Multi-dimensional graphing in two-dimensional space

Abstract

A method and system for displaying a function in two dimensions where the function is made up of numerous independent variables and at least one dependent variable. A new independent variable value can be defined having values which correspond to the multiple dependent variables. The variable values are read into a computer and the independent variables are then ranked by the user from fastest- to slowest-running variable. Each dependent variable corresponding to the new independent variable is plotted along the Y-axis. The independent variable values are plotted along the X-axis in a hierarchical manner. The hierarchical manner involves a nesting of fastest-running variables within slower-running variables. Rectangles are then drawn to correspond to each variable value. Each rectangle horizontally encloses the faster-running variables associated with it.

Description

sequentially displaying means, second sequentially displaying means, first repeating means, and second repeating means. Using the first repeating means, the system gets 140 the first independent variable, and sequentially displays 145 the results corresponding to the values associated with the fastest-running variable for a first value of the next fastest-running variable and a first value of each of the remaining independent variables. After the displaying 145 of the results for the fastest-running independent variable, a rectangle is displayed 150, using group means, by horizontally enclosing the displayed results associated with the fastest-running independent variable.

The system then gets 155 a next value of the next fastest-running independent variable and sequentially displays 160 the result, using the second sequentially displaying means, corresponding to the values associated with the fastest-running independent variable for a next value of the next fastest-running independent variable and a first value for each of the remaining independent variables. The first repeating means repeats 165 getting 155 a next value and displaying 160 the results for each remaining value of the ext fastest-running independent variable until there are no remaining values of the next fastest-running independent variable.

Using the group means, the system displays 170, on the display, a rectangle horizontally enclosing the displayed results associated with the next fastest-running independent variable, while also enclosing the previously displayed rectangle, thereby nesting the faster-running variables within the rectangles of the slower-running variables according to the hierarchical plotting of the results.

The system, using second repeating means, gets 180 a next fastest-running independent variable and repeats steps 145 to 170 of FIG. 1 until 175 there are not remaining next faster-running independent variables, i.e. the last value of the slowest-running independent variable is displayed. The system, using looping means, then gets 195 a next new independent variable and repeats steps 140 to 180 until 190 there are no remaining new independent variables, and the plotting of the variables and nesting of results in rectangles stops 200.

The multi-dimensional graphing and two-dimensional space method and system works by displaying hierarchical rectangles in different colors. The fastest-running variables are displayed in the present embodiment as "hash" marks. These hash marks can be thought of as rectangles having zero height. The next fastest variable then becomes a rectangle encompassing the fastest-running variable values throughout the range of the next fastest-running variable. This iteration of next fastest-running variables continues until the slowest-running variable is accounted for. This results in a nesting of rectangles as shown in FIGS. 3A, 3B and 3C. FIGS. 3A, 3B, and 3C are is a graphical representation using the present invention of the Ideal Gas Law. The Ideal Gas Law is described in the form P=nRT/V, where P is pressure, n is a number of moles, R is the gas constant, T is the temperature (in degrees Kelvin), and V is the volume occupied by the gas.

The ranking of the independent variables T, n and V is shown is FIG. 2 in their corresponding positions on the X-axis. In this particular case, T is designated as the fastest-running variable and is illustrated along the number line as the smallest set of hash marks; n is the next fastest-running variable, and this is shown as the next largest set of hash marks. Finally, V is shown as the slowest-running variable, and this is illustrated as the largest set of hash marks.

It can be seen in FIG. 2 that the fastest-running variables are nested within each next fastest-running variable and then repeated for the negative value of the next fastest-running variable. This translates to T values of 1, 2, 3 and 4, while n=0 and V=0. Then, T runs through 1, 2, 3 and 4, while n=1 and V=0, etc., until completed for all four values of n (1-4). This cycle is then repeated for all values of V from 1-4.

