Abstract:
A texture procedure allows the rendering of curve bounded objects to a graphics display device directly from a high level curve-based description. The method comprises receiving a curve-based description of the graphics object and dividing the graphics object into a rectangular mesh of texels. Each texel is then detailed by defining a combination of curved geometry functions and a boolean function. These function are then evaluated for each pixel of the graphics display device thereby rendering the graphics object to a graphics display. The texture procedure features include being procedural based and not image-based. This allows a rendering with continued accuracy even under arbitrary magnification conditions. Furthermore, the texture procedure is defined as such that will allow it to function using conventional tri-linear interpolation hardware.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to computer graphics, and more particularly to a texture procedure for rendering curved bounded images on a graphics display device. 
     2. Related Art 
     A typical computer generated image of a curve bounded object utilizes a number of line segments to approximate the curved boundary. A pixel is a picture element of a graphics display device. Each pixel may represent unique attributes such as color, lighting, texture, etc. As is well known in the relevant art(s), texture procedures may be used to provide visual detail for displayed graphical objects. In a line segment technique, the more line segments used in the representation, the greater accuracy the ultimate display may contain. Each segment may contribute to one or more pixels of the final image. However, the number of line segments that can be used to render a display of a curved bounded region is limited. It is limited by the resolution of the intended graphics display device as well as the available memory space on the computer system. Furthermore, the more line segments used, the greater will be the calculation time, and thus the rendering time. 
     Conventional textures are image-based and composed of texels, which taken collectively form an image. A polygon is rendered with a mapping specified between the polygon&#39;s vertices and the texture. This mapping is usually specified in texture coordinates (s, t). When the polygon is rendered into pixels, the texture coordinates of each pixel are determined and used to lookup a value in the texture that is used in the drawing of the pixel. This value may be color, transparency, etc. Other conventional methods for rendering curved bounded regions have attempted various polygonal approximations and other incremental methods that result in the same limitations described above with respect to line segment techniques. Therefore, what is needed is a texture procedure for rendering curved based objects without the need to convert the graphics object into line segments or perform any type of tessellation. 
     SUMMARY OF THE INVENTION 
     The present invention is a system and method for rendering a curve bounded object to a graphics display device directly from a high level curve-based description. The method includes receiving a curve-based description of the graphics object (e.g., character typeface) and dividing the graphics object into a rectangular mesh of texels. Each texel is detailed by defining a combination of curved geometry functions and a boolean function. Each texel contains a miniature resolution independent image of bounded complexity. Taken collectively, the texels form a continuous or resolution independent binary image. The result of the above steps transforms the graphics object into a geometric-texture. The method of the present invention, for each pixel to be displayed, then determines the Cartesian (s, t) coordinate pair within the geometric-texture. The curved geometry functions and the boolean function defined for the texel containing the (s, t) pair are then evaluated. This step is repeated for each (s, t) pair of each pixel of each polygon to be rendered. The result is an alpha value or color for each pixel and thus the display of the curve bounded object to a graphics display device. 
     In a preferred embodiment, the method of the present invention utilizes two horizontal axis functions, f 0 (s) and f 1 (s), and two vertical axis functions, g 0 (t) and g 1 (t), within each texel in creating the curved geometries. 
     One advantage of the present invention is that unlike conventional methods that used image-based textures, the invention uses procedural textures that have no inherent resolution and will remain accurate when subject to arbitrary magnification. 
     Another advantage of the present invention is that the method can be performed using conventional tri-linear interpolation hardware when the curved geometry functions defined for each texel are cubic polynomials. 
     Another advantage of the present invention is that the method may also be used to implement a class of procedural alpha texture for selectively drawing (trimming) graphic primitives. 
