Abstract:
An approach for positioning screen elements on a display screen is disclosed herein. The screen elements represent expressions in a formula and may include one or more glyphs. Optimal positions are determined for the screen elements based on analyzing positional characteristics associated with displaying the screen elements at the resolution supported by the display screen against positional characteristics based on an optimal resolution. This analysis indicates whether the degree of displacement between display of a screen element at the actual resolution and display of the screen element at the optimal resolution would exceed a predetermined threshold level. If not, the screen element is output for display on the display screen based on the positional characteristics associated with the display screen resolution. Otherwise, the positional characteristics based on the actual resolution are modified such that the degree of displacement conforms to the threshold level.

Description:
BACKGROUND 
   Conventional “type and see” computer applications such as word processors, spreadsheet programs, text editors, email programs and the like typically output information for display on a monitor having a vastly lower resolution than that provided by any printer. With that said, printer resolution is commonly referred to as “high” resolution whereas monitor resolution is often coined “low” resolution. This low resolution is due the physical limitations on the number of pixels that may be provided on display screens of conventional monitors. While new technologies such as plasma-based monitors and liquid crystal displays utilize smaller pixels and, consequently, provide more pixels per inch than conventional monitors, these still emerging technologies are drastically more expensive than conventional monitors but still don&#39;t provide the high resolution available with even mid-grade laser printers. 
   Ideally, these emerging technologies will be operable to display information in substantially the same high resolution provided by printers such that superimposing printed information on a display screen would yield the same size and positioning of the presented information. While such an “optimal” resolution is currently not available, application developers strive to at least maintain the positioning of information consistent between display screens and printed documentation even if it results in diminishing legibility. To accomplish this, it is common practice to scale down printer positioning of information for display on a display screen. However, by such reduction, the displayed information appears to run together thereby hindering the reader&#39;s ability to distinguish between characters. This problem is further exuberated in the construction of formulas, which typically include elements in close relation to each other such as the case with exponential elements and fractions. 
   It is with respect to these and other considerations that the present invention has been made. 
   SUMMARY 
   The present invention is generally related to determining optimal positions for elements being displayed on a display screen given resolution constraints associated with the display screen. More particularly, the present invention involves positioning elements of a formula optimally on a display screen taking into account such resolution limitations. For example, the present invention is applicable to determine an optimal position for a screen element representing a superscript relative to a screen element representing a base character to form an exponential formula expression. 
   In response to receipt of instruction to display a screen element on a display screen, an embodiment of the present invention involves determining positioning characteristics for displaying the screen element on the display screen. The positioning characteristics represent a screen layout for the screen element at the actual resolution of the display screen. The determined positioning characteristics of the screen element at actual resolution (AR) are analyzed against positioning characteristics associated with an optimal resolution (OR) to determine whether the AR-based positioning characteristics should be modified prior to outputting the screen element for display to the screen. 
   In an embodiment, this evaluation involves determining a degree of displacement between the screen element if displayed on the display screen based on the AR-based positioning characteristics and the screen element if displayed on the display screen based on the OR-based positioning characteristics. If the determined degree of displacement exceeds a predetermined threshold value, then the AR-based positioning characteristics are modified such that the displacement would substantially equal the threshold value and the screen element is displayed on the display screen based on the modified positioning characteristics. Otherwise, the screen element is displayed on the display screen based on the original AR-based positioning characteristics. 
   In accordance with another embodiment, the present invention relates to method for positioning a first screen element relative to a second screen element on a display screen. In response to receiving a request to display the first screen element adjacent to the second screen element, this method involves determining a first set of positioning characteristics for displaying the first screen element relative to the second screen element based on an actual resolution of the display screen. If the first set of positioning characteristics indicates that the first screen element would be separated from the second screen element by at least a predetermined amount of empty space, then method includes displaying the first screen element on the display screen based on a vertical position determined from the first set of positioning characteristics and a horizontal position determined from a second set of positioning characteristics that are based on an optimal resolution. 
   The various embodiments of the present invention may be implemented as a computer process, a computing system or as an article of manufacture such as a computer program product or computer readable media. The computer program product may be a computer storage media readable by a computer system and encoding a computer program of instructions for executing a computer process. 
