Abstract:
Apparatus and methods for dynamically rendering transparent ink strokes, in some situations such that the rendered ink stroke has transparency similar to physical ink while it is being drawn. For example, the ink stroke may be dynamically rendered as a stroke having uniform transparency while it is being drawn. Only the new ink segment that has most recently been added to the stroke may be drawn, and areas of the new ink segment that overlap older segments of the ink stroke may be frozen, or excluded from being re-painted.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application is a continuation of U.S. patent application Ser. No. 10/770,448, filed Feb. 4, 2004 and titled “Dynamic Rendering of Ink Strokes with Transparency,” now U.S. Pat. No. 7,091,963, which application is a continuation of U.S. patent application Ser. No. 09/918,484, filed Aug. 1, 2001, and titled “Dynamic Rendering of Ink Strokes with Transparency”, now U.S. Pat. No. 6,707,473. This application is related to U.S. patent application Ser. No. 10/972,390 , entitled “Dynamic Rendering of Ink Strokes with Transparency,” filed Oct. 26, 2004, U.S. patent application Ser. No. 09/918,721, entitled “Rendering Ink Strokes of Variable Width and Angle,” filed Aug. 1, 2001, and U.S. patent application Ser. No. 09/852,799, entitled “Serial Storage of Ink and its Properties,” filed May 11, 2001. All of said applications are hereby incorporated by reference as to their entireties. 

   FIELD OF THE INVENTION 
   The present invention is directed generally to rendering transparent digital ink, and more particularly to improved ways of rendering transparent digital ink dynamically. 
   BACKGROUND OF THE INVENTION 
   The term “digital ink” refers to one or more strokes that are recorded from a pointing device, such as a mouse, a stylus/pen on a digitizer tablet, or a stylus/pen on a display screen integrated with a digitizer tablet (e.g., a touch-sensitive display screen). As used herein, the term “ink” is shorthand for digital ink. Also, the term “pen” and “stylus” are used generically and interchangeably. Each stroke may be stored as one or more ink packets, in which each ink packet may contain coordinates (x, y) corresponding to the position of the pointing device. For example, a user may move a pen along a touch-sensitive display screen of a computer system so as to draw a line or curve, and the computer system may sample the coordinates (x, y) along the trajectory of the pen tip position over time (or on any other interval as known in the art) as the user moves the pen. These coordinates represent points along the curve or line and are stored as ink packets. 
   Ink may be either transparent or non-transparent, as used herein. Ink that is transparent means that the ink does not fully conceal the background behind it when displayed on a display or printed on a printer. Ink that is not transparent completely conceals or occludes the background behind it. Non-transparent ink may also be referred to as opaque ink. For instance,  FIG. 1  shows ink strokes  101 ,  102 , and  103 . Ink strokes  102  and  103  each overlay ink stroke  101 , but ink stroke  103  completely conceals its background, including the portion of ink stroke  101  that it overlays (i.e., the portion of ink stroke  101  that is a background behind ink stroke  103 ). Thus, ink stroke  103  is considered opaque. In contrast, ink stroke  102  allows some of ink stroke  101 , as well as some of the white background, to show through where ink stroke  102  overlays ink stroke  101 . Thus, ink stroke  102  is considered transparent. Ink can be of any transparency and still be considered transparent. Current graphics interfaces are capable of applying transparent paint with a prescribed degree of transparency. For example, ink may be 50% transparent, which means that 50% of the background is concealed, or ink may be 25% transparent, which means that 75% of the background is concealed. A transparent ink stroke can be analogized with a piece of glass, such as colored glass, in which objects behind the glass can be seen. A non-transparent ink stroke can be analogized with a brick wall that hides everything behind it. 
   It is often desirable to render a transparent ink stroke dynamically while the ink stroke is being drawn, in other words, to draw the ink stroke on the display screen while the pointing device moves and adds new points to the ink stroke or strokes. One way to accomplish this is to erase the entire screen and redraw everything on the screen each time a new point is added to the ink stroke. This is an imperfect solution, however, since in practice there is typically a short time interval between ink points, and repeatedly clearing and redrawing the screen uses massive amounts of processing power, not to mention causing the screen to flicker. A way to reduce the redrawing time would be draw each new segment of an ink stroke as it is drawn. The problem with this is that the transparencies of the overlapping portion of ink segments are reduced in an unexpected and unintended manner. The effect of redrawing transparent ink is shown in  FIG. 2 , where the darker circles of an ink stroke  200  represent the overlapping start and end points of the segments. These overlapping areas are darker because they are each drawn twice—once when a segment ending with a particular point is drawn, and again when the next segment beginning with the same point is drawn—thereby reducing the transparency at the overlap. The result is an unintentionally non-uniform ink stroke. This is analogous to repeatedly making a glass window thicker, thereby making objects on the other side of the glass more difficult to see by making the window darker. The variable transparency of the rendered ink is unexpected to the user who would expect transparent ink to be rendered as transparent physical ink as applied to paper and/or over other ink. 