    
    
     BRIEF DESCRIPTION OF THE FIGURES 
     The present invention will be described with reference to the accompanying drawings, wherein: 
     FIGS. 1A and 1B are a flowchart representing the overall preferred operation of the present invention; 
     FIG. 2 is an illustration of how Bézier curves divide a texel according to the present invention; 
     FIG. 3 is an illustration of how a texel is located within a texture according to the present invention; 
     FIG. 4 illustrates an example of the preprocessing and run-time transformations according to the present invention; 
     FIG. 5 is a block diagram of an exemplary computer system useful for implementing the present invention; 
     FIG. 6 is an illustration of a texture map on which the present invention would operate; 
     FIG. 7 is an illustration of a texture map divided into several texels according to the present invention; 
     FIG. 8 is an illustration of a texture map divided into several labeled texels according to the present invention; 
     FIG. 9 is an illustration of one texel of a texture map containing one curve according to the present invention; 
     FIG. 10 is an illustration of one texel of a texture map containing two curves according to the present invention; 
     FIG. 11 is an illustration of one texel of a texture map containing three curves according to the present invention; 
     FIG. 12 is an illustration of a texel containing a Bézier curve, g 0 (t), defined as a function of the vertical axis according to the present invention; 
     FIG. 13 is an illustration of a texel containing a Bézier curve, f 0 (s), defined as a function of the horizontal axis according to the present invention; 
     FIG. 14 is an illustration of a texel containing a Bézier curve, g 1 (t), defined as a function of the vertical axis according to the present invention; 
     FIG. 15 is an illustration of a texel containing a Bézier curve, f 1 (s), defined as a function of the horizontal axis according to the present invention; 
     FIG. 16 is an illustration of a texel containing four Bézier curves and the mapping done according to the present invention; and 
     FIGS. 17-24 illustrate how the texture procedure of the present invention can be performed using conventional tri-linear interpolation hardware. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Overview 
     The present invention relates to a texture procedure that uses conventional tri-linear interpolation hardware to compute whether a pixel is inside or outside a curved region. The method can be used to quickly render characters (e.g., PostScript™ typefaces) directly from curved based descriptions. 
     The present invention is described in terms of a character-rendering example. This is for convenience only and is not intended to limit the application of the present invention. In fact, after reading the following description, it will be apparent to one skilled in the relevant art how to implement the following invention in alternative embodiments (e.g., to implement a class of procedural alpha texture for selectively trimming graphic primitives). 
     Referring to FIGS. 1A and 1B, texture procedure  100  illustrates the overall operation of the present invention. Texture procedure  100  begins at step  102  with control passing immediately to step  104 . In step  104 , a computer stored (either digitized or synthesized) curve-based description of a curve bounded object is received and converted into a texture map. For example, a PostScript™ font (character) might be received. They are defined by curves and straight lines. The texture map is then divided into a rectangular mesh of regions known as texels in a step  106 . Next, in a step  108 , the interior of each texel, based on the shape of part of the curve bounded object appearing in the texel, is detailed by defining up to four Bézier curves. Use of the Bézier formulation for constructing curves to display curved bounded regions (surfaces) is well known in the relevant art. See Hearn, Donald and Baker, M. Pauline,  Computer Graphics,  (Prentice-Hall: USA 1986) pp. 195-98, which is incorporated herein by reference in its entirety. 
     The four Bézier curves are two functions, f 0 (s) and f 1 (s), of the horizontal axis (s), and two functions, g 0 (t) and g 1 (t), of the vertical axis (t). In a preferred embodiment, each of the four functions are defined using the Bézier formulation with four control points, P 0 , P 1 , P 2  and P 3 , as will be explained in detail below with reference to FIGS.  8 - 15 ). This process results in four cubic polynomial functions that display (approximately) the desired curved object. Each of these curves divides the texel into two regions, plus (+/0) and minus (−/1) (as shown in FIG. 2 with reference to a texel  202 ). The plus region is above or to the right of the curve, whereas the minus region is below or to the left of the curve. 
     In addition to the four Bézier functions, each texel is also defined, in step  110 , by a boolean function based on the shape appearing in each texel and the four Bézier curves. In step  112 , a 16-bit boolean vector is then obtained by evaluating the boolean function (defined in step  110 ) based on the shape of the part of the curve bounded object appearing in each texel and the four cubic polynomial Bézier functions (defined in step  108 ). 
     In step  113 , the resultant geometric-texture is stored (on a host computer memory as will be explained below with reference to FIG.  5 ). Steps  102  to  113  can be part of a preprocessing procedure for a set of characters (e.g., a PostScript™ font). Once created and stored, the geometric textures can be loaded into memory at run-time (step  113   b ) for continuation of the texture procedure  100 . The geometric-texture is used to draw a polygon that allows the curved object, defined by the geometric-texture, to be drawn. Because the geometric-texture of the present invention behaves like a conventional texture, many polygons could be used to draw the object, possibly mapping it onto a three-dimensional object. 