   These and various other features as well as advantages, which characterize the present invention, will be apparent from a reading of the following detailed description and a review of the associated drawings. 

   
     DESCRIPTION OF THE DRAWINGS 
       FIG. 1  illustrates display of screen elements positioned relative to one another in accordance with an embodiment of the present invention. 
       FIG. 2  shows an exemplary computer system upon which embodiments of the present invention may be implemented. 
       FIG. 3  is a flow diagram illustrating operational characteristics of a process for positioning a screen element shown in  FIG. 1  relative to another screen element in accordance with an embodiment of the present invention. 
       FIG. 4  illustrates a virtual representation of a screen layout for a screen element if displayed on a display screen at an actual resolution for the screen. 
       FIG. 5  illustrates a virtual representation of a screen layout for the screen element of  FIG. 4  if displayed on the display screen based on an optimal resolution. 
       FIG. 6  illustrates comparison of the virtual representation shown in  FIG. 4  with the virtual representation shown in  FIG. 5 . 
       FIG. 7  is a flow diagram illustrating additional operational characteristics for use with the positioning process of  FIG. 3  in accordance with an embodiment of the present invention. 
       FIG. 8  is a flow diagram illustrating additional operational characteristics for use with the positioning process of  FIG. 3  in accordance with an embodiment of the present invention. 
   

   DETAILED DESCRIPTION 
   The present invention will now be described more fully hereinafter with reference to the accompanying drawings, in which embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. Like numbers refer to like elements throughout. 
   In general, the present invention relates to positioning screen elements relative to one another on a display screen. In accordance with an embodiment, the screen elements are one or more image representations (“glyphs”) of characters and/or ligatures. In accordance with an exemplary embodiment, the present invention is described herein with reference to glyphs individually or collectively representing expressions or sub-expressions in a formula in a word processing document being displayed through a display screen. For example,  FIG. 1  illustrates a formula  112  having glyphs  114 - 142  displayed as content in a word processing document  104  presented to users through a display screen  102 . While some of these glyphs (e.g.,  122 ) individually represent a single formula expression (“=”), other glyphs (e.g.,  124  and  126 ) collectively represent a single formula expression (e.g., “X 2 ”) and yet other glyphs (e.g.,  148  and  150 ) collectively represent a single sub-expression for complete formula expression (e.g., “X a     b   ”). 
   Embodiments of the present invention are illustrated herein with reference to positioning the glyphs  114 - 142  relative to one another to form the expressions and sub-expressions of the formula  112  in accordance with an exemplary embodiment of the present invention.  FIG. 1  illustrates the formula  112  at a point in time after the glyphs  114 - 142  have already been positioned and, thus, the formula expressions and sub-expressions already formed.  FIGS. 3-7 , on the other hand, illustrate a process  300  for positioning the glyphs  114 - 142  to render this resultant formula  112  in accordance with an embodiment of the present invention. Prior to further describing the positioning process  300  with reference these figures, however, an exemplary computing environment (e.g., computing device  200 ) for practicing the process  300  is shown in  FIG. 2  and described in the following paragraphs below. 
   In its most basic configuration, the computing device  200  includes at least one processing unit  202  and memory  204 . Depending on the exact configuration and type of computing device  200 , the memory  204  may be volatile (such as RAM), non-volatile (such as ROM, flash memory, etc.) or some combination of the two. The most basic configuration of the computing device  200  is illustrated in  FIG. 2  by dashed line  206 . Additionally, computing device  200  may also have additional features/functionality. For example, computing device  200  may also include additional storage (removable and/or non-removable) including, but not limited to, magnetic or optical disks or tape. Such additional storage is illustrated in  FIG. 2  by removable storage  208  and non-removable storage  210 . 
   Computer storage media, as used herein, includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules, or other data. Memory  204 , removable storage  208  and non-removable storage  210  are all examples of computer storage media. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can accessed by device  200 . Any such computer storage media may be part of device  200 . 
   The computing device  200  may also contain communications connection(s)  212  for communicating with other devices. The communications connection(s)  212  is/are an example of communication media. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared, and other wireless media. 
