PATENT ABSTRACT
A graphics processor capable of rendering three-dimensional polygons with color, shading; and other visual effects also corrects interpolation errors that occur as a result of mapping the polygon to a pixel grid display. The processor renders polygons using an Incremental Line-Drawing algorithm and features an error correction circuit capable of adjusting the initial and incremental gradient parameters for each pixel characteristic and then rendering each scan line with the proper orthogonal adjustment. The error correction circuit includes an ortho correction engine for correcting errors in the initial and incremental pixel parameters and an ortho adjust engine to accommodate overflows in the x-coordinate calculations. The processor is able to render the polygons with monotonic gradients in color, shading, depth, and other visual characteristics without interpolation error.

PATENT DESCRIPTION
CROSS-REFERENCE TO RELATED APPLICATIONS 
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     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
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     REFERENCE TO MICROFICHE APPENDIX 
     This patent document includes a Microfiche appendix consisting of 1 microfiche with 61 frames. 
     COPYRIGHT AUTHORIZATION 
     A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent files or records, but otherwise reserves all copyrights whatsoever. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to computer systems and particularly to graphics processors. More particularly, the present invention relates to a graphics processor adapted to remove interpolation errors from parameters defining a polygon and render the polygon on a computer display. 
     2. Background of the Invention 
     Recent advances in graphics processing technology have allowed computer display devices to deliver higher resolution, greater rendering precision, and faster processing speed. Such advances have enabled computers to better perform the complex instructions demanded by graphics-intensive software applications offering movie playback, interactive video, multimedia, games, drawing or drafting capabilities, and other video-intensive tasks. One important feature of these applications is the capability to quickly and accurately render complex graphic objects on-screen, at the same time incorporating visual effects (also known as “pixel characteristics”) such as shading, specular lighting, three-dimensional (3D) perception, texture-mapping, fog or haze effects, alpha blending, depth, and other effects. Such visual effects make the graphics seem more realistic and improve the overall quality of the images. 
     Shading consists of varying an image color along the span of the image, while the lighting effect is accomplished by multiplying the color intensities of an image by a constant value. Other techniques exist to create 3D effects such as depth and texture-mapping by translating two-dimensional (2D) patterns and shapes so that images appear to have a depth component, even though the images are tendered on a 2D screen. Fog and alpha blending change the appearance of an image in subtler ways. Fog creates the illusion of a mist, or haze throughout the object and may be used in conjunction with other 3D effects to render images that appear to be at far distances. Alpha blending may be used to mesh together, or blend, screen images. 
     Computer systems typically incorporate raster display systems for viewing graphics, consisting of a rectangular grid of pixels aligned into columns and rows. Typical displays may incorporate screens with 640×480 pixels, 800×600 pixels, 1024×768 pixels, 1280×1024 pixels, or even more pixels. The display device is usually a cathode ray tube (CRT) capable of selectively lighting the pixels in a sweeping motion, moving across each consecutive pixel row “scan line”), from left to right, top to bottom. Accordingly, an entire screen of pixel values is known as a “video frame,” and the display device usually contains a frame buffer consisting of Dynamic Random Access Memory (DRAM or Video Random Access Memory (VRAM) which holds the pixel intensity values for one or more video frames. The frame buffer, updated regularly by the computer or graphics processor, is read by the display device periodically in order to excite the pixels. Frame buffers in color displays typically hold 24-bit values (3 bytes) for each pixel, each byte holding the pixel intensity value for one of the three primary colors, red, green, or blue. Accordingly, the three primary colors are combined to produce a wide spectrum of colors. Liquid crystal display (LCD) systems operate in a similar fashion as do CRT devices. 
     The pixel intensity values usually are computed and placed into the frame buffer by a graphics processor that is controlled by a software application known as a display driver. The display driver typically handles all of the graphics routines for the software applications running on the host computer by sending parameters to the graphics processor which describe the geometries of the graphics. One common technique for rendering screen images is to partition the images into simple constituent polygons such as triangles or quadrangles and to then render the constituent polygons on the display. Such a technique has two distinct advantages. First, since even very large polygons can be defined in terms of relatively few parameters, the software driver may send only the necessary polygon parameters, as opposed to transmitting a distinct intensity value for each pixel to the graphics processor. By sending a minimum of data per pixel, the software driver has more time in which to transmit increasingly detailed information to the processor about the polygon, including the parameters to describe the visual effects listed above. In one method of defining a polygon via parameters, the software driver uses the polygon vertex coordinates to calculate through interpolation (or “interpolate”) the widths of the polygon along each scan line as well as the slopes of the edges between the vertices. A relatively small number of parameters which completely define the polygon may then be transmitted to the graphics processor to define the polygon for rendering. 
     Second, graphics processors have been developed which are highly successful at implementing elementary polygon-rendering routines. A typical polygon-rendering algorithm uses an initial polygon coordinate along with the polygon height and width and the slopes of the polygon edges to incrementally render the polygon. Beginning at the initial coordinate, the graphics processor enters into the frame buffer a horizontal line of pixels spanning the width of the polygon on the initial pixel row. Using the initial coordinate along with the polygon height and edge slopes, the graphics processor can compute the polygon coordinates along one vertical or slanted edge, called the “main slope,” of the polygon. For each consecutive scan line, the graphics processor then uses the width values of the polygon to draw each horizontal row of polygon pixels into the frame buffer. Such an algorithm is known as the Incremental Line-Drawing algorithm, or Digital Differential Analyzer (DDA). 
     An incremental algorithm for rendering pixels at discrete positions on a pixel grid generally begins at a starting point and proceeds for some number of iterations, calculating the location of a single pixel during each iteration. The location of the current pixel in the scan line during any iteration is calculated by adding an increment, or delta, to the previous coordinate. The number of iterations needed for one scan line is the number of points in that scan line, or the distance to be spanned. Using such an algorithm, a graphics processor can draw polygons that are random triangles of any orientation or quadrangles with at least one flat top or bottom. Setting aside trivial triangles and colinear triangles, which are either points or lines, any random triangle or quadrangle can be partitioned into upper and lower triangles with a common horizontal side. The common horizontal side intersects the center, or opposite, vertex of the random triangle or quadrangle. The edge of the triangle or quadrangle opposite this center vertex, or the main slope, always spans the entire height of the triangle or quadrangle. The random quadrangle or triangle may be constructed by invoking an Incremental Line-Drawing algorithm twice-first to draw the upper polygon and again to draw the lower polygon. 
     Referring now to FIG. 1, triangles  100 ,  120 ,  140 , and  160  represent the four general orientations of a random triangle. Triangle  100  may be partitioned into two constituent triangles  102  and  104  having common horizontal side  106  and opposite vertex  108 . Main slope  110  spans the entire height of triangle  100 , while first opposite slope  112  and second opposite slope  114  constitute the other two edges. Triangle  100  can be rendered using the Incremental Line-Drawing algorithm by drawing constituent triangles  102  and  104  separately, as will be explained in greater detail below. Triangles  120 ,  140 , and  160  may be partitioned similarly into triangles  122  and  124  (constituting triangle  120 ), triangles  142  and  144  (constituting triangle  140 ), and triangles  162  and  164  (constituting triangle  160 ). Accordingly, these triangles have main slope  130  (triangle  120 ), main slope  150  (triangle  140 ), and main slope  170  (triangle  160 ) with opposite slopes  132  and  134  (triangle  120 ), opposite slopes  152  and  154  (triangle  140 ), and opposite slopes  172  and  174  (triangle  160 ). Triangles  120 ,  140  and  160  also have opposite vertex  128  (triangle  120 ), opposite vertex  148  (triangle  140 ), and opposite vertex  168  (triangle  160 ). 