   There is also a need for providing various artistic features not provided by current systems, such as dynamically rendering ink responsive to variable width, pressure, speed, and angle of the pen. 
   SUMMARY OF THE INVENTION 
   Apparatus and methods are disclosed for dynamically rendering transparent ink strokes that solves at least one of the problems associated with rendering transparent ink. Using the present invention, the rendering of electronic ink (or ink as used herein) is improved. For example, the ink stroke may be dynamically rendered as a stroke having uniform transparency while it is being drawn. This may be performed without having to clear and redraw the entire screen. 
   To dynamically draw a transparent ink stroke, a computer system may draw only the segment that has most recently been added to the stroke. The system may further exclude areas of the new segment that overlap older portions of the stroke from being painted more than once, which would otherwise make the older segments less transparent. For instance, the color settings of pixels in the overlapping areas may be frozen before painting the new segment. Freezing the color settings may reduce or prevent unintended non-uniformities in the ink stroke. 
   These and other features of the invention will be apparent upon consideration of the following detailed description of preferred embodiments. It will be apparent to those skilled in the relevant technology, in light of the present specification, that alternate combinations of aspects of the invention, either alone or in combination with one or more elements or steps defined herein, may be used as modifications or alterations of the invention or as part of the invention. It is intended that the written description of the invention contained herein covers all such modifications and alterations. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing summary of the invention, as well as the following detailed description of preferred embodiments, is better understood when read in conjunction with the accompanying drawings, which are included by way of example, and not by way of limitation with regard to the claimed invention. In the accompanying drawings, elements are labeled with reference numbers, wherein the first digit of a three-digit reference number, and the first two digits of a four-digit reference number, indicates the drawing number in which the element is first illustrated. The same reference number in different drawings refers to the same or a similar element. 
       FIG. 1  is an exemplary embodiment of both transparent and non-transparent digital ink as they may be displayed, according to at least one aspect of the present invention. 
       FIG. 2  is an exemplary embodiment of transparent digital ink as it may be displayed, showing non-uniformities due to blending of multiple segments. 
       FIG. 3  is an exemplary embodiment of transparent digital ink as it may be displayed, without the non-uniformities of the ink shown in  FIG. 2 , and according to at least one aspect of the present invention. 
       FIG. 4  is a functional block diagram of an exemplary embodiment of a computer system according to at least one aspect of the present invention. 
       FIG. 5  is a functional block diagram of an exemplary embodiment of an ink rendering system according to at least one aspect of the present invention. 
       FIG. 6  is an exemplary flowchart showing steps that may be performed in order to render transparent ink according to at least one aspect of the present invention. 
       FIG. 7  is an exemplary geometrical representation of stroke segments including circular pen tip instances and connecting quadrangles according to at least one aspect of the present invention. 
       FIG. 8  is an exemplary embodiment of digital ink corresponding to the stroke segments of  FIG. 7  as it may be displayed, according to at least one aspect of the present invention. 
       FIG. 9  is an exemplary geometrical representation of a frozen region within a series of stroke segments, according to at least one aspect of the present invention. 
       FIG. 10  is a functional block diagram of an exemplary embodiment of another ink rendering system according to at least one aspect of the present invention. 
       FIG. 11  is an exemplary geometrical representation of a stroke including differently-sized circular pen tip instances and connecting quadrangles according to at least one aspect of the present invention. 
       FIG. 12  is an exemplary embodiment of digital ink corresponding to the stroke of  FIG. 11  as it may be displayed, according to at least one aspect of the present invention. 
       FIG. 13  is an exemplary geometrical representation of a stroke including differently-sized and differently-angled oval pen tip instances and a connecting quadrangle according to at least one aspect of the present invention. 
       FIG. 14  is an exemplary embodiment of digital ink corresponding to the stroke of  FIG. 13  as it may be displayed, according to at least one aspect of the present invention. 
       FIG. 15  is an exemplary geometrical representation of a stroke including differently-sized and differently-angled rectangular pen tip instances and a connecting quadrangle according to at least one aspect of the present invention. 
       FIG. 16  is an exemplary embodiment of digital ink corresponding to the stroke of  FIG. 15  as it may be displayed, according to at least one aspect of the present invention. 
       FIGS. 17A and 17B  are exemplary geometrical representations of a stroke including differently-sized and differently-angled rectangular pen tip instances and two different possible connecting quadrangles according to at least one aspect of the present invention. 
       FIG. 17C  is an exemplary representation of the stroke of  FIGS. 17A and 17B  including all of the possible corner-connecting quadrangles according to at least one aspect of the present invention. 
       FIG. 18  is an exemplary embodiment of digital ink corresponding to the stroke of  FIG. 17C  as it may be displayed, including all possible connecting quadrangles, according to at least one aspect of the present invention. 