   The computing device  200  may also have input device(s)  214  such as keyboard, mouse, pen, voice input device, touch input device, etc. Output device(s)  216  such as a display, speakers, printer, etc. may also be included. The devices may help form the user interface  104  discussed above. All these devices are well know in the art and need not be discussed at length here. 
   The computing device  200  typically includes at least some form of computer readable media. Computer readable media can be any available computer storage media that can be accessed by processing unit  202 . 
   The computing device  200  may operate in a networked environment using logical connections to one or more remote computers (not shown). The remote computer may be a personal computer, a server computer system, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer device  200 . The logical connections between the computer device  200  and the remote computer may include a local area network (LAN) or a wide area network (WAN), but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets, and the Internet. 
   When used in a LAN networking environment, the computer device  200  is connected to the LAN through a network interface or adapter. When used in a WAN networking environment, the computing device  200  typically includes a modem or other means for establishing communications over the WAN, such as the Internet. The modem, which may be internal or external, may be connected to the computer processor  202  via the communication connections  212 , or other appropriate mechanism. In a networked environment, program modules or portions thereof may be stored in the remote memory storage device. By way of example, and not limitation, a remote application program may reside on memory device connected to the remote computer system. It will be appreciated that the network connections explained are exemplary and other means of establishing a communications link between the computers may be used. 
   With the computing environment of  FIG. 2  in mind, logical operations of the various exemplary embodiments described below in connection with the positioning process  300  may be implemented (1) as a sequence of computer implemented acts or program modules running on a computing system and/or (2) as interconnected machine logic circuits or circuit modules within the computing system. The implementation is a matter of choice dependent on the performance requirements of the computing system implementing the invention. Accordingly, the logical operations making up the embodiments of the exemplary embodiments described herein are referred to variously as operations, structural devices, acts or modules. It will be recognized by one skilled in the art that these operations, structural devices, acts and modules may be implemented in software, in firmware, in special purpose digital logic, and/or any combination thereof without deviating from the spirit and scope of the present disclosure as recited within the claims attached hereto. 
   Referring now to  FIG. 3 , the positioning process  300  embodies operational characteristics practiced by an application program implemented on the computing device  200 . Exemplary application programs include, but certainly are not limited to, word processing applications, spreadsheet applications, presentation applications, application development applications, text editor applications, electronic mail applications and web browsers. With that said, the positioning process  300  is operable to position any type of screen element relative to one another on a display screen (e.g.,  102 ). 
   In accordance with an exemplary embodiment, the positioning process  300  illustrates construction of the formula  112  of  FIG. 1  by positioning glyphs (e.g.,  114 - 142 ) relative to one another to form and arrange the expressions and sub-expressions of the formula  112 . For illustrative purposes, the positioning process  300  is described with reference to a point in time during the construction of the formula  112  when a screen element (i.e., superscript glyph  126  (“ 2 ”)) is being positioned relative to another screen element (i.e., base glyph  124  (“X”)) to render the exponential formula expression “X 2 ” on the display screen  102 . As such, to illustrate performance of the positioning process  300 , reference to the term “screen element” in the text of the  FIG. 3  refers to the superscript glyph  126  (“ 2 ”) while reference to term “expression” in the text of this figure refers to the complete expression “X 2 .” Even further,  FIGS. 4-6  sequentially depict a conceptual illustration of performance of the positioning process  300  in relation to this exemplary illustration and are thus described in connection therewith. 
   While only a single iteration of the positioning process  300  is described below, this process  300  is repetitively practiced to accomplish display of the entire formula  112 . The positioning process  300  therefore involves a flow of operations (i.e., an “operation flow”) that are invoked either concurrently (e.g., concurrent process threading) or sequentially to position the various glyphs  114 - 142  to form expressions and sub-expressions for the formula  112 . For example,  FIG. 8 , described in more detail below, illustrates the recursive nature of the positioning process  300  in connection with positioning glyphs  140  and  142  relative to glyph  138  and glyphs  148  and  150  relative to glyph  146  to form nested exponential expressions in accordance with an exemplary embodiment of the present invention. 