     At run-time processing, for each pixel to be displayed of each graphics primitive (i.e. polygon) to be rendered using the geometric-texture, the texture (s, t) coordinate pair is determined (step  114 ). This is done using any form of conventional interpolation. Then, the texel into which the (s, t) pair falls is located. In step  115 , the texture (s local , t local ) coordinate pair local to the texel must be computed. Since there are normally a  2   n  number of texels in each dimension of the texture map, this computation is not costly. For example, step  114  and the computation of step  115  are illustrated in FIG.  3 . FIG. 3 shows a texture map  300  divided into sixteen texels (4×4 array). The (s, t) coordinates within the texture  300  range from 0.0 to 1.0. Thus the coordinates of any pixel within the texture map  300  will be expressed in as a (s, t) pair where s and t are fractions. A pixel  302  is first located within the texture map  300 . Its (s, t) coordinate pair, relative to texture  300 , is (⅜, {fraction (9/16)}). This completes step  114 . 
     In step  115 , a local (s local , t local ) coordinate pair of pixel  302 , relative to texel  202 , is computed. Texel  202  has an origin whose (s, t) coordinate pair is (¼, ½). The origin is simply the (s, t) coordinate pair, relative to texture  300 , of the bottom left corner of the texel  202 . Then, the (s local ,t local ) coordinate pair of pixel  302  is computed as follows: 
     
       
         (s local , t local )=(j*(s—s origin ), k*(t—t origin ))  
       
     
     where j and k are the number of texels which divide texture map  300  in the s and t direction respectively. In FIG. 3, the result of the above calculation is an (s local, t   local ) coordinate pair of (½, ¼) for pixel  302 . 
     In step  116 , for each (s local , t local ) pair, the four functions of the texel where the (s, t) pair lies are evaluated in parallel. As will be explained below (with reference to FIGS.  17 - 24 ), these four evaluations may use the same hardware that is required for tri-linear mapped textures. The difference between each function and the opposing texture coordinate is used to determine whether the (s local , t local ) pair is in the plus (+/0) or minus (−/1) region. The evaluations are illustrated in step  116   b  as follows: 
     Curve  0 : sign(t −f 0 (s local )) yields Bit  0   
     Curve  1 : sign(t −f 1 (s local )) yields Bit  1   
     Curve  2 : sign(s −g 0 (t local )) yields Bit  2   
     Curve  3 : sign(s −g 1 (t local )) yields Bit  3   
     The resultant 4-bit “outcode” (bits  0 - 3  concatenated), corresponds to the (s local , t local ) coordinate pair&#39;s relationship with the plus or minus regions with respect to each curve. The outcode is then used as an index into the boolean vector (step  118 ) (as will be further explained below with reference to FIG.  16 ). Step  120  can then determine, for example, the alpha value or “in” or “out” for each (s local , t local ) pair. If the boolean vector bit pointed to by the outcode is set to 0, then the pixel is transparent (step  122 ). Alternatively, if the boolean vector bit pointed to by the outcode is set to 1, the pixel is opaque (step  124 ). Steps  114 - 124  are thus repeated for every pixel to be displayed of each geometric-texture to be rendered (this recursion is not shown in FIGS  1 A and  1 B). The process is thus completed, as indicated by step  126 , when the entire curve bounded object is rendered to the graphics display device pixel by pixel. 
     The run-time processing steps of  113   b  to  126  can be repeated any number of times to produce different transformations of the geometric-texture. The texture procedure  100  is illustrated for a set of characters in FIG.  4 . FIG. 4 shows the division between preprocessing (steps  102  to  113 ) and run-time processing (steps  113   b  to  126 ). In the case of drawing a character, the polygon normally specifies (s, t) coordinates that completely surround the character to be drawn. The recursion mentioned above would thus be performed for each (s, t) coordinate pair of the polygon&#39;s pixels. By applying transformations to the polygon (only four points) the entire character is transformed. In FIG. 4, an entire alphabet and a polygon with the mapping into the geometric-texture is defined. The polygon can then be drawn several times by applying different transformations, according to the present invention, that result in four different renderings  402   a - 402   d.    
     Furthermore, it is important to note that the result of texture procedure  100  (as shown in FIGS. 1A and 1B) is a single number (a zero or a one). Therefore, it will be apparent to one skilled in the relevant art how to implement the method of the present invention to use the result for various purposes (e.g., color) other than transparency. 