       FIG. 19  is a functional block diagram of an exemplary embodiment of yet another ink rendering system according to at least one aspect of the present invention. 
       FIG. 20  is a geometric representation of an exemplary ink stroke illustrating the sample points therein as well as a fitting curve for position, width, and rotation, in accordance with at least one aspect of the present invention. 
       FIG. 21  is a representation of a rendered exemplary ink stroke according to at least one aspect of the present invention. 
   

   DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
   Improved transparent ink rendering systems and methods are disclosed. The various embodiments of the invention are described in the following sections: General Purpose Computing Environment, Ink Rendering System, and Ink Smoothing. 
   General Purpose Computing Environment 
     FIG. 4  illustrates a schematic diagram of an exemplary general-purpose digital computing environment that may be used to implement various aspects of the present invention. In  FIG. 4 , a computer  400  such as a personal computer includes a processing unit  410 , a system memory  420 , and/or a system bus  430  that couples various system components including the system memory to processing unit  410 . System bus  430  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. System memory  420  includes read only memory (ROM)  440  and random access memory (RAM)  450 . 
   A basic input/output system  460  (BIOS), containing the basic routines that help to transfer information between elements within computer  400 , such as during start-up, is stored in ROM  140 . The computer  400  also includes a hard disk drive  470  for reading from and writing to a hard disk (not shown), a magnetic disk drive  480  for reading from or writing to a removable magnetic disk  490 , and an optical disk drive  491  for reading from or writing to a removable optical disk  492  such as a CD ROM or other optical media. Hard disk drive  470 , magnetic disk drive  480 , and optical disk drive  491  are connected to the system bus  430  by a hard disk drive interface  492 , a magnetic disk drive interface  493 , and an optical disk drive interface  494 , respectively. The drives and their associated computer-readable media provide nonvolatile storage of computer readable instructions, data structures, program modules and other data for personal computer  400 . It will be appreciated by those skilled in the art that other types of computer readable media that can store data that is accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks, Bernoulli cartridges, random access memories (RAMs), read only memories (ROMs), and the like, may also be used in the example operating environment. 
   A number of program modules can be stored on hard disk drive  470 , magnetic disk  490 , optical disk  492 , ROM  440 , and/or RAM  450 , including an operating system  495 , one or more application programs  496 , other program modules  497 , and program data  498 . A user can enter commands and information into computer  400  through input devices such as a keyboard  401  and pointing device  402 . Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner or the like. These and other input devices are often connected to processing unit  410  through a serial port interface  406  that is coupled to the system bus, but may be connected by other interfaces, such as a parallel port, game port or a universal serial bus (USB). Further still, these devices may be coupled directly to system bus  430  via an appropriate interface (not shown). A monitor  407  or other type of display device is also connected to system bus  430  via an interface, such as a video adapter  408 . In addition to the monitor, personal computers typically include other peripheral output devices (not shown), such as speakers and printers. In one embodiment, a pen digitizer  465  and accompanying pen or stylus  466  are provided in order to digitally capture freehand input. Although a direct connection between pen digitizer  465  and processing unit  410  is shown, in practice, pen digitizer  465  may be coupled to processing unit  410  via a serial port, parallel port, and/or other interface and system bus  430  as known in the art. Furthermore, although digitizer  465  is shown apart from monitor  407 , in some embodiments the usable input area of digitizer  465  be co-extensive with the display area of monitor  407 . Further still, digitizer  465  may be integrated in monitor  407 , or may exist as a separate device overlaying or otherwise appended to monitor  407 . 
   The computer  400  can operate in a networked environment using logical connections to one or more remote computers, such as a remote computer  409 . Remote computer  409  can be a server, 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 computer  400 , although only a memory storage device  411  has been illustrated in  FIG. 4 . The logical connections depicted in  FIG. 4  include a local area network (LAN)  412  and a wide area network (WAN)  413 . Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet. 
   When used in a LAN networking environment, the computer  400  is connected to local network  412  through a network interface or adapter  414 . When used in a WAN networking environment, the computer  400  typically includes a modem  415  or other device for establishing a communications over wide area network  413 , such as the Internet. Modem  415 , which may be internal or external, is connected to system bus  430  via the serial port interface  406 . In a networked environment, program modules depicted relative to the computer  400 , or portions thereof, may be stored in a remote memory storage device. 
   It will be appreciated that the network connections shown are exemplary and other techniques for establishing a communications link between the computers can be used. The existence of any of various well-known protocols such as TCP/IP, Ethernet, FTP, HTTP and the like is presumed, and the system can be operated in a client-server configuration to permit a user to retrieve web pages from a web-based server. Any of various conventional web browsers can be used to display and manipulate data on web pages. 