The result of this is shown in FIG. 3A. In FIG. 3A, three colors are used. Color selections are made by the operator. White (14) designates the fastest-running variable T, while blue (15) represents the next fastest-running variable n and, finally, orange (16) represents the slowest-running variable V. When looking at the graph, the value P is plotted along the Y-axis while the independent variable values are plotted along the X-axis.

The white hash marks designate the T value. The white hash marks are connected by splines (17), whose purpose is to aid in the visual interpretation. They are not a requirement of the present method and system, but instead a useful interpretative tool. The splines are used in connecting groups of rectangles, i.e., fastest, next fastest and slowest-running variables.

The blue rectangles represent the value of n. Each blue rectangle encompasses four of the white hash marks (fastest-running variables), as there are four values of T for each value of n. The blue rectangles are also connected by splines.

Finally, the orange rectangles which represent the variable values V encompass four blue rectangles. This is a result of there being four values of n for each value of V.

Thus, a nesting of rectangles in a hierarchical fashion illustrates the graph of the Ideal Gas Law. One can view the different groupings of variables at any point to interpret the data set, while at the same time seeing conditions on either side of that location for the entire set of variable values.

FIG. 3B shows a graph representing the gaussian function w=e-(x**2+y**2+z**2) using the invention.

FIG. 4 shows an overview of the program structure of an embodiment of the present invention. This system chart of the program structure shows the available tools all connected to a main event loop 27. Main event loop 27 is shown in FIG. 5. All tools and operational commands are initiated via subroutine calls in the present embodiment of the invention. The tools are shown in FIGS. 6-13.

In FIG. 5, there is shown a flow chart of the basic operation of main event loop 27. The main event loop is a central point of program flow. After the program embodying the present invention is initialized, it enters the main event loop and all subsequent actions are dispatched from here. An event is usually some user input requesting some action of the program. Once the main event loop is entered, it continually scans for an event (i.e., tool). When an event is detected (received), the main event loop determines what action should be taken and issues the appropriate function calls subroutine calls). After the function (tool) completes its execution, the main event loop resumes scanning for user input.

In block 42, the main event loop awaits for user action. Then, after user action such as the toggling of buttons on a mouse or pressing of keys on a keyboard, an appropriate function call is made in block 43. The function (or subroutine) is called and completed and then processing returns to the main event loop in block 44. The main event loop then keeps cycling waiting for user input before making the appropriate function calls.

FIG. 6 shows a flow chart of the Zoom In tool (20 in unscaled mode or Zoom In tool 21 in scaled mode). The Zoom In tool reduces the dimensionality of the plotted space. One of the currently displayed second slowest-running variables is selected from the currently displayed slowest-running variable, and this selected second slowest-running variable becomes the currently displayed slowest-running variable. The net effect of this tool is to zoom in on one of the spaces or subspaces. This tool can be used for finding maxima and minima.

In the scaled version, the zoomed-in subspace is proportioned to the size of the display screen. In the unscaled Zoom In, the selected subspace is kept in its original proportion as in the space from which it was selected. Unscaled tools allow the user to see tendencies, such as decay and growth.

The Zoom In tool operates by grabbing the position of the variable space to be zoomed in, as shown in block 45. This can be accomplished by pointing to the variable space (rectangle) using a mouse or other pointing device. Next, the plotted space (displayed space) is set to the subspace selected in block 45. This is shown in block 46. In block 47, the display space is repainted in either scaled or unscaled version showing the zoomed-in subspace. In block 48, control returns to the main event loop to continue scanning for new events.

FIG. 7 shows a flow chart of the Zoom Out tool. The Zoom Out tool corresponds to block 22 (in unscaled form) and block 23 (in scaled form) of FIG. 4. The Zoom Out tool works inversely to the Zoom In tool and, as such, increases the dimensionality of the plot. Note that the dimensionality cannot be increased above the maximum starting value. The subspace which runs slower than the currently displayed slowest-running variable becomes the currently displayed slowest-running variable. This, again, is up to the maximum starting value.