     Environment 
     FIG. 5 is a block diagram of an exemplary computer imaging system  501  useful for implementing the present invention. Computer imaging system  501  includes a host computer  502 , geometry engine  504 , rasterizing unit  506 , texture engine  508 , texture memory  510 , attenuation unit  550 , and frame buffer  512 . Imaging system  501  further includes a separator unit  507 . Steps  106 - 120  are carried out in texture engine  508  and can be implemented in software, firmware, and/or hardware in one or more processing components. Steps  122 - 124  would take place on frame buffer  512 . For example, any host or graphics processor can be used to implement texture procedure  100  in software running on a processor(s). In the example of FIG. 5, host  502  can implement step  114  by controlling pixels passed to separator unit  507 . Separator unit  507  can be any type of processing logic (or program code executing on host  502 ). 
     The present invention is described in terms of an example computer graphics processing environment. As described herein, the present invention can be implemented as software, firmware, hardware, or any combination thereof. 
     Given the description herein, it would be apparent to one skilled in the art to implement the present invention in any computer graphics application, API, or any other system that supports a texture engine including, but not limited to, a computer graphics processor (single chip or multiple chips), high-end to low-end graphics workstations, gaming platforms, systems and consoles. 
     Description in these terms is provided for convenience only. It is not intended that the invention be limited to application in this example environment. In fact, after reading the following description, it will become apparent to a person skilled in the relevant art how to implement the invention in alternative environments. 
     The present invention can be implemented using software running (that is, executing) in an environment similar to that described above. In this document, the term “computer program product” is used to generally refer to a removable storage unit or a hard disk installed in a hard disk drive. These computer program products are means for providing software to a computer system (e.g., host  502 ). 
     Computer programs (also called computer control logic) are stored in main memory and/or secondary memory. Computer programs can also be received via a communications interface. Such computer programs, when executed, enable the computer system to perform the features of the present invention as discussed herein. In particular, the computer programs, when executed, enable a processor to perform the features of the present invention. Accordingly, such computer programs represent controllers of a computer system. 
     In an embodiment where the invention is implemented using software, the software may be stored in a computer program product and loaded into a computer system using a removable storage drive, hard drive, or communications interface . Alternatively, the computer program product may be downloaded to computer system over a communications path. The control logic (software), when executed by a processor, causes the processor to perform the functions of the invention as described herein. 
     In another embodiment, the invention is implemented primarily in firmware and/or hardware using, for example, hardware components such as application specific integrated circuits (ASICs). Implementation of a hardware state machine to perform the functions described herein will be apparent to persons skilled in the relevant art(s). 
     Detailed Example of Texture Procedure  100   
     FIG. 6 is an illustration of a texture map  600  on which the present invention would operate. In a preferred embodiment, the method of the present invention to directly render curve bounded objects would be used to render characters (e.g., PostScript™ typefaces). Accordingly texture map  600  is a computer stored image of the lowercase letter “b”. 
     FIG. 7 is an illustration of texture map  600  divided into a rectangular mesh of regions known as texels. Although FIG. 6 shows texture map  600  divided into 25 texels (5×5 rectangular grid), it should be understood that this is presented as an example and not a limitation. For reasons that will become clear, the method of the present invention allows texture map  600  to be divided into a lesser number of texels than those needed by conventional texture resolution methods. 
     FIG. 8 is an illustration of texture map  600  divided into texels as shown in FIG.  7 . However, in FIG. 8, five texels  601 - 605  have been labeled “A” through “E”, respectively, for purposes of the following explanation of a preferred embodiment of the present invention. 