     Examining the triangles from left to right, the main slopes  110  and  170  of triangles  100  and  160 , respectively, have downward gradients, while the main slopes  130  and  150  of triangles  120  and  140 , respectively, have upward gradients. The opposite vertices  108  and  148  both lie to the left of respective main slopes  110  and  150 , while opposite vertices  128  and  168  both lie to the left of respective main slopes  130  and  170 , respectively. Hence, triangles  100  and  140  are said to have negative opposite vertex directions, while triangles  120  and  160  are said to have positive opposite vertex directions. Thus, triangles  100 ,  120 ,  140 , and  160  embody all four combinations of main slope gradients and opposite vertex directions, thereby constituting the four general types of random triangles. It follows that any one of the four triangles  100 ,  120 ,  140 , and  160  can be uniquely identified by its main slope gradient and opposite vertex direction. 
     The parameters needed by a graphics processor to render a quadrangle with flat top and bottom edges or any randomly-oriented triangle typically comprise a set of fractional-valued parameters including a starting x-coordinate X MINT :X MFRAC , a delta X main ΔX MINT :ΔX MFRAC , a starting line width W MINT :W MFRAC , and a delta main width ΔW MINT :ΔW MFRAC . A software driver transmits the polygon parameters to the graphics processor, which renders the polygon as described below. Each fractional-valued parameter can be expressed as an integer plus a fraction, with the term “INT” denoting the integer portion and “FRAC” identifying the fractional portion. For example, if X MINT :X MFRAC =3.25, then X MINT =3 and X MFRAC =¼. For clarity, the fractional-valued parameters X MINT :X MFRAC , ΔX MINT :ΔX MFRAC , W MINT :W MFRAC , and ΔW MINT :ΔW MFRAC may be abbreviated as X M , ΔX M , W M , and ΔW M , respectively, all other fractional-valued parameters expressed herein using similar notation. A graphics processor also receives integer-valued parameters including an initial y coordinate Y M , a polygon height, and, the rendering direction X DIR , which defines whether the pixels are drawn from left to right or from right to left across each scan line. In the example of FIG. 1, a graphics engine draws pixels across a scan line from the main slope to the opposite slope, although the pixels may be rendered from opposite slope to main slope in some implementations. By convention, X DIR  may be thought of as negative if the main slope lies to the right of the opposite slope or positive if the main slope lies to the left of the opposite slope, and the graphics processor assigns X DIR =0 if X DIR  is positive and X DIR =1 if X DIR  is negative. Notice that the X DIR  parameter corresponds exactly to the “opposite vertex direction” defined with respect to the triangles of FIG.  1 . Hence, triangles  100  and  140  have X DIR =1 (negative) while triangles  120  and  160  have X DIR =0 (positive). 
     A drawing algorithm similar to the DDA commonly is used by graphics systems to compute and apply visual effects to the pixels of the rendered polygons. Along with the parameters that describe the polygon coordinates, the display driver transmits to the graphics processor a set of parameters that describe the visual effects, or pixel “characteristics,” throughout the polygon. The display driver typically calculates these parameters based on the values of the pixel characteristics at the vertices of the polygon. For instance, to display a polygon with red color, the display driver sends to the processor a starting red color value and a pair of gradient values, one gradient value defining the rate of change of red intensity along the main slope of the polygon and the other gradient value defining the rate of change of red intensity between adjacent pixels on a given scan line. In addition to computing the pixel coordinates using the Incremental Line-Drawing algorithm or the like, the graphics processor uses the starting and gradient parameters to assign a red intensity value to each pixel. The graphics processor typically computes the other pixel characteristics, including blue and green intensity and the other visual effects described previously, in the same manner as and concurrently with the polygon coordinate calculations. In fact, even though the pixel depth value is essentially a spatial characteristic like the x- and y-coordinates, the depth characteristic values are usually calculated in the same manner as the other visual effects, using a starting depth value and two gradient values to incrementally assign depth values to each pixel as the polygon is rendered. 
     A few problems arise when rendering polygon with visual effects onto a pixel grid, however. First, a pixel grid is inherently discrete, i.e. it is not possible to render images between the pixels of a pixel grid. Hence, although interpolation and other techniques may result in fractional-valued polygon parameters, screen images must be mapped to integer-valued pixel locations. One result of such a mapping is that the outlines of some shapes, notably those with slanted and curved edges, may appear jagged on-screen. Higher screen resolutions mitigate this jagged effect, since pixels which are closer together result in a smaller difference (or “error”) between the fractional-valued coordinates of the image and the integer-valued pixel coordinates used to display the image. Another problem with pixel-mapping is that some smooth changes, or monotonic gradients, in visual effects such as gradients in color, lighting, texture, fog, and alpha blending may appear uneven, or banded as a result of the mapping error. For instance, a polygon intended to change smoothly from light red at the top of the polygon to dark red at the bottom of the polygon may actually appear to have horizontal bands of single shades of red. Banding artifacts occur frequently in polygon images with steeply sloping side edges and can distort and ruin the intended appearance of these images. 
     Visual depth effects may also suffer from mapping errors. Depth effects create the illusion of three dimensions, wherein graphics images displayed on a 2D screen may actually appear as 3D objects. A sense of depth perception can make graphic objects look more realistic. Mapping errors, however, can cause objects which are intended to intersect smoothly along a line in 3D to appear to have a jagged intersection. Texture-mapping as well as other 3D effects may also suffer from this problem. 
     Because these interpolation errors can severely degrade the quality of computer display images, a number of correction schemes have been proposed. As mentioned above, increasing the screen resolution helps to dilute the effects of jagged lines and curves in 2D shapes. Special drawing techniques have also been used to combat jagged lines, such as unweighted area sampling, scan conversion, and interpolated shading techniques such as Gouraud shading. These enhancements do not prevent pixel characteristics from suffering interpolation errors in some images, however. Coplanar polygons, for example, in which each edge lies in a single plane (in contrast with polygons whose edges are curved in 3D), can exhibit considerable banding and other nonlinear artifacts due to interpolation errors, even when rendered on high-resolution screens and when using special drawing techniques. In particular, these errors are particularly noticeable in polygons with steeply sloping side edges and a large orthogonal (horizontal) gradient in one or more pixel characteristics. 
     For example, a steep slope in a line implies that that line changes slowly in the x-direction per unit change in y-direction. Because polygons are typically drawn in consecutive scan lines, the rate of change of the line in the y-direction is always one pixel per scan line. Hence, the main gradient slope parameter computed by the display driver more specifically defines the rate of change of the main slope in the x-coordinate. The slope parameter for a steep main slope may therefore have a small fractional component. Since the graphics processor typically rounds the pixel coordinates down before rendering, many consecutive pixels along one edge of a polygon may be rounded to the same x-coordinate. Because that edge is sloped, however, the difference, or “error,” between the true x-coordinates and the rounded x-coordinates varies from scan line to scan line. The visual effects added to the pixels by the graphics processor thus become shifted in value by varying amounts, each value shifted by a degree proportional to the interpolation error, caused by rounding, of the corresponding pixel coordinate. This uneven shifting of visual characteristics on consecutive scan lines produces the unintended banding effects and jagged intersections mentioned above. Moreover, such problems occur in any visual effect applied to the pixels, including color, lighting, depth, texture-mapping, fog, alpha, depth, and other visual effects. 
     For example, FIG. 2 illustrates a shaded polygon  200  to be rendered onto a pixel grid. Because FIG. 2 illustrates polygon  200  as an ideal quadrangle superimposed onto coordinate system  203 , the graphics controller must translate the parameters of polygon  200  to fit an integer-valued pixel grid. Parameters X M  and Y M  define the starting x and y pixel grid coordinates from which polygon  200  will be rendered. By convention, X M  and Y M  identify the x- and y-coordinates of the main slope upper vertex, although other implementations may define the lower main slope vertex as the initial point. Since polygon  200  has main slope  201 , the coordinate pair (X M , Y M )=(2.75, 2) defines initial point  205 . Accordingly, X MINT =2 and X MFRAC =0.75. The polygon height parameter defines the vertical height of the polygon, determining the number of scan lines needed to render the polygon. For polygon  200 , the polygon height is 6 pixels, since the polygon spans rows (or “scan lines”) 2 through 7 of the pixel grid  203 . 