An index is kept when zooming into subspaces so that the control system of the present invention can monitor the level of display of the current displayed space.

In block 49, the previous (zoomed-in) subspaced index is retrieved. In block 50, the plotted space (displayed space) is set to this subspace index value. In block 51, the plotted space is repainted to the subspace corresponding to this index value. Then, in block 52, control returns to the main event loop and scans for new events. This can be accomplished in numerous ways, and in the present embodiment is operated by clicking on one of the mouse buttons. It could just as easily be configured to work via keyboard commands.

FIG. 8 shows a flow chart for the Animate tool. The Animate tool sequentially displays each subspace in the currently displayed slowest-running variable. In the present embodiment, the sequential display cycles continually over the subspaces until the user terminates the animation. It is possible in other embodiments to set the cycling to a designated number. It is also possible to have a manually operated cycling operated by a pointing device such as a mouse, or through keyboard commands.

The Animate tool can be operated in unscaled mode 24 or scaled mode 26. As with the other tools, scaled mode proportionally adjusts the currently displayed subspace to fill the display screen, while unscaled mode maintains the sizing of the designated subspace without adjustment.

The Animate tool operates by first retrieving the subspace index in block 53. At decision block 54, it is determined whether to continue with the Animation process. Should the user desire to continue, processing moves along to block 55, where the plotted space is set to the subspace index. In block 56, the plotted space is repainted according to the display for the subspace index. This will be in either scaled or unscaled mode, depending on the user's selection. In block 57, the next subspace index is obtained. Processing then returns to decision block 54, and the user determines whether or not to continue with the animation. Should the decision be "NO," processing continues to block 58 where the plotted space is set to the last subspace index. The screen is then repainted in block 59, and processing returns to the main event loop in block 60.

FIG. 9 shows a flow chart for the Expander tool. The Expander tool is applied about a particular point in the multi-dimensional space, and displays the variation along each independent variable using a homogeneous horizontal increment for each variable rather than the hierarchical increment which is the basis for the multi-dimensional graphing and two-dimensional spaced method and system. The Expander tool takes a section of each variable through the point expanded upon, but does not sample all points in the display space. This tool is useful for tasks such as finding minima and maxima.

The expander tool allows one to view how the dependent variable changes as one moves away from the point in question in the white independent variable direction until one reaches the edges of the data domain, similarly for the blue, red, etc., independent variables. This tool can clearly be generalized by showing variations as one moves away from the point of expansion in more complex ways that involve non-parallel moves. For example, one could show the variations that occur when, in addition to the standard expander tool moves, one also displays moves about each point that correspond to incrementing all of the other colored variables by ± one. Further generalizations can involve all possible moves about the new points until, in fact, one could show all possible paths through the N dimensional space.

In block 61, the position or point is grabbed. In block 62, a new window is created for displaying the results of the expansion. In block 63, the lines representing the expansion through the point are painted. The painting of the lines is completed in the colors representing the corresponding independent variables. In block 64, processing returns to the main event loop.

FIG. 10 shows a flow chart for the operation of the General Zoom tool. This General Zoom tool sets the limits, left and right, of the currently displayed slowest-running variable. The General Zoom tool does not change the currently displayed slowest-running variable. This tool is useful for showing portions of the currently displayed subspace. The General Zoom tool can be used in scaled (block 37) or unscaled (block 38) mode. The scaling and unscaling is exactly the same as has been described for the previous tools. This, as always, is a user designation.

In block 65, the position on the X-axis is obtained for the left and right boundaries. In block 66, these boundaries are set as left and right limits. In block 67, the subspace is set to the left and right limits which were set in block 66. In block 68, the displayed subspace in repainted with the new left and right boundaries. In block 69, processing returns to the main event loop.

In the present embodiment of the invention, the General Zoom tool applies for all subspaces that the user now goes into and out of. This is a design choice, and is not a limitation of this tool in the present invention. Also, the General Zoom can be reset to the original limits in block 33 of FIG. 4.