   Ink Rendering System 
   An exemplary ink rendering system  500  is illustrated in  FIG. 5 . Some or all of the ink rendering system  500  may be software, hardware, and/or firmware, and may be a part of the computer system  400  or a separate unit. For instance, some or all of the ink rendering system  500  may be embodied as computer code stored in the RAM  450  as part of the operating system  495 , an application program  496 , and/or another program module  497 . The ink rendering system  500  may include an ink storage  501  coupled to a rendering environment  502 , which in turn may be coupled to a graphics toolbox  503 , which in turn may be coupled to an output device  504  such as a display screen (e.g., monitor  407 ) and/or printer. The ink storage  501  may include information relating to ink including a file structure having data points representing points of the ink. The file structure may also include alternatively (or in addition to the data points) other ways to represent the ink including vectors between points, data points, stroke width information, and/or any other ink storage scheme. 
   Stored ink may be rendered by calling the graphics toolbox  503  to perform various functions. The ink storage  501  may maintain a list of rendering environments, one for each view in which the application renders dynamically. Each rendering environment may maintain a list of the states, one for each stroke that is currently being dynamically rendered. Each state may represent the last pen tip position (e.g., point) recorded and/or a queue of geometric regions that are further described below. In at least one embodiment, the graphics toolbox  503  has transparent painting capabilities, such as does Microsoft WINDOWS GDI+. 
     FIG. 6  illustrates an example of the operation of the ink rendering system  500 . When a user draws a stroke, the ink rendering system  500  may receive a new pen tip position (step  601 ). More particularly, the ink storage  501  may receive the new pen tip position. Pen tip positions may be sampled and determined according to the position of the stylus  466  upon the digitizer  465 . Pen tip positions may further be determined according to the position of the stylus  466  within a known input window or area that defines a portion of the digitizer  466  surface. For instance, where the digitizer  465  and the monitor  407  are combined or co-extensive, there may be a predefined window displayed on the digitizer  465  within which input from the stylus  466  may be accepted, e.g., for drawing an object and/or for entering text. 
   Pen tip positions may be sampled at a particular rate. The sampling rate may be set at a rate at least high enough to capture sufficient pen tip positions based on the anticipated speed of a normal user. Once the new pen tip position is captured and received, the ink rendering system  500  (e.g., in particular, the ink storage  501 ) may determine the area (and/or the contour that outlines and defines the area) that is associated with the pen tip at the new position based on the size and/or shape of the virtual pen tip. This area is also known as a “pen tip instance.” For example, where the virtual pen tip is considered to be a 3-millimeter diameter circle, then the pen tip instance may be the 3-millimeter diameter circle centered at the new pen tip position. Or, where the virtual pen tip is considered to be a rectangle of 2 millimeters by 4 millimeters, then the pen tip instance may be the 2 by 4 millimeter rectangle centered at the new pen tip position. Examples of circular pen tip instances  701 ,  702 ,  703 ,  704  are shown in  FIG. 7 . The size and shape of the pen tip instance are considered properties of the pen tip position. Where the entire stroke has the same size and/or shape, then the size and/or shape may be a property of the entire stroke as opposed to each pen tip position. Of course, any shape may be used for a pen tip. Circular pen tip instances are used here for simplicity. 
   Each time a pen tip instance is determined, that pen tip instance (and/or the associated pen tip position) may be stored for later retrieval. Pen tip instances and/or positions may be stored as data in, e.g., RAM  450 . Data representing the position (e.g., (x, y) coordinate position), shape, and/or rotation of the pen tip instance may further be stored. Previous pen tip instances and/or positions may further be stored as part of digital ink storage such as in the serialized format described in U.S. patent application Ser. No. 09/852,799, entitled “Serial Storage of Ink and its Properties,” filed May 11, 2001. 
   Referring still to  FIG. 6 , the ink rendering system  500  may render an ink segment that connects between the previous pen tip instance and a new pen tip instance in an ink stroke. To do so, the ink rendering system  500  may compute the new pen tip instance and/or one or more connecting quadrangles that connect between the new pen tip instance and a previous pen tip instance (step  602 ). Both the pen tip instances and the connecting quadrangles are referred to herein as “regions.” 
   The new pen tip instance is associated with the new pen tip position, and may be centered about the new pen tip position. The new connecting quadrangle may be determined in a variety of ways, and the method for determining the connecting quadrangle may depend upon the shapes of the new and previous pen tip instances. Various methods for determining connecting quadrangles will be discussed herein. Examples of connecting quadrangles  705 ,  706 ,  707  are shown in  FIG. 7 . A new region may be defined as the new pen tip instance, the new connecting quadrangle, or the combination (e.g., union) of the new pen tip instance and the new connecting quadrangle. For example, the new region may be pen tip instance  704 , connecting quadrangle  707 , or the union of pen tip instance  704  and connecting quadrangle  707 . Conventional graphics toolboxes are capable of performing such a combination/union when provided with the shapes to be combined. In alternative embodiments, more than one new pen tip instance and/or new connecting quadrangle may be the new region. For instance, two consecutive new pen tip instances and their two corresponding new connecting quadrangles may be all unioned together as the new region. In this way, the method of  FIG. 6  does not necessarily need to be performed between each and every pen tip instance. 