     W M  represents the number of pixels along the initial scan line  2  and corresponds to the initial distance between the main slope  201  and opposite slope  202  of polygon  200 . For polygon  200 , W M =2.0, since the width between initial point  205  and endpoint  207  along the initial scan line is 2 units. Referring still to FIG. 2, X DIR =0 for polygon  200 , since main slope  201  is situated to the left of opposite slope  202 . Finally, the parameter ΔX M  defines the gradient of the main slope in terms of the change in x-coordinate per scan line, while ΔW M  defines the change in the horizontal width of the triangle along the main slope. Thus, ΔX M  and ΔW M  for polygon  200  are −0.25 and +0.25, respectively. Accordingly, ΔX MINT =0, ΔX MFRAC =−0.25, ΔW MINT =0, and ΔW MFRAC =+0.25. 
     The pixel characteristics of a polygon may be sent to the graphics processor in a format similar to the polygon coordinate parameters as described above. For each type of visual effect, the graphics processor receives a starting characteristic parameter which defines the value of the pixel characteristic at the initial polygon pixel (i.e., at (X M , Y M )), a “delta main” parameter which defines the difference in the characteristic values of adjacent pixels along the main slope of the polygon, and a “delta ortho”parameter which defines difference in the characteristic values of adjacent pixels. These three parameters allow the graphics processor to render polygons with a smooth, or monotonic, change in characteristic values along each scan line. 
     Still referring to FIG. 2, polygon  200  may be rendered with a gradient in one or more characteristic values. The graphics processor receives a set of parameters for each of the different pixel characteristics, including parameters for red color, green color, blue color, specular red, specular blue, and specular green, depth, and the three texture-mapping coordinates u, v, and w. In the example of FIG. 2, the software driver transmits to the graphics processor parameters R M =60 (a starting red intensity parameter), ΔR M =5 (delta red main), and ΔR O =20 (delta red ortho), which define the desired shading effect along polygon  200 . The parameter R M  indicates the initial red color intensity at the starting coordinates (X M , Y M ). The parameter ΔR M  defines the change in red color intensity between each pixel along the main slope, and ΔR O  defines the change in red color intensity per pixel in the orthogonal (horizontal) direction, or across each scan line. Given these parameters, a graphics processor can compute red color intensity values for each pixel when polygon  200  is rendered onto a display. 
     If polygon  200  were rendered on an infinitely precise pixel grid, applying the three red color parameters would result in a smooth, monotonic color change across the surface of polygon  200 . The numbers in parentheses throughout FIG. 2 indicate the resulting red color intensities at various points on polygon  200 . For instance, the red color value is 60 at the starting point  205 . After applying the parameter R M =60 to the starting point  205 , the graphics controller can vary the color monotonically along the main slope  201  according to ΔR M . The points along the main slope  201  thus take red color values of 65 (point  210 ), 70 (point  215 ), 75 (point  220 ), 80 (point  225 ), and 85 (point  230 ), a monotonic increase of 5 red color intensity values per main slope pixel. In the orthogonal (horizontal) direction, each point in the interior of polygon  200  and along the opposite slope  202  take on values proportional in ΔR O  to their distances from the main slope points on the same scan lines. For instance, point  207  lies 2 integer units to the right of corresponding main slope point  205 , which has red color intensity 60. Because ΔR O =20 units per x-coordinate, point  207  has a red color intensity of 100, which is 2*ΔR O =40 units higher than that of point  205 . The color gradients along orthogonal scan lines  3  through  7  also exhibit a constant shift of 20 units of color intensity per integer change in the orthogonal direction, as indicated by the red color intensity values in parentheses corresponding to points  210  through  232 . Polygon  200  further exhibits a monotonic color gradient of 10 color units per integer change in the vertical direction, as indicated by the colors of point  207  (red=100), point  212  (red=110), point  217  (red=120), point  222  (red=130), point  227  (red=140), and point  232  (red=150) and by the colors of point  206  (red=85), point  211  (red=95), point  216  (red=105), point  221  (red=115), point  226  (red=125), and point  231  (red=135). The vertical gradient follows naturally from the main slope and ortho gradients. 
     FIG. 3 illustrates the result of a graphics processor using the Incremental Line-Drawing algorithm to interpolate polygon  200  onto pixel grid  303  using the polygon parameters X M , ΔX M , W M , ΔW M , Y M , polygon height, and X DIR . The graphics processor used to draw polygon  300 , however, does not include any type of error correction, and the red shading in polygon  300  appears banded rather than monotonic. Upon receiving the polygon parameters from the software driver, the graphics processor first computes the initial pixel values corresponding to point  205 . First, the graphics processor determines the x-coordinate of initial pixel  305  on scan line  2  by rounding X M  down to the nearest integer. Thus, pixel  305  is drawn at (x, y)=(2, 2). The red color intensity for initial pixel  305  is R M =60, by definition. Since X DIR =0, the graphics engine renders the remaining pixels in the positive direction (to the right) across the initial scan line. Because the initial scan line width W M =2.0, pixels  306  and  307  are rendered to complete the initial scan line. The graphics processor determines the red color value for each pixel by adding delta red ortho (ΔR O =20) to each of the preceding pixels. Thus, pixel  306  has a red color value of 80, and pixel  307  has a red color value of 100. 
     After completing the initial scan line, the graphics processor advances to scan line  3  and computes the main slope x-coordinate by adding ΔX M  to the previous main slope x-coordinate. Thus the new x-coordinate is 2.75−0.25=2.50, and, rounding down the x-coordinate, the graphics engine draws a new main slope pixel  210  at (x, y)=(2, 3). The red color value for pixel  310  may be determined by adding ΔR M =5 to the red value of the previous main slope pixel  305 . Thus, pixel  310  has red color value 65. The red intensity values along scan line  3  are determined by adding ΔR O =20 to the value of the preceding pixel. Hence, pixels  311  and  312  have red color intensities 85 and 105, respectively. The graphics processor continues to compute the pixel coordinates and red color values in this manner, rendering each consecutive row of pixels from scan line  4  through scan line  7 . Accordingly, main slope pixels are assigned red color values of 70 (pixel  315 ), 75 (pixel  320 ), 80 (pixel  325 ), and 85 (pixel  330 ). 
     Because no error correction was used for polygon  300 , however, the red color values appear banded. While the red color gradient in polygon  200  was smooth, the red color “jumps” between scan lines  5  and  6 . For example, the red color values for pixels  305  (red=60),  310  (red=65),  315  (red=70), and  320  (red=75) progress gradually in steps of 5. The difference between pixels  320  and  326 , however, is 25 units of red intensity. This same 25-unit shift in color gradient is also evident between pixels  321  and  327  and between pixels  322  and  328 . Thus, instead of having a smooth color gradient throughout, polygon  300  appears to have two distinct red bands. 
     The source of this banding effect lies in the difference, or error, between the fractional-valued x-coordinates of polygon  300  and the integer-valued pixels which the graphics processor actually renders. Notice that the difference between the x-coordinate of pixel  305  (x=2.75) and the actual, rendered location of pixel  305  (x=2) is ¾ pixel but that ideal starting color and actual starting color are both 60. Thus, the software driver calculated the initial red color value as if the first pixel  305  would be rendered at x=2.75. However, the graphics processor rounded each x-coordinate along scan line  2  by ¾ of a pixel. Comparing the ideal quadrangle of FIG. 2 to the rendered pixels of FIG. 3, it can be seen the rendered pixels along scan line  2  of FIG. 3 have the wrong color values for the x-coordinates at which they were rendered. For instance, point  206  of FIG. 2 was intended to have red color value 85. The pixel in FIG. 3 corresponding to the coordinates of point  206  in FIG. 2, however, has red color 100, a difference of 15 color values. Likewise, the graphics processor rounded the x-coordinate of each pixel on scan line  3  by ½ pixel, effectively shifting each color value on scan line  3  by 10 color values. Also, the graphics processor rounded the x-coordinate of each pixel on scan line  4  by ¼ pixel, effectively shifting each line  4  red color value by 5 color values. Because the fractional x-coordinate for scan line  5  (x=5.0) had no fractional portion, the x-coordinate for pixel  320  needed no rounding. Therefore, the red color values were not shifted on scan line  5 . Comparison of pixel  320  with the corresponding pixel  220  of polygon  200  verifies that both pixels have the same color value. It should be noted that although the example of FIG. 3 is directed toward the error induced when shading a polygon with the color red, a similar banding effect may occur with respect to any pixel characteristic that is applied to the polygon. Hence, the example FIG. 3 is representative of interpolation error that may occur in green color, blue color, specular red, specular green, specular blue, u-texel, v-texel, w-texel, alpha, fog, depth, and other pixel characteristics. 