FIG. 11 shows a flow chart for the Decimate/Undecimate tools 32, 34, 35 and 36. As with the other tools, the Decimate and Undecimate tools operate in a scaled or unscaled mode.

The Decimate tool decreases the number of currently displayed slowest-running variable subspaces by only plotting every Nth subspace, where N is the level of decimation. The Undecimate tool operates in the opposite manner, but is limited to undecimating only decimated subspaces.

Without the Decimator tool, an obvious drawback to this embodiment of the invention is that each data point uses at least one horizontal pixel. Since work station monitors generally have about 10³ pixels horizontally, this obviously limits the number of total data points displayed at any one time to 10³. This is despite the fact that multi-dimensional problems tend to require large numbers of data points.

The Decimator tool allows a fraction of the total distinct values for each variable to be displayed. In many cases, this still allows useful interpretation of the data and functions. For example, in a data set that has 10⁶ data points, one can show only the first, fourth, seventh and tenth values for each variable, hence, reducing the total number of points that need to be displayed to 4⁶, or 4096. This makes it necessary to scroll only four frames, instead of one thousand, to see the "entire" data set.

In order to get a detailed look at a particular subspace, the zoom tool can be used. It is also possible to decimate certain variables in certain increments, while other variables in other increments. In another possible embodiment of this invention, a combination Zoom and Decimator tool is used for handling large data sets.

In block 70, the decimate level is set. This can be done through clicking the buttons on a pointing device, such as a mouse (in the present embodiment), or through keyboard input. In block 71, the subspace is repainted incorporating the decimation or undecimation level. Finally, in block 72, processing control returns to the main event loop.

FIG. 12 shows a flow chart for the Permute tool. The Permute tool changes the hierarchical assignment of the independent variables. The starting assignment is used as a reference for all future assignments. The functional dependence remains unchanged after using the Permute tool. It is only the order in which the data are plotted which is changed. In short, the Permute tool allows for the exchange of the rankings of the independent variables. This is very useful for determining which ranking gives the most useful or most beneficial visual results.

A related tool is the Array Plot tool 31 of FIG. 4. The Array Plot tool can show all or some combinations of rankings of independent variables in the display space. This allows the user to select which ranking gives the best or desired visual results.

In the present embodiment of the invention, the Permute tool works between pairs of variables. This pairwise exchange has been found to be a very practical way of using the Permute tool, but is not a limitation of the present invention.

In block 73, the subspaces are set to permute. In block 74, the data is rearranged according to the new ranking of the independent variables. In block 75, the displayed space is repainted according to the permutation. Finally, in block 76, processing control returns to the main event loop.

FIG. 13 shows a flow chart for the Cloning tool 30. The Cloning tool simply makes a copy of the currently displayed plot and places it in a window in another part of the screen. This allows the concurrent display of various subspaces. These displayed subspaces can be operated on by various tools to show an overall picture for the user. It can also be used to show different "zooms" at the same time for the user.

In block 77, the current plot is painted into the cloning space. In block 78, the clone spot is repainted onto screen. In block 79, processing control returns to the main event loop.

There are other tools in the program structure of FIG. four 4, such as a Resize Tool 40 and Resize Panel 39. The Resize Tool is used for changing the size of the display space. The Resize Panel is used for changing the size of the display panel which monitors the operation of the display space and the various tools operating on it at any given time.

The Splines tool 29 draws lines between rectangles to be used as a guide for the eye. The Splines that are drawn between rectangles are drawn according to the following criteria:

Y=ymin if |ymin|>=|ymax|

Y=ymax if |ymin|<|ymax|

where ymin and ymax are the minima and maxima of the rectangle through which the spline is drawn. The splines are drawn hierarchically, joining rectangles of the same subspace together.