     FIG. 9 is a detailed illustration of texel  602  (labeled “B”) of texture map  600 . The shape appearing in texel  602 , which is simply the part of the lowercase letter “b” of FIG. 7 that falls into texel  602 , defines one Bézier curve  0 . Curve  0  is defined through the Bézier formulation using four control points P 0 , P 1 , P 2  and P 3 . This curve is simply a function of the vertical axis (t). Thus, the 16-bit boolean vector would contain half ones corresponding to the Bézier curve  0 . The boolean vector would thus be: 
     0101010101010101 
     FIG. 10 is a detailed illustration of texel  603  (labeled “C”) of texture map  600 . The shape appearing in texel  603 , which is simply the part of the lowercase letter “b” of FIG. 7 that falls into texel  603 , defines two Bézier curves  0  and  1 . Bézier curve  0  is simply a function of the vertical axis (t) as in texel  602 . Bézier curve  1  is a function of the horizontal axis (s). Thus, the 16-bit boolean vector reflects the union of the two half spaces of Bézier curves  0  and  1 . The boolean vector would thus be: 
     Curve  1 : 1100110011001100 
     Curve  0 : 1010101010101010 
     Boolean Function=curve  0  V curve  1   
     Boolean Vector=1110111011101110 
     FIG. 11 is a detailed illustration of texel  604  of (labeled “D”) of texture map  600 . The shape appearing in texel  604 , which is simply the part of the lowercase letter “b” of FIG. 7 that falls into texel  604 , defines three Bézier curves  0  , 1  and  2 . Two Bézier curves are function of the vertical axis (t) and one is a function of the horizontal axis (s). The boolean vector is such that the vertical curves are split by the horizontal curve. The boolean vector would thus be: 
     Curve  1 : 1100110011001100 
     Curve  0 : 1010101010101010 
     Curve  2 : 1111000011110000 
     Boolean Function  (curve0Λcurve1)V(curve2Λ{overscore (curve)}1)   
     Boolean Vector=1011100010111000 
     Referring to FIG. 8, it can be seen that texel  601  (labeled “A”) has no curves defined as all the pixels within the texel need to be filled during rendering. Thus, no matter what curves are evaluated, according to the present invention, the result would always be a logical  1  (i.e. inside the curve). Thus the boolean vector for such a texel is all TRUE (“1”s). Still referring to FIG. 8, texel  605  (labeled “E”) also contains no curves defined. Because no pixels within the texel need to be filled during rendering, no matter what curves are evaluated, according to the present invention, the result would always be a logical  0  (i.e. outside the curve). Thus the boolean vector for such a texel is all FALSE (“0”s). 
     Now referring to FIG. 12, a detailed illustration of a texel  900  containing a Bézier curve  0  defined as a function of the vertical axis, g 0 (t) is shown. If a (s local , t local ) coordinate pair evaluates in the shaded region, then bit  0  is set to logical TRUE (“ 1 ”). 
     FIG. 13 is an illustration of texel  1200  containing a Bézier curve  1  defined as a function of the horizontal axis, f 0 (s). If a (s local, t   local ) coordinate pair evaluates in the shaded region, then bit  1  is set to logical TRUE (“ 1 ”). 
     FIG. 14 is an illustration of texel  1200  containing a Bézier curve  2  defined as a function of the vertical axis, g 1 (t). If a (s local , t local ) coordinate pair evaluates in the shaded region, then bit  2  is set to logical TRUE (“ 1 ”). 
     FIG. 15 is an illustration of texel  1200  containing a Bézier curve  3 , defined as a function of the horizontal axis, f 1 (s). If a (s local, t   local ) coordinate pair evaluates in the shaded region, then bit  3  is set to logical TRUE (“ 1 ”). 
     Now referring to FIG. 16, texel  1200  is shown containing all four Bézier curves  0 - 3  (shown individually in FIG. 12-15 respectively). Also shown in FIG. 16 is the mapping done according to the present invention is illustrated. The resulting four bits ( 0 - 3 ) are concatenated to form a 4-bit outcode. The 4-bit outcode is then used as an index into the earlier evaluated 16-bit boolean vector for texel  1200  (see step  112  of FIG.  1 A). The bit in the 16-bit boolean vector that corresponds to the 4-bit outcode is then used to render the pixel on the graphics display. The value read from the boolean vector is the alpha value for the pixel. More specifically, if the boolean vector bit pointed to by the outcode is set to 0, then the pixel is transparent. Alternatively, if the boolean vector bit pointed to by the outcode is set to 1, the pixel is opaque. In general, the boolean vector will contain 2 n  bits for n curves in each texel. This is because each curve provides one bit of the index into the boolean vector. 
     Advantages of the Present Invention 
     An advantage to the above described texture procedure is that by modeling the part of the texture map that falls into each texel with four Bézier curves, more detail is modeled by each texel. Thus, a texture map can be divided into fewer texels thereby providing a very significant performance increase in rendering objects to graphics display devices. 
     Another advantage of the present invention is that conventional tri-linear interpolation hardware can be used to implement the texture procedure  100 . Tri-linear interpolation, as is well known in the relevant art, is normally used to compute the weighted average of eight texels. This technique provides some rudimentary filtering. Tri-linear interpolation is briefly described to illustrate the analogy to the present invention that permits re-use of the tri-linear interpolation hardware for rendering curve bounded regions in accordance with the present invention. 