     One solution to such a problem has been to implement an error-correction algorithm that selectively alters the visual characteristics along each scan line. U.S. Pat. No. 5,625,768 assigned to Cirrus Logic, Inc. discloses a display driver that both generates polygon rendering parameters and calculates error adjustment terms for each pixel characteristic. The error adjustment terms are transmitted to a graphics processor along with the normal pixel parameters and stored into a register file. To compute the pixel characteristic values, the graphics processor first uses a set of interpolation circuits to compute an uncorrected version of each pixel characteristic. However, these uncorrected pixel characteristic values are subject to interpolation errors, as discussed previously. With the error adjustment terms stored in the register file, the graphics processor uses a second set of interpolation circuits to compute the accumulated error for each pixel, adjusting each pixel characteristic to correct the interpolation error. Hence, the graphics processor essentially renders the pixels and then uses the error terms to correct the error that occurred while rendering the pixels. 
     The corrected values allow the display to avoid the visual defects noted earlier, such as banding and 3D intersection problems. However, such a configuration requires a very complex display driver, which places an extra burden on the host computer system. In addition, the software driver must transmit extra correction parameters to the graphics processor, requiring roughly twice the amount of communications bandwidth between the computer and the graphics processor as is needed to transmit polygon parameters only. Further, the graphics processor requires a large amount of register file space to store both the polygon parameters and the error correction parameters. 
     In addition, the graphics processor uses an error correction circuit coupled to each pixel characteristic interpolator, the error correction circuit adapted to correct each characteristic of a pixel after that characteristic has been calculated. Such a configuration not only results in an excessive amount of hardware but also increases the amount of calculation time, or latency, required to generate the pixel characteristic values, since the pixel characteristic value must essentially pass through two computations: one calculation to determine the uncorrected pixel value, and another calculation to correct the pixel value for interpolation error. 
     In light of the foregoing reasons, there remains a need for an effective yet efficient error correction system capable of adjusting pixel characteristic values in polygons rendered by a graphics processor. Such a system should be capable of rendering a polygon from a standard set of interpolation parameters without excessive hardware complexity or latency. Further, such an error correction system should integrate seamlessly with the CPU, using minimal computer memory while requiring as few CPU calculations as possible. To date, no such system is known that incorporates such features. 
     SUMMARY OF THE INVENTION 
     Accordingly, there is provided herein a graphics processor capable of receiving parameters defining a polygon from a display driver, adapting the parameters to anticipate and prevent interpolation error, and rendering a polygon using the corrected parameters. The present invention is capable of rendering polygons with any number of pixel characteristics, including color, specular lighting, depth, texture-mapping, fog, alpha blending, and any other suitable visual screen effects. Because the graphics processor implements all of the error correction calculations, an accompanying display driver need only transmit the basic polygon parameters. Thus, a simple, efficient software driver may be used, saving valuable computer memory, CPU time, and communications bandwidth between the CPU and graphics processor. 
     The graphics processor receives from the software display driver parameters with both integer and fractional portions, or “fractional-valued parameters,” describing a polygon from a software driver, including an initial x-coordinate, an initial y-coordinate, a polygon height, an initial line width, a rendering direction, an X main slope, and a delta X width, storing these parameters into a register file. The software driver also transmits to the graphics processor a set of parameters for each visual effect (or “pixel characteristic”), including a starting characteristic value, a “delta main” slope value, and a “delta ortho” slope value. The delta main and delta ortho parameters define the gradients of the characteristic values along the main slope and in the orthogonal (horizontal) direction, respectively. The characteristic parameters are also stored in the register file. Since the graphics processor is capable of rendering pixels with a plurality of different pixel characteristics, the graphics processor receives a starting parameter, a delta main parameter, and a delta ortho parameter for each pixel characteristic. 
     The graphics processor also includes an ortho correction engine capable of reading the characteristic parameters from the register file, updating the parameters to correct for interpolation error, and writing the parameters back into the register file. To correct the parameters, the ortho correction engine exploits an inherent linearity in the pixel interpolation error. As explained previously, the polygon parameters describe the starting points and main slopes of the polygon as well as the pixel characteristics throughout the polygon. Since the parameters that describe the coordinates of the polygon may be fractionally-valued, some of the polygon pixel coordinates and characteristic values must be rounded down to fit the pixel grid. 
     Because the ideal pixel characteristics along a given scan line are related to the pixel characteristics of the starting pixel, or main slope pixel, of that scan line by multiples of the orthogonal gradient parameters, shifting the position of the main slope pixel (by rounding its x-coordinate) shifts the locations of the remaining pixel characteristic values along the same scan line by an equivalent distance. To correct the shift in position, each characteristic value must be shifted in a direction opposite to that in which the pixel coordinate is rounded by an amount proportional to the degree of rounding in the x-coordinate. Because the fractional-valued x-coordinate for any pixel along the main slope of a polygon is always a fixed distance from the x-coordinate of the preceding scan line, the amount of rounding necessary for each main slope pixel changes linearly each successive scan line, in proportion to the fractional portion of the delta X main parameter. 
     The ortho correction engine exploits this linearity by correcting each characteristic parameter once in the register file. When the characteristic parameters have been adjusted, the graphics engine renders the polygon using the Incremental Line-Drawing algorithm. Because the starting and delta main parameters are corrected before rendering, the pixels along the main slope have no interpolation error. Thus the main slope pixels are error-free, and the remaining pixels along each scan line become shifted accordingly, requiring no error correction. First, the graphics engine calculates the x- and y-coordinates and characteristic values of the initial pixel, followed by the remaining pixels along the initial scan line. The graphics engine next calculates the x- and y-coordinates and characteristic values of the main slope pixel of the following scan line, rendering the rest of the scan line by incrementing the characteristic values of each subsequent pixel along that scan line by the corresponding delta ortho values. The graphics engine renders remaining scan lines likewise. 
     Due to the nature of polygon interpolation, a few selected main slope pixels will be rendered error-free even without prior error correction. In the present invention, these selected main slope pixels will cause the main slope interpolator to overflow (carry) or underflow (borrow) during interpolation. To remove the error correction introduced into these selected pixels, an ortho adjust engine shifts each characteristic value of these selected pixels by the proper delta ortho value to account for the overflow or underflow. 
     By correcting each characteristic parameter in the register file before rendering the polygon, the graphics engine is able to draw the polygon very efficiently, only requiring periodic ortho-adjustments to the characteristic values along the main slope. The error correction may thus be thought of as “error prevention,” since correcting the parameters before rendering the polygon eliminates the need for error correction after rendering. By avoiding error correction post rendering, the present invention achieves a design that is simpler and faster than previous methods. 
     Because the parameter errors are proportional to the degree of rounding of the x-coordinates, the ortho correction engine calculates a corrected starting error term for a starting characteristic parameter by multiplying the delta ortho parameter by the fractional portion of the starting x-coordinate. A bank of multiplexers couples to the register file so that the parameters may be selectively shifted into the correction engine. If the pixels are drawn across each scan line from left to right, the graphics engine computes the corrected starting characteristic parameter as the sum of the resulting error term and the uncorrected initial characteristic parameter. If the pixels are drawn across each scan line from right to left, the graphics engine computes the corrected initial characteristic parameter as the resulting error term subtracted from the uncorrected initial characteristic parameter. The ortho correction engine calculates a delta main error term for a delta main characteristic parameter by multiplying that delta main characteristic parameter by the fractional portion of the delta X main parameter. If the X direction is negative, the graphics engine computes the corrected delta main characteristic parameter as the sum of the resulting error term and the uncorrected delta main characteristic parameter. If the X direction is positive, the graphics engine computes the corrected delta main characteristic parameter as the resulting error term subtracted from the uncorrected delta main characteristic parameter. Accordingly, the ortho error correction engine consists primarily of a multiplier and an adder/subtractor with operands shifted in directly from the register file. The delta ortho characteristic parameters do not need error correction, since the Incremental Line-Drawing algorithm propagates the main-slope parameter corrections across each scan line. 