   The combination (e.g., union) of some or all of a plurality of previous regions may also be determined (step  603 ). These previous regions may be stored in a queue. A queue is an ordered list of items and is of a fixed, dynamic, maximum, or other controlled length. For example, a queue may have a maximum enforced length of 2, 3, or 4 items, although any length may be used. The queue may be configured as a first-in-first-out (FIFO) type queue, as in a pipeline. Where the maximum length of the FIFO queue is surpassed by adding another item to the queue, the oldest item is pushed out of the queue. The queue may separately store the actual items, or may have pointers that point to the items stored elsewhere. Where the items are stored elsewhere, they may be stored in a serialized or other format. In alternative embodiments, the items in the queue may be any items of data that represent some or all of the characteristics of pen tip positions and/or connecting quadrangles. In still further embodiments, each item in the queue may be a combined pen tip position and connecting quadrangle. 
   For example, referring to  FIG. 9 , where the new pen tip instance is pen tip instance  905 , the new connecting quadrangle is connecting quadrangle  909 , and the queue has a maximum length of 4 regions, the queued regions to be combined may be pen tip instances  903 ,  904  (two regions) and connecting quadrangles  907 ,  908  (two more regions, for a total of four regions). The union of these queued regions is shown as the shaded area in  FIG. 9 . The arrow  910  indicates the direction of movement of the pen tip, such that the pen tip instance  905  is the most recent and the pen tip instance  901  is the earliest in time. Note that although connecting quadrangle  906  and pen tip instance  901  may have been in the queue at an earlier time, these two regions were later pushed out of the queue due to the enforcement of its maximum length. 
   The ink rendering system  500  (in particular, e.g., the rendering environment  502 ) may freeze the color settings of the pixels (step  604 ) within the region defined by the combination (e.g., union) of the queued regions (e.g., the shaded area in  FIG. 9 ). The combined queued regions thus become an excluding clip region that may be sent to the graphics toolbox  503 . Freezing the color settings means preventing the color and intensity of the pixels from changing. Thus, any further attempts at painting the frozen pixels will have no effect on the color and intensity of the frozen pixels. This is important where the colors are transparent, since the new connecting quadrangle (e.g., quadrangle  909 ) is likely to overlap with the union of the queued regions (e.g., the shaded area in  FIG. 9 ). Without freezing the pixels in the queued regions, the overlapping portion will undergo a change in transparency when the new regions are painted. Conventional graphics toolboxes are capable of freezing the color settings of a group of pixels. An alternative to determining the union of the queued regions and then freezing the determined union region is to simply freeze each of the queued regions individually. This alternative provides the benefit of avoiding the step of determining the union. However, it increases the number of regions that need to be sent to the graphics toolbox for freezing. 
   The new region may be sent to the graphics toolbox  503  for painting (step  605 ). The new region may be painted in a transparent or nontransparent color as desired. After the new regions are painted, some or all of the pixels in the excluding clip region may be unfrozen (step  606 ). This step allows the color settings of the formerly frozen pixels to again be modified. More generally, the ink rendering system  500  may determine whether pixels within the new pen tip instance and/or new connecting quadrangle are also within the previous regions (such as those regions in the queue). For those pixels that are, the color settings of those pixels may not be changed. For those pixels that are in the new pen tip instance but not within any of the previous queued regions, the color settings may be changed. 
   The new region (e.g., connecting quadrangle  909 ) may then be pushed into the queue (step  607 ). Where the queue has rules that determine the queue length, one or more of the oldest regions may be pushed out of the queue as appropriate according to the queue rules. For example, referring to  FIG. 9 , a queue having a maximum of 4 regions may currently contain the following regions in the following order:  907 ,  903 ,  908 , and  904  (wherein  904  is the oldest). When connecting quadrangle  909  (in this example, the new region) is pushed into the queue, then region  907  is pushed out the queue in order to maintain no more than 4 regions within the queue. Thus, the new queue would contain regions  903 ,  908 ,  904 , and  909  (in that order, with  903  being the oldest and  909  being the newest). The queue may have any maximum length, such as 1 region, between 2 and 4 regions inclusive, between 5 and 10 regions inclusive, or between 10 and 20 regions inclusive, 10 and 100 regions inclusive, or more. If the queue length is too short, then it is likely that a new pen tip instance from a slow-moving pen may overlap a region recently dropped from the queue, resulting in an unintended decrease in transparency in the overlapping area. This results in an unexpected rendering of ink. However, as processing time increases with queue length, using a long queue length may require the system to group numerous regions, objects, or shapes, thereby slowing the system during the rendering process and/or requiring higher processor speed to maintain adequate representation of ink in real-time. 