   The operation flow of the positioning process  300  begins with a start operation  302  and concludes with a terminate operation  320 . The start operation  302  is initiated in response to receipt of an instruction to display a formula expression. As such, with respect to the exemplary illustration provided herein, such an instruction involves a request to position the superscript glyph  126  adjacent to the base glyph  124  to form the exponential expression “X 2 .” From the start operation  302 , the operation flow passes to a first create operation  304 . 
   The first create operation  304  determines the actual resolution for the display screen  102  and then creates a pixilated representation of the exponential expression “X 2 ” at the determined actual resolution. The display screen  102  is operable to display glyphs at various resolutions and, as such, the actual resolution may vary depending on times at which the positioning process  300  is implemented. With that said, an embodiment of the present invention involves determining the current actual resolution based on evaluating current display settings. 
   The pixilated representation is a virtual, in-memory representation of the superscript glyph  126  as would be displayed adjacent the base glyph  124  at the actual resolution. With that said, though, this representation is embodied in a data structure maintained internal to the application program for use in administering the positioning process  300  and is not displayed as output to the display screen  102  at least at this point in time. The data structure includes data identifying positioning characteristics (i.e., screen layout) for the superscript glyph  126  and the base glyph  124  at the actual resolution and, more particularly, which pixels on the display screen  102  are to be colored to provide an image of the exponential formula expression “X 2 .” To illustrate,  FIG. 4  illustrates the superscript glyph  126  as it would be displayed on the display screen  102  adjacent the base glyph  124  based on the pixilated representation  125  created by the first create operation  304 . From the first create operation  304 , the operation flow passes to a second create operation  306 . 
   The second create operation  306  determines an optimal resolution and then creates a representation of the exponential formula expression “X 2 ” at the determined optimal resolution. The optimal resolution representation is a virtual, in-memory representation of the superscript glyph  126  as would be displayed adjacent the base glyph  124  if the display screen  102  displayed glyphs at the optimal resolution. As such, this representation is embodied in a data structure maintained internal to the application program for use in administering the positioning process  300  and is not displayed as output to the display screen  102 . The data structure includes coordinates that define hypothetical positioning characteristics (i.e., hypothetical screen layout) for the superscript glyph  126  and base glyph  124  at the optimal resolution. 
   In an embodiment, the optimal resolution is defined by the developer of the application program administering the positioning process  300 . In accordance with another embodiment, the optimal resolution may be defined based on the resolution associated with a printer attached to the computing device  200 . After the optimal resolution is determined and the associated representation of the exponential formula expression “X 2 ” has been created, the operation flow passes from the second create operation  306  to a scale operation  308 . 
   The scale operation  308  scales down the optimal resolution representation to conform to resolution constraints associated with the display screen  102 . To accomplish this, the scale operation  308  applies a resolution factor to the positioning characteristics (i.e., coordinates) defining the hypothetical screen layout of the superscript glyph  126  and the base glyph  124  at optimal resolution. In accordance with an embodiment, the resolution factor is based on a ratio of the defined optimal resolution to the actual resolution of the display screen  102 . Alternatively, the resolution factor may be defined by other mathematical considerations that relate the optimal resolution to the actual resolution. 
   Once defined, the resolution factor is applied to the optimal resolution coordinates thereby rendering a scaled, or “rounded,” down set of coordinates that represent a scaled-down optimal resolution representation of the superscript glyph  126  and the base glyph  124 . Like the optimal resolution representation, the scaled-down optimal representation is embodied in a data structure maintained internal to the application program for use in administering the positioning process  300  and is not displayed as output to the display screen  102 .  FIG. 5  illustrates the superscript glyph  126  as it would be displayed on the display screen  102  adjacent the base glyph  124  based on the scaled-down optimal resolution representation  127 . From the scale operation  308 , the operation flow passes to a compare operation  310 . 