The nested hierarchical rectangles of the present invention correspond to the behavior of the dependent variable W over independent variable subspaces of various dimensionalities. The following formulas are for the vertical and horizontal locations and extents of these rectangles in world coordinates (not screen coordinates). The corresponding screen coordinates would be measured from the lower left corner of the X window and would, in general, be offset in the scale for each independent variable to reflect the fact that the starting values of each variable may not be zero, and the increment value may vary from one independent variable to the next.

It is useful to denote the independent variables as X₁, X₂ . . . X_n (instead of Xwhite, Xblue, where the colors pertain to the rectangle colors). Here, X₁ is the fastest-running variable, X₂ is the second fastest-running variable, and so on. Associated with each value of X₁ is an independent variable subspace of dimension d=0, i.e., a point. Associated with each value of X₂ is an independent variable subspace of dimension d=1, i.e., a line (of points). In general, each value of X_L corresponds to an independent variable subspace of dimension d=L-1, and has a corresponding rectangle (which may be thought of as corresponding to a subspace of dimension L, i.e., L-1 independent variables along the horizontal and one dependent variable, namely w, along the vertical).

Each independent variable X_L takes on values

X_L,i =X_Ls +(i-1)ΔX_L

with i=1 to N_L

In general, the starting values X_Ls may differ, as may the increments ΔX_L and the total number of values NL. In the formulas given below, we will set X_Ls =0 and X_L =1 ΔX_L =1 which, in fact, corresponds more closely to the actual screen displayed rectangles and is essential in order to obtain correct formulas for the locations and extents of the rectangles.

Claims

1. A method using a system including a computer with a display for displaying a function in two dimensions, said display defined by an X-axis and a Y-axis wherein said function is comprised of a plurality of at least three discrete independent variables and at least one dependent variable, each independent variable having associated values, the method comprising the steps of:

a. defining a new independent variable for said dependent variables, said new independent variables variable having associated values, each of said values representing one dependent variable;

b. reading the values associated with all of the independent variables;

c. selecting which independent variable is a fastest-running independent variable, and ranking the selected independent variable as the fastest-running independent variable;

d. selecting which independent variable is a next fastest-running independent variable, and ranking the selected independent variable as the next faster fastest-running independent variable;

e. repeating step (d) until all independent variables are ranked, with a last ranked independent variable being a slowest-running independent variable;

f. determining a result for each value of a the new independent variable and for all combinations of the values of all independent variables other than the new independent variables variable;

g. plotting the results determined in step (f) along for at least one value of the new independent variable in the Y-axis direction, with the corresponding values associated with all independent variables along the X-axis in a hierarchical manner, said plotting comprising the steps of:

(1) sequentially displaying horizontally next to one another the result results corresponding to the values associated with the fastest-running independent variable for a first value of the next faster fastest-running independent variable and a first value of each of the remaining independent variables;

(2) sequentially displaying horizontally next to one another, in a same manner as for the first value of the next fastest-running independent variable so as to preserve horizontal spacing of the fastest-running independent variable's values, the result results corresponding to the values associated with the fastest-running independent variable for a second value for the next fastest-running independent variable and a first value for each of the remaining independent variables, the horizontally displaying next to one another of the results being in such a manner that the ranking of the independent variables remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking;

(3) repeating steps (g)(2) for each remaining value of the next fastest-running independent variable;

(4) repeating steps (g)(1)-(g)(3) for each value of any remaining independent variable, preserving horizontal spacings of any given independent variable's values for every occurrence in a nested hierarchy, until completed for a last value of the slowest-running independent variable, the horizontally displaying next to one another of the results being in such a manner that the ranking of the independent variables from fastest to slowest remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking;

h. displaying a group of rectangles, each rectangle having a first vertical extreme and a second vertical extreme, said rectangles horizontally enclosing the plotted results associated with an independent variable, said group of rectangles containing a nesting of rectangles of faster-running independent variables horizontally within the rectangles of slower-running independent variables according to the hierarchical manner in which the results associated with the independent variables are plotted; and

i h. repeating steps (f) and (g) for each remaining new independent variable value.