   Further, a queue length that is allowed to be too long may prevent certain desirable overlapping of transparent ink, such as when writing the script letter “e” as in  FIG. 21 . For example, if the queue length were long enough to include all of points A through P of  FIG. 21 , then the overlapping as shown would not occur since all of the pixels in the shown segments would be part of the excluding clip region. But if the maximum queue length were set to, e.g., 4 regions, then at point M, as the overlap begins to occur, the queue would contain only the regions of the connecting region between L and M, the region defined by the pen tip instance at point L, the connecting region between K and L, and the pen tip instance at point K. In such a case, the portion of the ink to be overlapped would not be part of the excluding clip region. It is thus desirable to use a queue length that balances the above considerations. For example, a queue with a length of 4 regions is a reasonable compromise between quality and speed for a digitizer having a resolution of about 12,000 by 9,000 pixels with a sampling rate of about 130 samples per second. The maximal queue length may depend upon the resolution of the input digitizer, the display resolution, the sampling rate, the pen speed, user settings, application settings, and/or other considerations. For instance, a larger maximal queue length may be desirable with a higher digitizer resolution and/or a higher sampling rate. 
   The exemplary method of  FIG. 6  may be repeated for each new region. Following the example discussed above, after the new region  909  is pushed into the queue, the method of  FIG. 6  may be practiced where the new region is pen tip instance  905 . Once pen tip instance  905  is painted in step  605  and the excluding clip region is unfrozen in step  606 , then the pen tip instance  905  may be pushed into the queue and pen tip instance  903  may be pushed out of the queue. This results in the queue containing regions  908 ,  904 ,  909 , and  905 . 
   As an alternative to determining the union of the queued regions and/or freezing the pixels in the union, the intersection (i.e., overlap) between the new region and one or more of the queued regions may be determined. Instead of freezing the entire union of the queued regions, it may be desirable to freeze only those pixels in the intersection. For instance, where connecting quadrangle  909  is the new region, the intersection between the new region and the union of regions  907 ,  903 ,  908 , and  904  may be determined (as an alternative to step  603 ), and only those pixels in the intersection would be frozen (as an alternative to step  604 ). 
   It is understood that one or more of the steps illustrated in  FIG. 6  may be performed in a different order, combined with another step(s), and/or divided into further sub-steps as appropriate. For example, step  603  may be performed prior to step  602  or even prior to step  601 . Also, while embodiments of the present invention are described with the connections between pen tip instances being line segments, it is appreciated that the ink between the pen tip instances do not have to be actual line segments or quadrilaterals. The ink may include groupings of triangles, be bowed in shape, or assume a variety of shapes. One example of using curved lines is the advantage of being able to provide a degree of smoothing to an ink stroke. 
   The generation of connecting quadrangles is now discussed. Referring to  FIG. 7 , a particular exemplary ink stroke may include four circular pen tip instances  701 ,  702 ,  703 ,  704 , and three connecting quadrangles  705 ,  706 ,  707 . Connecting quadrangle  707  (for example) has four corners A, B, C, D, and four sides. The notation for an edge will refer to the end points of the edge. Thus, for example, the edge between corners A and B will be referred to as edge (or line or chord) AB. 
   The calculations for determining a connecting quadrangle may vary depending upon the relative shapes and sizes of the pen tip instances. Where the pen tip instances are both perfectly circular and of the same size, as in  FIG. 7 , the connecting quadrangle  707  that connects pen tip instance  703  and  704  may be defined by lines AC, BD that are tangent to the outer edges of both pen tip instances, closed by the chords AB, CD that connect them. Note that in this example where the pen tip instances are of the same size and are circular, the chords AB, CD each defines the geometric diameter of its respective pen tip instance. Also note that in this example, the connecting quadrangles are each rectangles with orthogonal sides. However, as will be seen in further examples, the connecting quadrangles are not necessarily rectangles. They may be any type of quadrangle such as parallelograms and trapezoids. 
   Thus far the exemplary pen tip instances have all been identically sized circles. However, this is not always the case. Pen tip instances may be of any shape, such as circles, rectangles (including squares), triangles, ovals, blobs, stars, lines, arcs, points, or polygons. Pen tip instances may be symmetric or asymmetric. An example of an asymmetric pen tip instance is one configured to emulate the tip of a calligraphy pen. Pen tip instances may also be of varying size, such that two consecutive pen tip instances in the same set of ink may be of different sizes. Pen tip instances may further be of varying shape, such that two consecutive pen tip instances in the same set of ink may be of different shapes. Pen tip instances may further be of varying rotation, such that two consecutive pen tip instances in the same set of ink may be rotated at different angles. Of course, where the pen tip instance is an exact circle, the angle of rotation is meaningless. The rotation of a pen tip instance is also considered a property of each pen tip position and/or the entire stroke. To account for these potential variations in pen tip instance characteristics, another exemplary ink rendering system  1000  is shown in  FIG. 10 . The ink rendering system  1000  includes, or is coupled to, a pen device  1000  that feeds the (x, y) coordinates of the pen tip to a contour generator  1002 . The pen device  1000  may also feed the pen tip instance size and/or rotation (e.g., angle) for each pen tip instance. The contour generator  1002  may be configured to generate a contour defining the outline of the pen tip instance based on the information provided by the pen device  1000 , as well as information about the particular pen tip instance shape selected. Alternatively, there may be a plurality of contour generators  1002  each specializing in a different shape or family of shapes. For example, there may be a first contour generator that is configured to generate contours for circular pen tip instances and a second contour generator that is configured to generate contours for rectangular (including square) pen tip instances. 