     The rendering circuitry in the graphics processor comprises an x-coordinate interpolator, an interpolator for each characteristic value, and draw engine for asserting control signals. In response to the sequence of control signals from the draw engine, the interpolators calculate the x-coordinate and characteristic values for each pixel, beginning at the initial main slope pixel and proceeding to each consecutive pixel across the initial scan line. The characteristic interpolators assign the error-corrected initial characteristic parameters to the initial pixel, adding the appropriate delta ortho parameter value to the characteristic value of each consecutive pixel across the scan line. After rendering the first scan line, the interpolators calculate the x-coordinate and characteristic values for the main slope pixel on the following scan line by adding the X main slope and error-corrected delta main characteristic parameters to the initial pixel values. If the fractional portion of the x-coordinate interpolator overflows or underflows as described below, an ortho adjust engine adjusts each main slope pixel characteristic value by the appropriate delta ortho parameter. The interpolators then proceed to render the rest of the scan line, incrementing successive pixel characteristic values by the corresponding delta ortho value. The interpolators draw the remainder of the polygon likewise, first rendering a main slope pixel followed by the rest of the scan line. 
     The ortho adjust engine generally comprises a logic circuit coupled to the x-coordinate interpolator, the x-interpolator consisting primarily of an accumulator. The accumulator computes the x-coordinate of each pixel along the main slope by adding the fractional-valued delta X main parameter to the x-coordinate of the previous main slope pixel. The ortho adjust engine detects when the fractional portion or the x-coordinate overflows or underflows, indicating whether or not to add or subtract a delta ortho from the characteristic values of the current main slope pixel. If an overflow or underflow condition occurs, the ortho adjust engine asserts a delta ortho add signal if the rendering direction points from the main slope of the polygon into the body of the polygon or asserts a delta ortho subtract signal if the rendering direction points from the main slope away from the body of the polygon. 
     Hence, the present invention discloses a efficient, yet powerful design to correct polygon interpolation errors and render the pixels to be displayed on a pixel grid. Further, the present invention implements the error correction algorithm and rendering entirely, requiring only the intepolated polygon parameters from a display driver. The present invention exploits an inherent linearity in the characteristic error values, using an efficient error-correction engine to correct the irked characteristic parameters before rendering, allowing the rendering apparatus to implement the basic Incremental Line-Drawing algorithm without concurrent error correction. Compared to conventional methods, such a scheme results in lower complexity and reduced latency for rendering corrected pixel characteristics. 
     Thus, the present invention comprises a combination of features and advantages that enable it to substantially advance the rendering art. These and other characteristics and advantages of the present invention will be readily apparent to those skilled in the art upon reading the following detailed description of the preferred embodiments of the invention and by referring to the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A better understanding of the present invention can be obtained when the following detailed description of the preferred embodiment is considered in conjunction with the following drawings, in which: 
     FIG. 1 is an exemplary diagram of the four types of random triangles; 
     FIG. 2 is an example of a random quadrangle with monotonic shading; 
     FIG. 3 is an example of a the random quadrangle of FIG. 2 rendered onto a pixel grid using an Incremental Line Drawing algorithm; 
     FIG. 4 is a block diagram of a preferred embodiment of a graphics processor; 
     FIG. 5 is a flowchart of the Incremental Line-Drawing algorithm with error correction in accordance with the preferred embodiment; 
     FIG. 6 is a preferred embodiment of the random quadrangle of FIG. 2 rendered onto a pixel grid using error-correction; 
     FIG. 7 is a block diagram of a preferred embodiment of the polygon engine of the graphics processor of FIG. 4; 
     FIG. 8A is a block diagram of a preferred embodiment of the ortho correction engine of the polygon engine of FIG. 7; 
     FIG. 8B is a block diagram of an interpolator from the polygon engine of FIG. 7; 
     FIG. 8C is a block diagram of the X interpolator and ortho adjust engine of the polygon engine of FIG. 7; 
     FIG. 9A is a flowchart of an algorithm for processing and error-correcting graphics pixels suitable for display onto a pixel grid; 
     FIG. 9B is a flowchart of an error-correction program used in the algorithm of FIG. 9A for a pixel characteristic other than the z-coordinate; 
     FIG. 9C is a flowchart of an error-correction program used in the algorithm of FIG. 9A for the z-coordinate pixel characteristic; 
     FIG. 9D is a flowchart of the steps for updating an accumulator used for a pixel characteristic other than the z-coordinate in the algorithm of FIG. 9A; 
     FIG. 9E is a flowchart of the steps for updating the accumulator used for the z-coordinate pixel characteristic in the algorithm of FIG. 9A; 
     FIG. 9F is a flowchart of the steps for ortho-adjusting an accumulator in the algorithm of FIG. 9D; and 
     FIG. 9G is a flowchart of the steps for ortho-adjusting an accumulator in the algorithm of FIG.  9 E. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Referring to FIG. 4, the preferred embodiment of a graphics processor for rendering graphics onto a display unit  450  includes a host interface (HIF)  405 , a polygon engine  410 , a register file  415 , a pixel pipe  420 , a memory controller  425 , a display controller  426 , and Rambus™ Access Circuitry (RAC)  428 . Graphics processor  450  preferably also includes circuits for implementing graphics operations such as texture-mapping, video synchronization, bus interfacing, and other typical graphics operations (not shown). The graphics processor  450  preferably communicates with a host computer (not shown) through a host bus  400 . The host interface  405  receives data such as polygon parameters over the host bus  400  with the graphics processor  450  and transmits control signals to the host computer as well. The polygon engine  410  and register file  415  couple to the host interface  405  via HIF bus signals, which couple to the register file  415  via 32-bit input signals. The polygon engine  410  feeds address and control signals to the register file  415  and receives data signals from the register file  415 . The polygon engine  410  sends pixel coordinates and characteristic values to a pixel pipe  420  via output signals. The pixel pipe  420  uses known techniques to combine the pixel coordinates and characteristic values into pixel information suitable for rendering onto a graphics display. The pixel pipe  420  then transmits the pixel information to the memory controller  425  via pixel_data signals. 
     The memory controller  425  and display controller  426  couple to the RAC  428  via a Rambus™ Interface (RIF) bus  427 . The RAC  428  is adapted to transact memory reads and writes to the RDRAM  430  over the Rambus™  431 . It should be understood, however, that any suitable type of memory technology and, such as dynamic random access memory (DRAM), synchronous DRAM, or extended data output DRAM, may be substituted for the RDAM  430 , along with a suitable bus protocol replacement for the Rambus™  431 . Upon receiving the pixel data from the pixel pipe  420 , the memory controller  425  writes the pixel data the memory device  430  by passing the pixel data to the RAC  428  via the RIF bus  427 . The display controller  426  generates the appropriate signals to drive the display  435  based on the pixel values defined by the pixel pipe. These pixel values are stored in the RDRAM  430 , as described above. The display controller, therefore, also accesses the memory device  430  over the RIF bus  427  via the RAC  428 . The memory controller  425  and display controller  426  must therefore share the RIF bus  427  and may use any suitable arbitration protocol. To ensure timely rendering, however, the display controller  426  preferably has priority of the memory controller  425  for use of the RIF bus  427 . 
     The register file  415  generally comprises a bank of registers adapted to store the polygon parameters which are received from the host computer. Table I illustrates a preferred embodiment of the structure of register file  415 . The first column of Table I identifies the registers which hold the polygon parameters. The second and third columns provide a description and recommended format, respectively, for the register values. In the Format column, an “I.F” value indicates that the register contents hold an I-bit integer and an F-bit fractional. Hence, an “I.F”—formatted register holds I+F bits. Also in the Format column, an ‘x’ indicates a reserved portion of the register, and an ‘s’ denotes a sign bit portion of the integer value. The Address Offset column, written in hexadecimal notation, describes the addresses of the registers within the register file. The register file is preferably loaded by the software driver in order, from Address Offset 0h to Address Offset FCh. 
     