2. The method of claim 1 29 wherein said group of rectangles displayed in step (h) are comprised of different colored rectangles.

3. The method of claim 2 wherein each color represents a different independent variable.

4. The method of claim 1 29 wherein said first vertical extreme of each rectangle is on the Y-axis line X-axis (Y=0) and the second vertical extreme of the rectangle representing the fastest-running variable is at the point corresponding to the value a result of each particular fastest-running variable value.

5. The method of claim 4 wherein the second vertical extreme of each horizontally enclosing rectangle is equal to the sum of the second vertical extremes of the horizontally enclosed rectangles associated with the next fastest-running any faster-running independent variable.

6. The method of claim 5 wherein the values of the dependent variables are both can include positive and negative results.

7. The method of claim 1 29 comprising the further step of selecting which of the rectangles displayed in step (h) is to be displayed first.

8. The method of claim 7 comprising the further step of selecting the order in which the remaining rectangles are displayed.

9. The method as set forth in claim 1 29 wherein the second vertical extreme of each rectangle is determined by a predetermined function.

10. The method of claim 9 wherein the predetermined function defines a subset of the nested rectangles.

11. The method of claim 10 wherein the nested rectangles correspond to the next fastest-running any faster-running independent variable.

12. The method of claim 1, wherein more than one new independent variable is defined in step (a), each of said more than one new independent variables corresponding to a set of dependent variables, and each new independent variable having associated new values relating to a property of said corresponding dependent variable variables.

13. A system using a computer for displaying a function in two dimensions on a display, said display defined by an X-axis and a Y-axis, wherein said function is comprised of a plurality of at least three discrete independent variables and at least one dependent variable, each independent variable having associated values, the system comprising:

a. defining means for defining a new independent variable for said dependent variables, said new independent variables variable having associated values, each of said values representing one dependent variable;

b. reading means for reading the values associated with all of the independent variables;

c. first selecting means for selecting which independent variable is a fastest-running independent variable and ranking the selected independent variable as the fastest-running independent variable;

d. second selecting means for selecting which independent variable is a next fastest-running independent variable and ranking the selected independent variable as the next faster fastest-running independent variable;

e. ranking means for repeating said selection of the next fastest-running independent variable until all independent variables are ranked with a last ranked independent variable being a slowest-running independent variable;

f. determining means for determining a result for each value of a the new independent variable and for all combinations of the values of all remaining independent variables other than the new independent variable;

g. plotting means for plotting said results along for at least one value of the new independent variable in the Y-axis direction with the corresponding values associated with the independent variables along the X-axis in a hierarchical manner, said plotting means comprising:

(1) first sequentially displaying means for sequentially displaying horizontally next to one another the result results corresponding to the values associated with the fastest-running independent variable for a first value of the next fastest-running independent variable and a first value of each of the remaining independent variables;

(2) second sequentially displaying means for sequentially displaying horizontally next to one another, in a same manner as for the first value of the next fastest-running independent variable so as to preserve horizontal spacing of the fastest-running independent variable's values and in such a manner that the ranking of the independent variables remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking, the result results corresponding to the values associated with the fastest-running independent variable for a second value of the next fastest-running independent variable and a first value for each of the remaining independent variables;

(3) first repeating means for repeating the sequential display of the results, horizontally next to one another, by the second sequentially displaying means for each remaining value of the next fastest-running independent variable;

(4) second repeating means for repeating the sequential display by the first sequentially displaying means, by the second sequentially displaying means, and by the first repeating means for each value of any remaining independent variable, preserving horizontal spacings of any given independent variable's values for every occurrence in a nested hierarchy and continuing in such a manner that the ranking of the independent variables from fastest to slowest remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking, until completed for a last value of the slowest-running independent variable;

h. group means for displaying a group of rectangles, each rectangle having a first vertical extreme and a second vertical extreme, said rectangles horizontally enclosing the plotted results associated with an independent variable, said group of rectangles containing a nesting of rectangles of faster-running independent variables within the rectangles of slower-running independent variables according to the hierarchical manner in which the results associated with the independent variables are plotted; and

i h. looping means for repeating the determining of the results by said determining means, and for repeating the plotting of the results by said plotting means for each remaining new independent variable value.