   The compare operation  310  analyzes the screen layouts specified by the pixilated representation  125  and the scaled-down optimal resolution representation  127  to determine the degree of displacement of the superscript representation  126  between these representations ( 125 ,  127 ) if displayed on the display screen  102 . As noted above, the screen layouts specified by both representations are embodied in data structures maintained internal to the application program administering the positioning process  300 . The compare operation  310  therefore renders analysis between the two representations by comparing the positioning of the two screen layouts relative to one another. In an embodiment, the compare operation  310  involves selecting common points (e.g., bottom-left corner) of the superscript glyph  126  on both screen layouts and measuring the number of pixels that would exist between the common points. In this embodiment, the compare operation  310  involves an analysis that can be logically represented by displaying both the pixilated representation  125  and the scaled-down optimal resolution representation  127  together on the display screen  102  and determining displacement of the superscript glyph  126  therebetween, as shown in  FIG. 6 . 
   After the compare operation  310  has determined the degree of displacement (in number of pixels) between the superscript glyph  126  on the pixilated representation  125  and the superscript glyph  126  on the scaled-down optimal resolution representation  127 , the operation flow passes to a query operation  312 . The query operation  312  determines whether the degree of displacement determined by the compare operation  310  exceeds a maximum pixel displacement value defined for optimal positioning. The maximum pixel displacement value represents a maximum number of pixels that may exist between the actual resolution representation of an evaluated glyph on the display screen  102  and the scaled-down optimal resolution representation of that glyph on the display screen  102 . The maximum pixel displacement value, which may vary between different fonts, may be defined by either the developer of the application program or a user of the application program (e.g., by customizing display properties through an “options” dialog or the like). 
   If the degree of displacement exceeds the maximum pixel displacement, the query operation  312  passes the operation flow to a shift operation  314 . The shift operation  314  shifts the superscript glyph  126  on the pixilated representation  125  toward the superscript glyph  126  on the scaled-down optimal resolution representation  127  such that the degree of displacement therebetween is equal to the maximum pixel displacement value and, therefore, within an “allowed” pixel range. As such, the data identifying a screen layout for the superscript glyph  126  at the actual resolution is modified based on the shift in the pixilated representation  125 . From the shift operation  314 , the operation flow passes to a first output operation  316 . 
   The first output operation  316  specifies the appropriate pixels for coloring to display the superscript glyph  126  on the display screen  102  based on the shifted screen layout now specified for the pixilated representation  125 . The first output operation  316  therefore results in the superscript glyph  126  being output to the display screen  102  for display to the user, as illustrated in  FIG. 1 . From the first output operation  316 , the operation flow concludes at the terminate operation  320 . 
   Referring back to the query operation  312 , if the degree of displacement determined by the compare operation  310  does not exceed the maximum pixel displacement value defined for optimal positioning, the pixilated representation  125  is considered in the “allowed” pixel range relative to the scaled-down optimal resolution representation  127 . Consequently, the operation flow passes to a second output operation  318 . The second output operation  318  specifies the appropriate pixels for coloring to display the superscript glyph  126  on the display screen  102  based on the screen layout specified for the pixilated representation  125 . Thus, like the first output operation  316 , the second output operation  318  results in the superscript glyph  126  being output to the display screen  102  for display to the user, as illustrated in  FIG. 1 . 
   Turning now to  FIG. 7 , an operation flow  330  for use with the positioning process  300  of  FIG. 3  is shown in accordance with an embodiment of the present invention. In this embodiment, the positioning process  300  includes a query operation  334  that is invoked upon completion of the scale operation  308 . The query operation  334  determines whether the glyph that is requested to be displayed is preceded by at least a predetermined amount of empty space (i.e., “white space”). In an embodiment, such a predetermined amount of empty space is determined by the developer of the application program. For example, the predetermined amount of empty space may be defined to be the amount of white space conventionally known to be associated with an operator. Exemplary operators include the equals sign, the plus sign and the minus sign. With that said, glyphs representing operators typically include sufficient white space for positioning any glyphs therearound and, thus, obviate the need for the determining whether the position of these operator-based glyphs or any glyphs following these operator-based glyphs should be shifted. In fact, with respect to fractional formula expressions having sub-expressions such as, for example, a+b/x+y, this abundance of white space typically results in the operators (e.g., +) in the sub-expressions aligning in a vertical orientation such that one operator is situated substantially above the other. 