       
         
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 Preferred structure of register file 415. 
               
             
          
           
               
                 Register 
                   
                   
                 Address 
               
               
                 Name 
                 Description 
                 Format 
                 Offset 
               
               
                   
               
             
          
           
               
                 X M   
                 initial x-coordinate 
                 11.16 
                 0h 
               
               
                 Y M   
                 initial y-coordinate 
                 11.16 
                 4h 
               
               
                 R M   
                 initial red color 
                 8.16 
                 8h 
               
               
                 G M   
                 initial green color 
                 8.16 
                 Ch 
               
               
                 B M   
                 initial blue color 
                 8.16 
                 10h 
               
               
                 ΔX M   
                 delta X main 
                 12.16 
                 14h 
               
               
                 main count 
                 polygon 1 height 
                 ×.11 
                 18h 
               
               
                 opp count 
                 polygon 2 height 
                 ×.11 
                 1Ah 
               
               
                 W M   
                 initial width for polygon 1 
                 11.16 
                 1Ch 
               
               
                 W O   
                 initial width for polygon 2 
                 11.16 
                 20h 
               
               
                 ΔW M   
                 polygon 1 delta width 
                 s.12.16 
                 24h 
               
               
                 ΔW O   
                 polygon 2 delta width 
                 s.12.16 
                 28h 
               
               
                 ΔR M   
                 red delta main 
                 s.9.16 
                 2Ch 
               
               
                 ΔG M   
                 green delta main 
                 s.9.16 
                 30h 
               
               
                 ΔB M   
                 blue delta main 
                 s.9.16 
                 34h 
               
               
                 ΔR O   
                 delta red ortho 
                 s.9.16 
                 38h 
               
               
                 ΔG O   
                 delta green ortho 
                 s.9.16 
                 3Ch 
               
               
                 ΔB O   
                 delta blue ortho 
                 s.9.16 
                 40h 
               
               
                 Z M   
                 initial z-coordinate 
                 s.16.16 
                 44h 
               
               
                 ΔZ M   
                 delta Z main 
                 s.16.16 
                 48h 
               
               
                 ΔZ O   
                 z delta ortho 
                 s.16.16 
                 4Ch 
               
               
                 V M   
                 initial V value 
                 s.10.16 
                 50h 
               
               
                 U M   
                 initial U value 
                 s.10.16 
                 54h 
               
               
                 ΔV M   
                 V delta main 
                 s.10.16 
                 58h 
               
               
                 ΔU M   
                 U delta main 
                 s.10.16 
                 5Ch 
               
               
                 ΔV O   
                 V delta ortho 
                 s.10.16 
                 60h 
               
               
                 ΔU O   
                 U delta ortho 
                 s.10.16 
                 64h 
               
               
                 W M   
                 initial W value 
                 s.10.16 
                 68h 
               
               
                 ΔW M   
                 W delta main 
                 s.10.16 
                 6Ch 
               
               
                 ΔW O   
                 W delta ortho 
                 s.10.16 
                 70h 
               