   The contour generator  1002  (or another specialized contour generator) may also generate contours that define the shape of the connecting quadrangles, based on the received and utilized pen tip instance characteristics and positions. The contour generator  1002  may then send the generated contours to a graphics toolbox  1004 . Where the ink is transparent, the contour generator  1002  may communicated with the graphics toolbox  1004  via a rendering environment  1003 , and the method of  FIG. 6  may be implemented. The graphics toolbox  1004  may fill or freeze the provided contours as appropriate and then output pixel values to an output device  1005  such as the monitor  407 . 
   Referring to  FIG. 11 , an exemplary ink stroke has four pen tip instances  1100 ,  1101 ,  1102 ,  1103  of different sizes. Since the pen tip instances are circular, rotation is less important in this example and will be ignored in the present example. As this ink stroke was drawn, the size of the pen tip instances changed from medium (pen tip instance  1100 ), to larger (pen tip instances  1101 ,  1102 ), and then smaller (pen tip instance  1103 ). The size, rotation, and/or pen tip shape may be adjusted automatically by a software application running on the computer  400  and/or by the user. For example, the user may have pressed then stylus/pen  466  down against the digitizer  466  with additional pressure, or may have moved the stylus/pen  466  more slowly, to select larger pen tip instances. Or the user may physically rotate the pen along its longitudinal axis in order to obtain different rotated pen tip instances. The connecting quadrangles for different-sized circular pen tip instances are, in some embodiments, generated by determining tangential lines (e.g., lines AC and BD in  FIG. 11 ) between the pen tip instances and then connecting those lines at the tangents with connecting chords (e.g., chords AB, CD in  FIG. 11 ). 
   Referring to  FIG. 13 , the same method may be used as in  FIG. 11  for determining connecting quadrangles (or other shapes). An exemplary ink stroke may include oval pen tip instances  1301 ,  1302 . The connecting quadrangle may, in one example, be determined by calculating the lines that run tangent between the two ovals. In this case, those tangential lines would be lines AC and BD in  FIG. 13 . The tangential lines would then be closed by connecting their endpoints at the tangents with lines AB, CD. Note that although these ovals are of different rotational angles, the rotation does not matter for ovals when determining the connecting quadrangles. 
   Next in  FIG. 15  is shown an exemplary embodiment of a connecting quadrangle between two rectangular pen tip instances  1501 ,  1502 , each having a different size and rotation. Although there are many possible connecting quadrangles, in this example, a connecting quadrangle  1503  connects corners A and E, corners A and C, corners C and G, and corners G and E. Another connecting quadrangle that could be used would connect corners A and H, corners D and F, corners B and G, and corners C and H. Another example is shown in  FIGS. 17A and 17B , showing two different connecting quadrangles  1703 ,  1704  that could be used to connect two pen tip instances  1701 ,  1702 . Connecting quadrangle  1703  connects corners A and A′, corners C and C′, corners A and C, and corners A′ and C′. Connecting quadrangle  1704  connects corners B and B′, corners D and D′, corners B and D, and corners B′ and D′. 
   It may be desirable to utilize a connecting quadrangle that connects between the outermost portions of the two pen tip instances to be connected. For instance, where the two pen tip instances are both polygons (i.e., closed shapes having only straight edges connected at corners), it may be desirable to connect the outermost corners together to provide for the largest area possible covered by the connecting quadrangle. Such an embodiment may in many cases provide a very smooth transition between pen tip instances and a higher-quality ink that is pleasing to the eye. Also, some or all of the possible connecting quadrangles (or a subset thereof) may be determined, and the determined quadrangles may be combined together (e.g., by taking their collective union) into a single connecting region. For example, referring to  FIG. 17C , all of the possible connecting quadrangles that connect the corners of the pen tip instances  1701 ,  1702  are shown. A result of this is that every corner of pen tip instance  1701  is connected to every corner of pen tip instance  1702  via an edge of at least one of the connecting quadrangles. This method may be extended to any polygon having any number of sides and corners. 
     FIG. 18  illustrates the resulting ink when all of the connecting quadrangles of  FIG. 17C  are combined together. 