14. The method of claim 1 29 further comprising, after the step (h) of displaying a group of rectangles, the step of:

(h2) graphically connecting the plotted result of each fastest running variable nested within a rectangle of the next fastest running variable.

15. The method of claim 14 further comprising, after the step (h) of graphically connecting, the step of:

(h3) graphically connecting the plotted result of a fastest running variable located within a rectangle of the next fastest running variable with the plotted result of a fastest running variable located within all other rectangles of the next fastest running variable.

16. The method of claim 1 29 further comprising, after the step (h) of displaying a group of rectangles, the step of:

(h2) graphically connecting the plotted result of a fastest running variable located within a rectangle associated with a particular independent variable with a corresponding plotted result of a fastest running variable located within all other rectangles associated with the same independent variable.

17. The method of claim 16 further comprising, after the step (h2) of graphically connecting, the step of repeating the step (h2) of graphically connecting for all independent variables.

18. The method of claim 14 wherein the step (h2) of graphically connecting further includes graphically connecting the plotted results using a line.

19. The method of claim 14 wherein the step (h2) of graphically connecting further includes graphically connecting the plotted results using a curve.

20. The method of claim 1 29 wherein the step (h) of displaying a group of rectangles further includes displaying the rectangles with each rectangle having a height equal to the difference between a maximum enclosed result and a minimum value of the independent variable associated with the rectangle enclosed result.

21. The method of claim 1 29 wherein the step (h) of displaying a group of rectangles further includes graphically connecting the rectangles associated with a given independent variable.

22. The method of claim 21 wherein the step (h) of graphically connecting further includes graphically connecting the rectangles at a corresponding point in each rectangle.

23. The method of claim 22 wherein the corresponding point is a top center point of each rectangle.

24. The method of claim 1 wherein the values read in step (b) comprise a grid.

25. The method of claim 24 wherein step (b) further includes the steps of:

b1. reading a first value;

b2. reading a last value;

b3. reading an incremental value; and

b4. generating the grid using the incremental value to generate values between the first value and the last value to form the grid.

26. The method of claim 1 wherein step (g) further includes, after step (g)(4), the step of:

(5) displaying the results of steps (g)(1) to (g)(4) together.

27. A method using a system including a computer with a display for displaying a function of at least one three independent variable variables and at least one dependent variable, each independent variable having associated values, the method comprising the steps of:

a. defining a new independent variable for said dependent variables, said new independent variables variable having associated values;

b. reading the values associated with each independent variable;

c. selecting which independent variable is a fastest-running independent variable, and ranking the selected independent variable as the fastest-running independent variable;

d. selecting which independent variable is a next fastest-running independent variable, and ranking the selected independent variable as the next fastest-running independent variable;

e. repeating step (d) until al independent variables are ranked;

f. determining a result for each value of a the new independent variable and for all combinations of the values of all independent variables;

g. plotting the results determined in step (f) for at least one value of the new independent variable, said plotting comprising the steps of:

(1) sequentially displaying horizontally next to one another the result results corresponding to the values associated with the fastest-running independent variable for a first value of the next fastest-running independent variable and a first value of each of the remaining independent variables;

(2) sequentially displaying horizontally next to one anther the result results corresponding to the values associated with the fastest-running independent variable for a second value of the next fastest-running independent variable and a first value for each of the remaining independent variables, the horizontally displaying next to one another of the results being in such a manner that the ranking of the independent variables remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking;

(3) repeating steps (g)(2) for each remaining value of the next fastest-running independent variable;

(4) repeating steps (g)(1)-(g)(3) for each value of any remaining independent variable, the horizontally displaying next to one another of the results being in such a manner that the ranking of the independent variables from fastest to slowest remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking;

h. displaying a group of rectangles, said group of rectangles containing a nesting of rectangles of faster-running independent variables within the rectangles of slower-running independent variables; and

i h. repeating steps (f) and (g) for each remaining new independent variable value.