     In standard tri-linear interpolation texture mapping, a texture map is stored in varying degrees of pre-filtering. For example, FIG. 17 shows a sample texture pattern  1702  and several pre-filtered or lower level of detail (LOD) versions  1704 - 1714  of the same map. Each of the lower LOD maps is one half the height and width of the next higher LOD map and is made by averaging together each group of four texels of the next higher LOD map. During a mapping operation, the size and shape of a pixel image mapped into the texture map is used to determine which level of detail (LOD) is appropriate for use in the texture mapping operation. Each LOD is useful for a different mapped pixel size. 
     When a pixel size corresponds exactly to an existing LOD map (e.g., texture map  1702 ), the tri-linear interpolation operation simplifies to a bilinear interpolation. For example, to determine the contribution of the texels of LOD  1702  to the color of a display pixel on a display screen, the location of the pixel is mapped (i.e., transformed) to the texture map. The texture is then “sampled” at the exact point where the pixel center mapped into the texture map. However, because the pixel center may not coincide exactly with a texel value, a weighted average of the four nearest texel values is taken. This is illustrated in FIG.  18 . The mapped pixel center is indicated at  1802 . Note that pixel center  1802  falls between texel centers A, B, C and D of texture map  1702 . One way to take the weighted average of these four texel values is by doing a bilinear interpolation (i.e., a linear interpolation in two dimensions). 
     In the case where the pixel size does not correspond exactly to any existing LOD map, it will likely fall between two maps (e.g., texture maps  1702  and  1704 ). In this case, a bilinear interpolation is performed in both maps and a linear interpolation is used to blend the two results. The two bilinear interpolations followed by a linear interpolation yields a “tri-linear interpolation.” For example, if the pixel size fall between a size corresponding to map  1702  and map  1704 , then a bilinear interpolation operation would be performed in map  1702  as discussed above. In addition, a bilinear interpolation operation would be performed in map  1704  as shown in FIG.  18 . The two resulting values would then be linearly blended based on the actual pixel size relative to the two bracketing LOD maps to yield a color value for the pixel. FIG. 19 graphically depicts the tri-linear interpolation operation between the two LOD maps  1702  and  1704 . 
     Normally, the tri-linear interpolation hardware simultaneously computes the four component values (R, G, B, α) for a pixel, effectively requiring four tri-linear interpolation engines as graphically depicted in FIG.  20 . In the case of the present invention, the objective is to use the same structure(s) to compute cubic polynomials. 
     The inventor discovered that a cubic polynomial in Bézier form (using four control points, P 0 , P 1 , P 2  and P 3 ), as shown in FIG. 21, can be computed using nested linear interpolation. The mapping of the four scalar control points between conventional Bézier polynomial calculation (FIG. 21) and tri-linear interpolation (FIG. 20) is shown in FIG.  22 . The mapping of the intermediate terms in the interpolation are shown in FIGS. 23 a  and  23   b . The complete mapping (a combination of FIGS. 22-23) is shown in FIG.  24 . 
     Thus, the four Bézier curves used to describe the detail in each texel can be computed using the four tri-linear interpolation engines (R, G, B, and α) of typical tri-linear interpolation hardware. Normally the four tri-linear interpolators use the same sets of weighting values in all four engines. However, to implement the present invention, two sets of weighting values must be used. Two engines will use the offset of the s texture coordinate within the texel, and the other two engines will use the offset of the t texture coordinate within the texel. As will be apparent to one skilled in the relevant art, the conventional tri-linear interpolation hardware will need to be augmented with logic to perform the various operations of texture procedure  100  (e.g., subtraction, looking up the outcode, etc.). 
     Another advantage of the present invention is that the four Bézier curves are defined by sixteen scalar values (four control points, P 0 , P 1 , P 2  and P 3 , for each curve defined in a texel) and the boolean vector is defined by a 16-bit value. The boolean vector may be stored as a separate 16-bit value (e.g., luminance texture word) or as part of the curve textures. Therefore, if the latter implementation is chosen, the 16-bit boolean vector can be stored as the low bit for each of the sixteen scalars (P 0 , P 1 , P 2  and P 3  for four curves) without using any additional computer memory resources. 
     Yet still, an additional advantage of the current invention is that complex geometry can be transformed with very little computational overhead. As illustrated in FIG. 4, a complex figure can have any projective transform applied to it at the cost of transforming the polygon that uses the figure as a texture which is usually only four points. Normally, the transformation of such a figure would require that all of the points describing the figure be transformed—a much more costly operation. 
     Conclusion 
     While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example, and not limitation. It will be apparent to persons skilled in the relevant art that various changes in form and detail can be made therein without departing from the spirit and scope of the invention. Thus the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.