   If the query operation  334  determines that the glyph that is requested to be displayed is preceded by the predetermined amount of empty space, then the operation flow of the positioning process  300  is passed directly to a partial shift operation  336 . The partial shift operation  336  determines a position for the glyph based in part on the pixilated representation and in part on the scaled-down optimal resolution representation. For example, in an embodiment, the partial shift operation  336  shifts the horizontal positioning of the glyph specified by the pixilated representation to match the horizontal positioning of the glyph specified by the scaled-down optimal resolution representation while maintaining the vertical positioning of the glyph specified by the pixilated representation. The x-coordinates for use in displaying the requested glyph are therefore determined based on the scaled-down optimal resolution representation while the y-coordinates are determined based on the pixilated representation. The operation flow then passes from the partial shift operation  336  to the first output operation  316  via a first transfer operation  336 . 
   If, however, the query operation  334  determines that the glyph that is requested to be displayed is not preceded by the predetermined amount of empty space, the operation flow of the positioning process  300  is passed to the compare operation  310  by a second transfer operation  338  thereby continuing the positioning process  300  as described above. 
   Although the present invention has been described in language specific to structural features, methodological acts, and computer readable media containing such acts, it is to be understood that the present invention defined in the appended claims is not necessarily limited to the specific structure, acts, or media described. One skilled in the art will recognize other embodiments or improvements that are within the scope and spirit of the present invention. 
   For example, while exemplary screen elements are described herein as glyphs that represent characters and ligatures embodying formula expressions, it should be appreciated that the present invention is applicable to position any type of screen element. Moreover, embodiments of the present invention are described herein with reference to the positioning of these exemplary screen elements in a word processing document. However, the present invention is equally applicable to other types of electronic documents and, for that matter, any electronic document operable to provide a medium for displaying screen elements. 
   Furthermore, the computing device  200  is only one example of a suitable operating environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention. Other well known computing systems, environments, and/or configurations that may be suitable for use with the invention include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems, microprocessor-based systems, programmable consumer electronics, network PCs, minicomputers, mainframe computers, distributed computing environments that include any of the above systems or devices, and the like. 
   In addition, while described in connection with positioning a screen element formed of a single glyph (i.e., superscript glyph  126  “ 2 ”) relative to another screen element also formed of a single glyph (base glyph  124  “X”), the positioning process  300  is also applicable to position a screen element formed of multiple glyphs relative to another screen element formed of one or more glyphs. For example, the positioning process is applicable to position glyphs (e.g.,  130 ,  132  and  134 ) of a fractional expression relative to one another as well as nested superscript glyphs (e.g.,  140 ,  142 ,  148  and  150 ) relative to base glyphs (e.g.,  138  and  146 ). With regard to the latter, embodiments of the present invention involve the positioning of screen elements that constitute sub-expressions of an expression. To accomplish this, the positioning process  300  is administered in conjunction with an additional set of operations that define a recursive procedure for practicing the process  300 , as shown in an exemplary manner in  FIG. 8 . Specifically,  FIG. 8  illustrates a process  800  for recursively positioning screen elements, to define expressions having sub-expressions such as, for example, nested exponents ( 140 / 142  and  148 / 150 ). 
   The recursive positioning process  800  is performed using an operation flow that begins with a first transfer operation  802  and that concludes with a terminate operation  822 . The first transfer operation  802  is initiated in response to initiation of the start operation  302  and serves to transfer the operation flow of the positioning process  300  from the start operation  302  to a first query operation  804  prior to invoking the first create operation  304 . As such, the start operation  802  is triggered to initiate the recursive positioning process  800  in response to receipt of an instruction to display a formula expression, as described above with reference to the start operation  302 . The first query operation  804  then determines whether the requested formula expression includes any sub-expressions. If so, the first query operation  804  passes the operation flow to a select operation  808 . Otherwise, the first query operation  804  passes the operation flow to a second transfer operation  806  and the positioning process is resumed at the first create operation  304  as described above. 