               
                 V off   
                 V offset of nearest vertex 
                 s.10.16 
                 74h 
               
               
                 U off   
                 U offset of nearest vertex 
                 s.10.16 
                 78h 
               
               
                 F M   
                 initial fog value 
                 8.8 
                 7Ch 
               
               
                 ΔF M   
                 fog delta main 
                 8.8 
                 B4h 
               
               
                 ΔF O   
                 fog delta ortho 
                 8.8 
                 B8h 
               
               
                 A M   
                 initial alpha value 
                 8.8 
                 C0h 
               
               
                 ΔA M   
                 alpha delta main 
                 s.9.8 
                 C4h 
               
               
                 ΔA O   
                 alpha delta ortho 
                 s.9.8 
                 C8h 
               
               
                 sR M   
                 initial specular red color 
                 8.16 
                 D0h 
               
               
                 sG M   
                 initial specular green color 
                 8.16 
                 D4h 
               
               
                 sB M   
                 initial specular blue color 
                 8.16 
                 D8h 
               
               
                 ΔsR M   
                 specular red delta main 
                 s.9.16 
                 DCh 
               
               
                 ΔsG M   
                 specular green delta main 
                 s.9.16 
                 E0h 
               
               
                 ΔsB M   
                 specular blue delta main 
                 s.9.16 
                 E4h 
               
               
                 ΔsR O   
                 specular red delta ortho 
                 s.9.16 
                 E8h 
               
               
                 ΔsG O   
                 specular green delta ortho 
                 s.9.16 
                 Ech 
               
               
                 ΔsB O   
                 specular blue delta ortho 
                 s.9.16 
                 F0h 
               
               
                 Opcode 
                 opcode for execution/polygon engine 
                 32.0 
                 FCh 
               
               
                   
                 (includes X DIR ) 
               
               
                   
               
             
          
         
       
     
     Graphics processor  450  preferably supports the rendering and error correction of color, specular lighting, depth, and texture-mapping pixel characteristics using an Incremental Line Drawing (ILD) algorithm. As discussed previously with respect to FIGS. 2 and 3, a polygon drawn using the ILD algorithm may exhibit visual aberrations such as banding or other types of distortion if no error correction is used. The source of this banding effect lies in the difference, or interpolation error, between the fractional-valued x-coordinates of polygon  300  and the integer-valued pixels which the graphics processor actually renders. In general, the interpolation error in the delta ortho gradient across any given scan line is directly proportional to the degree of rounding in the x-coordinates of the pixels on that scan line. For instance, the x-coordinates of the pixels on scan line  2  of FIG. 3 are rounded by ¾ of a pixel. Accordingly, the interpolation error is ¾*ΔR O =15 red color values along scan line  2 . Similarly, the x-coordinates of the pixels along scan line  3  are rounded by ½ of a pixel, resulting in an interpolation error of ½*ΔR O =10 red color values. Also, the x-coordinates of the pixels along scan line  4  are rounded by ¼ of a pixel, resulting in an interpolation error of ¼*ΔR O =5 red color values. Because the pixels of scan line  5  were not rounded, the interpolation error along scan line  5  is 0*ΔR O =0 red color values (i.e., line  5  has no interpolation error). The interpolation error along scan lines  6  and  7  is again proportional to the x-coordinate error, which is ¾ pixel for line  6  and ½ pixel for line  7 . 
     Thus, the red color shift of the pixels in polygon  300  generally changes linearly between each scan line and in proportion to the degree of rounding in the x-coordinates. The present invention exploits this linearity property to perform error correction on the characteristic parameters and then renders the pixels using the corrected parameter values. Because all pixels in the polygon are rendered with error-corrected characteristic parameters, the scan lines containing no interpolation error are additionally “ortho-adjusted” by the associated delta ortho value to counteract the error correction, which is not needed for the pixels on these scan lines. 
     As discussed previously, interpolation error may occur with respect to any pixel characteristic applied to the rendered polygon. Because the error correction algorithm presented herein is applied in substantially the same manner to all pixel characteristics, a “generic” pixel characteristic “C” may be used in some examples and equations to illustrate the general method. It therefore should be understood that the “C” characteristic may be interpreted to mean any pixel characteristic, including red color, green color, blue color, specular red, specular green, specular blue, u-texel, v-texel, w-texel, alpha, fog, or any other pixel characteristic. 
     In the preferred embodiment, the graphics processor calculates corrected initial and delta main parameters for a generic pixel characteristic “C” according to the following equations: 
     
       
         C MEC =C M ±ΔC O X MFRAC   (1) 
       
     
     
       
         ΔC MEC =ΔC M ±ΔC O *ΔX MFRAC   (2) 
       
     
     where C M  is the uncorrected “C” starting value, ΔC O  is the “C” delta ortho value, ΔC M  is the uncorrected “C” delta main value, and C MEC  and ΔC MEC  are the error-corrected “C” starting and “C” delta main parameters, respectively. The graphics processor preferably uses the addition operator in equations (1) and (2) if X DIR =1 and uses the subtraction operator if X DIR =0. 
     The graphics processor preferably ortho-adjusts selected main slope pixels by ΔC O , according to the value of X DIR  and according to the change in the fractional x-coordinate values between successive scan lines. In general, and ortho-adjustment is required for main slope pixels not needing prior error correction (such as pixel  320  FIG.  3 ). A preferred embodiment calculates a final main slope characteristic value by adding ΔC O  or −ΔC O  to the preliminary main slope characteristic value whenever calculation of the x-coordinate of the current main slope pixel results in either a fractional carry or a fractional borrow and if the following logical expression is true: 
     
       
         (ΔX M &lt;0)XNOR(X DIR =1)  (3) 
       
     
     If calculation of the x-coordinate of the current main slope pixel results in either a factional carry or a fractional borrow and if expression (3) is false, however, then a preferred embodiment calculates a final main slope characteristic value by subtracting ΔC O  from the preliminary main slope characteristic value. 
     Thus, if calculating the current main slope x-coordinate results in a borrow or carry in the fractional x-coordinate and if expression (3) is true, then the graphics processor preferably calculates the current main slope characteristic parameter C(i) from the previous main slope characteristic parameter C(i−1) using equation (4): 
     
       
         C(i)=C(i−1)+C M +ΔC O   (4) 
       
     
     If calculating the current main slope x-coordinate results in a borrow or carry in the fractional x-coordinate and if expression (3) is false, however, then the graphics processor preferably calculates the current main slope characteristic parameter C(i) from the previous main slope characteristic parameter C(i−1) using equation (5): 
     
       
         C(i)=C(i−1)+C M −ΔC O   (5) 
       
