identified fixed coefficient by the distance of that cluster from the common origin; andthe orientation module to reorient each of the clusters away from the common origin based on the further determined distance.

6. A system according to claim 2, wherein the span is proportionately increased relative to the reorientation of the clusters.

7. A system according to claim 2, further comprising: a representation module to increase the distance of the clusters as a similarity of concepts represented by each cluster decreases.

8. A system according to claim 1, wherein the clusters each comprise a non-circular shape having a convex volume and the center comprises a center of mass.

9. A system according to claim 8, further comprising: a segment module to determine a segment from the center of mass to a point along the span drawn between the centers of mass for the pair as the radii for each of the non-circular clusters in the pair.

10. A system according to claim 1, wherein the radii of each cluster signifies a number of documents attracted to that cluster.

11. A computer-implemented method for correcting a rendering of clusters, comprising the steps of: selecting a pair of clusters within a representation;determining a span between centers of the clusters;identifying radii for each of the clusters in the pair;summing the radii; andmoving at least one of the clusters in the pair within the representation when the span exceeds the sum,wherein the steps are performed on a suitably-programmed computer.

12. A method according to claim 11, wherein each of the clusters is oriented within the representation at a distance from a common origin.

13. A method according to claim 12, further comprising: further orienting each cluster in the representation at a fixed angle along a common axis drawn through the common origin.

14. A method according to claim 12, further comprising: increasing the distance of the at least one cluster by multiplying the distance of that cluster by a variable coefficient.

15. A method according to claim 12, further comprising: identifying a fixed coefficient;determining a further distance for each of the clusters in the pair by multiplying the identified fixed coefficient by the distance of that cluster from the common origin; andreorienting each of the clusters away from the common origin based on the further determined distance.

16. A method according to claim 12, further comprising: proportionately increasing the span relative to the reorientation of the clusters.

17. A method according to claim 12, further comprising: increasing the distance of the clusters as a similarity of concepts represented by each cluster decreases.

18. A method according to claim 11, wherein the clusters each comprise a non-circular shape having a convex volume and the center comprises a center of mass.

19. A method according to claim 18, further comprising: determining a segment from the center of mass to a point along the span drawn between the centers of mass for the pair as the radii for each of the non-circular clusters in the pair.

20. A method according to claim 11, wherein the radii of each cluster signifies a number of documents attracted to that cluster.