28. A system using a computer for displaying a function of at least one three independent variable variables and at least one dependent variable, each independent variable having associated values, the system comprising:

a. defining means for defining a new independent variable for said dependent variables, said new independent variables variable having associated values;

b. reading means for reading the values associated with each independent variable;

c. first selecting means for selecting which independent variable is a fastest-running independent variable and ranking the selected independent variable as the fastest-running independent variable;

d. second selecting means for selecting which independent variable is a next fastest-running independent variable and ranking the selected independent variable as the next fastest-running independent variable;

e. ranking means for repeating said selection of the next fastest-running independent variable until all independent variables are ranked;

f. determining means for determining a result for each value of a the new independent variable and for all combinations of the values of all remaining independent variables;

g. plotting means for plotting said results for at least one value of the new independent variable, said plotting means comprising:

(1) first sequentially displaying means for sequentially displaying horizontally next to one another the result results corresponding to the values associated with the fastest-running independent variable for a first value of the next fastest-running independent variable and a first value of each of the remaining independent variables;

(2) second sequentially displaying means for sequentially displaying horizontally next to one another in such a manner that the ranking of the independent variables remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking, the result results corresponding to the values associated with the fastest-running independent variable for a second value of the next fastest-running independent variable and a first value for each of the remaining independent variables;

(3) first repeating means for repeating the sequential display by the second sequentially displaying means for each remaining value of the next fastest-running independent variable;

(4) second repeating means for repeating the sequential display by the first sequentially displaying means, by the second sequentially displaying means, and by the first repeating means for each value of any remaining independent variable in such a manner that the ranking of the independent variables from fastest to slowest remains unchanged when using an extreme left of each result to determine the ranking as when using an extreme right of each result to determine the ranking, until completed for a last value of the slowest-running independent variable;

h. group means for displaying a group of rectangles, said group of rectangles containing a nesting of rectangles of faster-running independent variables within the rectangles of slower-running independent variables; and

i h. looping means for repeating the determining of the results by said determining means, and for repeating the plotting of the results by said plotting means for each remaining new independent variable value.

29. The method as set forth in claim 1, further comprising the step of:

displaying a group of rectangles, each rectangle having a first vertical extreme and a second vertical extreme, with at least one of said first vertical extreme and said second vertical extreme being determined by a property of at least one horizontally enclosed result, said rectangles horizontally enclosing the plotted results associated with a faster-running independent variable. 30. The system as set forth in claim 13, further comprising:

group means for displaying a group of rectangles, each rectangle having a first vertical extreme and a second vertical extreme, with at least one of said first vertical extreme and said second vertical extreme being determined by a property of at least one horizontally enclosed result, said rectangles horizontally enclosing the plotted results associated with a faster-running independent variable. 31. The method as set forth in claim 27, further comprising the step of:

displaying a group of rectangles, each rectangle having a first vertical extreme and a second vertical extreme, with at least one of said first vertical extreme and said second vertical extreme being determined by a property of at least one horizontally enclosed result, said rectangle horizontally enclosing the plotted results associated with a faster-running independent variable. 32. The system as set forth in claim 28, further comprising:

group means for displaying a group of rectangles, each rectangle having a first vertical extreme and a second vertical extreme, with at least one of said first vertical extreme and said second vertical extreme being determined by a property of at least one horizontally enclosed result, said rectangles horizontally enclosing the plotted results associated with a faster-running independent variable.