   Illustrating the first query operation  804  relative to the exemplary formula  112 , the formula expression “X 2 ”, formed using the superscript glyph  126  and the base glyph  124  does not include any sub-expressions and, thus, this operation  804  would pass the operation flow to the second transfer operation  806  in this circumstance. However, the formula expression “X a     b   ” formed using glyphs  146 ,  148  and  150  includes one sub-expression (i.e., “ a     b   ”) and, therefore, the first query operation  804  would pass the operation flow to the select operation  808  in this circumstance. Therefore, to illustrate the recursive positioning process  800 , this latter formula expression, i.e., “X a     b   ,” is illustrated herein. 
   The select operation  808  selects the glyphs embodying the outermost sub-expression in relation to the base screen element of the expression for initial evaluation by the recursive positioning process  800 . With respect to both subscript and superscript sub-expressions, the outermost sub-expression refers to sub-expression furthest to the right, i.e., the “right-most” sub-expression in the expression. With respect to the present illustration, the formula expression “X a     b   ” only has one sub-expression, i.e., “a b ,” and therefore, this sub-expression embodies the outermost sub-expression that will be applied to the recursive positioning process  800 . The select operation  810  therefore selects glyphs  148  and  150  as the screen elements for initial evaluation by the recursive positioning process  800 . After such selection, the operation flow of the recursive positioning process  800  passes to a first create operation  810 . 
   The first create operation  810  determines an optimal resolution and then creates a representation of the selected sub-expression “a b ” at the determined optimal resolution. The optimal resolution representation is a virtual, in-memory representation of the glyph  150  (“b”) as it would be displayed adjacent the glyph  148  (“a”) if the display screen  102  displayed glyphs at the optimal resolution. This representation is embodied in a data structure maintained internal to the application program for use in administering the recursive positioning process  800  and is not displayed as output to the display screen  102 . The data structure includes coordinates that define hypothetical positioning characteristics (i.e., hypothetical screen layout) for the glyph  150  (“b”) relative to the glyph  148  (“a”) at the optimal resolution. After the optimal resolution is determined and the associated representation of the selected sub-expression “a b ” has been created, the operation flow passes from the first create operation  810  to a scale operation  812 . 
   The scale operation  308  scales down the optimal resolution representation to conform to resolution constraints associated with the display screen  102 . To accomplish this, the scale operation  812  applies a resolution factor to the positioning characteristics (i.e., coordinates) defining the hypothetical screen layout of the glyph  150  (“b”) relative to the glyph  148  (“a”) at optimal resolution. The optimal resolution coordinates thereby rendering a scaled, or “rounded,” down set of coordinates that represent a scaled-down optimal resolution representation of the sub-expression “a b .” Like the optimal resolution representation, the scaled-down optimal representation is embodied in a data structure maintained internal to the application program for use in administering the recursive positioning process  800  and is not displayed as output to the display screen  102 . From the scale operation  812 , the operation flow passes to a third transfer operation  814 . 
   The third transfer operation  814  transfers the operation flow of the recursive positioning process  800  back to the positioning process  300 . In addition, the third transfer operation  814  specifies the scaled-down optimal representation of the sub-expression “a b ” to be the “screen element” positioned relative to the screen element to which the sub-expression is adjacent, which, in the present illustration is the base glyph  146  (“X”) of the formula expression “X a     b   .” The positioning process  300  is then administered as described above such that the sub-expression “a b ” is optimally positioned relative to the base glyph  146  (“X”). At the completion of the positioning process  300 , the operation flow passes from the terminate operation  320  and is passed back to the recursive positioning process  800  via a fourth transfer operation  816  for further evaluation. The fourth transfer operation  816  accepts the operation flow of the positioning process  300  and resumes the operation flow of the recursive positioning process  800  at a fifth transfer operation  820 . 
   The fifth transfer operation  820  re-initiates the positioning process  300  for each of the sub-expressions iterated through the recursive positioning process  800  such that each of the glyphs contained therein are optimally positioned relative to one another. For example, with respect to the present illustration, the glyph  150  (“b”) is optimally positioned adjacent the glyph  148  (“a”) using the positioning process  300 . In this regard, the order through which glyphs in the formula expression are evaluated is reversed relative to initial iterations of the recursive positioning process  800 . After each of the sub-expressions have been applied to the positioning process  300 , the operation flow of the recursive positioning process  800  concludes at the terminate operation  822 .