     FIGS. 8 ,  12 ,  14 ,  16 , and  18  illustrate the rendered ink that corresponds to the pen tip instances and connecting quadrangles in  FIGS. 7 ,  11 ,  13 ,  15 , and  17 C respectively. The rendered ink in these figures is a result of using the rendering system  1000  as described. 
   Ink Smoothing 
   The ink-rendering process may also include smoothing the ink. Smoothing may be performed by the rendering system  500 ,  1000 , such as by the graphics toolbox  503 ,  1004 , using known smoothing functions. Another example of an ink rendering system  1900  is illustrated in  FIG. 19 . The ink rendering system  1900  includes a pen device  1901 , a smoothing application or subroutine  1902 , a curve-sampling application or subroutine  1903 , a contour generator application or subroutine  1904 , and/or a recipient  1905 , which may be a graphics toolbox. In operation, the pen device  1901  (e.g., a digitizer and pen) may measure the pen&#39;s (x, y) location on the digitizer. The pen device  1901  may further determine the intended rotation angle and/or size of the pen tip. The smoothing application  1902  may receive a plurality of sampled pen tip positions, pen tip instance sizes, and/or angles of pen tip instance rotation and may smooth the position, size, and/or rotation amongst the plurality of pen samples. The curve-sampling algorithm  1902  may sample the smoothed (x, y) curve, the smoothed size function, and/or the smoothed rotation function and may output samples of these smoothed functions to the contour generator  1904 . The contour generator  1904  may then generate the desired contours such as the pen tip instances and/or the connecting quadrangles, and forward these contours on to the recipient  1905 . 
   Smoothing may be performed on the size and/or rotation parameters. The rendering system  1900  may use any smoothing technique such as least squares fitting. To smooth ink, samples of the ink may need to be taken. These samples may be taken anywhere along the ink stroke, but at least one sampling technique is to sample the locations that were originally sampled from the pen (i.e., the sampled pen tip locations). 
   An exemplary smoothing function may be implemented by the ink rendering system  1900  (more particularly, by, e.g., the smoothing application  1902 ) as follows for each sample along the ink stroke:
 
(smoothed width) i   =A   1 *(original width) i−l   +A   2 *(original width) i   +A   3 *(original width) i+l ,   (1)
 
where A 1 , A 2 , and A 3  are constants that may be chosen as desired, i is the sample number along the sampled ink stroke, “smoothed width” is the width of the ink stroke at sample i after smoothing, and “original width” is the width of the ink stroke at sample i before smoothing. In some examples, the sum of these three constants should equal unity. A combination of A 1 =0.25, A 2 =0.5, and A 3 =0.25 works well. Angle of rotation can also be smoothed using any of the method for smoothing width, including substituting “smoothed width” and “original width” in equation 1 with “smoothed angle” and “original angle,” respectively. In another embodiment, both size and angle may be smoothed for the same ink stroke.
 
   Referring to  FIG. 20 , an exemplary ink stroke is shown having sampled points C l  through C n  (each denoted with an “x”). Each sampled point C also has an associated size and/or rotation. Size, or width, at sample point C i  will be denoted as W i , and its associated rotation will be denoted as R i . It may be desirable to smooth the sampled ink as to the (x, y) positions of the sample points, the size or width of the sampled points, and/or the rotation of the sampled points. 
   For example, using a least-squares method for smoothing, the following algorithm may be used such that the fitting curve P minimizes the following:
 
minΣ{a(C i −P i ) 2 +b[W(C i )−W(P i )] 2 +c[R(C i )−R(P i )] 2 },   (2)
 
where a, b, and c are optional weighting constants; P i  are the locations of the points on the fitting curve P; W(P i ) are the sizes/widths for each point P i ; and R(P i ) are rotations for each point P i . The fitting curve P may be any curve desired, such as one chosen from the family of parametric or Bezier curves. In effect, width/size and/or rotation are treated as additional dimensions other than position. Any subcombination of the dimensions in fitting a curve may also be used. For example, the third term c[R(C i )−R(P i )] 2  may be dropped from equation 2 so that rotation is not considered. Or, the second term b[W(C i )−W(P i )] 2  may be dropped from equation 2 so that width or size is not considered. Or, the first term a(C i −P i ) may be dropped from equation 2 so that sample position is not considered. Alternatively, both the first and second terms, or both the first and third terms, may be dropped from equation 2 so that only rotation or only width are considered in determining the fitting curve parameters.
 
   While exemplary systems and methods embodying the present invention are shown by way of example, it will be understood, of course, that the invention is not limited to these embodiments. Modifications may be made by those skilled in the art, particularly in light of the foregoing teachings. For example, each of the elements of the aforementioned embodiments may be utilized alone or in combination with elements of the other embodiments. For example, while connecting quadrangles are discussed herein as a particularly advantageous shape, any shape of connecting regions other than quadrangular-shaped regions may be used. Also, while the above description discussed pen tip positions as being defined by (x, y) in a rectilinear coordinate system on the digitizer, any other coordinate system, such as polar, may be used.