     
     If ΔX M  is integer-valued (i.e., if ΔX MFRAC =0) or if calculating the current main slope x-coordinate does not result in a borrow or carry, then the graphics processor preferably calculates the current main slope characteristic parameter C(i) from the previous main slope characteristic parameter C(i−1) using equation (6): 
      C(i)=(i−1)+C M   (6) 
     For instance, if ΔX M  is an integer, no ortho adjustments are necessary. If ΔX M  is not an integer, the graphics processor may first calculate a preliminary characteristic value for a main slope pixel by adding ΔC M  to the corresponding characteristic value of the previous main slope pixel. If X DIR =0 and ΔX M &lt;0 (as in triangle  120 ), the graphics processor preferably determines a final characteristic value by subtracting ΔC O  from the preliminary characteristic value of the current main slope pixel if calculating the current x-coordinate requires a borrow in the fractional portion of the previous x-coordinate. For example, the x-coordinate of main slope pixel  320  is 2.00. To calculate the x-coordinate for the next main slope pixel  325  requires the graphics processor to perform a subtraction of |ΔX M |, or 0.25, from 2.00, necessitating a borrow by the fractional portion of 2.00 from the integer portion of 2.00. Thus, a preferred embodiment of a graphics processor rendering polygon  300  preferably subtracts ΔR O  from the preliminary red color value of pixel  325 , resulting in a final red color  65  for pixel  325 . This and other error-corrections calculations will become clear with respect to FIG. 6, below. 
     If X DIR =0 and ΔX M &gt;0 (as in triangle  160 ), the graphics processor preferably determines a final characteristic value by adding ΔR O  to the preliminary red color of the current main slope pixel if calculating the current x-coordinate results in a carry of the fractional portion of the x-coordinates during addition. For instance, if a starting main slope pixel has x-coordinate =1.75, X DIR =0, and ΔX M =+0.25, then the following main slope pixel will have x-coordinate 1.75+0.25=2.00. Because adding 1.75 to 0.25 requires carrying a value across the radix point (which is a decimal point in the present example but is a binary point in a preferred embodiment), the graphics processor preferably adds ΔR O  to the preliminary characteristic value of the main slope pixel on the following scan line. 
     If X DIR =1 and ΔX M &lt;0 (as in triangle  140 ), the graphics processor preferably determines a final characteristic value by adding ΔR O  to the preliminary characteristic value of the current main slope pixel if calculating the current x-coordinate requires a borrow in the fractional portion of the previous x-coordinate. For instance, if a main slope pixel has x-coordinate =2.00, X DIR =1, and ΔX M =−0.25, then the following main slope pixel will have x-coordinate 2.00−0.25=1.75. Because subtracting 1.75 from 2.00 requires a borrow by the fractional portion of 2.00 from the integer portion of 2.00, the graphics processor preferably adds ΔR O  to the preliminary characteristic value of the main slope pixel on the following scan line. 
     If X DIR =1 and ΔX M &gt;0 (as in triangle  100 ), the graphics processor preferably determines a final characteristic value by subtracting ΔR O  from the preliminary characteristic value of the current main slope pixel if calculating the current x-coordinate results in a carry of the fractional portion of the x-coordinates during addition. For instance, if the main slope pixel of a first scan line has x-coordinate 1.75, X DIR =0, and ΔX M =+0.25, then the main slope pixel on the second scan line will have x-coordinate 1.75+0.25=2.00. Because adding 1.75 to 0.25 requires carrying a value across the radix point, the graphics processor preferably subtracts ΔR O  from the preliminary characteristic value of the main slope pixel on the second scan line. 
     FIG. 5 illustrates a flowchart of the steps used to render error-corrected pixels onto a pixel grid using the equations above. Beginning with step  500 , the graphics processor  450  receives the starting characteristic parameters from the software driver. As described above, these parameters preferably include the initial x-coordinate X M , initial y-coordinate Y M , delta x main parameter ΔX M , x width parameter W M , delta x width parameter ΔW M , polygon height, rendering direction X DIR , and the starting, delta main, and delta ortho characteristic parameters for each type of pixel characteristic. Preferably, these parameters are stored in the register file  415 . 
     Moving next to step  505 , the graphics processor  450  corrects the starting parameter of each pixel characteristic (C MEC ) using equation (1). The processor  450  then replaces each uncorrected pixel characteristic (C M ) with the error-corrected pixel characteristic (C MEC ) in the register file  415 . Next in step  510 , the graphics processor  450  corrects the delta main value of each pixel characteristic (ΔC M ) using equation (2) and replaces the uncorrected characteristic value (ΔC M ) with the corrected characteristic value (ΔC MEC ) in the register file  415 . 
     The processor  450  next proceeds to step  515 , which represents the first step for rendering each scan line of the polygon. If the graphics processor  450  is rendering the first scan line during step  515 , then the characteristic value for the main slope pixel is C MEC , as computed in step  505 . If the graphics processor is rendering a subsequent scan line, then the main slope characteristic value will have been computed as in step  550  or  555 , as described below. 
     Next moving to step  520 , the graphics processor determines whether or not the main slope pixel is at the end of a scan line, which will occur if the main slope pixel is the only pixel on the scan line. If the main slope pixel is not at the end of the scan line, then the processor moves to step  525  to render the next pixel on the same scan line. In step  525 , the characteristic value of the next pixel of the same scan line is calculated by adding ΔC O  to the current characteristic value. Next in step  530 , the pixel is rendered onto the display  435 , and the graphics processor  450  again moves to step  520  to determine whether the current pixel is at the end of the scan line. If the current pixel is the last pixel on the scan line in step  520 , then the processor  450  moves to step  535 . 
     In step  535 , the processor determines whether or not the final scan line has just been rendered in steps  515  through  530 . If the final scan line has been rendered, then the processor moves to the “end” block. If the final scan line has not yet been rendered, then the processor  450  moves to the next scan line in step  540 . 
     Next in steps  545 ,  550 , and  555 , the graphics processor  450  calculates the characteristic value of the main slope pixel. Beginning with step  545 , the processor  450  determines whether or not the main slope pixel need to be ortho-adjusted. As described above, the main slope pixel will need to be ortho-adjusted if calculating the current main slope characteristic value will require a borrow or carry in the fractional portion of the characteristic value. If the main slope pixel does not need to be ortho-adjusted, then the processor in step  550  calculates the characteristic value of the main slope pixel using equation (6). If the main slope pixel does need to be ortho-adjusted, however, then the processor moves to step  555  and calculates the main slope characteristic value using equations (4) and (5). As described above, the processor  450  uses equation (4) in step  555  if equation (3) is true. If equation (3) is false, then the processor  450  uses equation (5) in step  555 . From steps  550  and  555 , the graphics processor returns to step  515  to begin rendering a new scan line. 
     From examining FIG. 5 in detail, it should be noted that blocks  500 ,  505 , and  510  essentially comprise the error correction steps of the present invention, since the remaining steps are directed mainly toward rendering the polygon. The error correction steps of blocks  500 ,  505 , and  510  therefore may be thought of as “error prevention” steps, since correcting the parameters before rendering the polygon eliminates the need for error correction after rendering. By avoiding error correction post rendering, the present invention may be implemented using simpler and faster components and algorithms. 
     FIG. 6 illustrates an error-corrected version of polygon  200  rendered onto a pixel grid by graphics processor  450 . The graphics processor  450  receives parameters X M =2.75, Y M =2, X DIR =0, W M =+2.0, ΔW M =+0.25, ΔX M =−0.25, R M =60, ΔR O =20, ΔR M =5, and polygon height=6 pixels, as in the previous example. Before rendering polygon  400 , however, the preferred graphics processor corrects the red characteristic parameters R M  and ΔR O  according to equations (7) and (8), which follow from equations (1) and (2), respectively: 
     
       
         R MEC =R M −ΔR O *X MFRAC =45  (7) 
       
     
     
       
         ΔR MEC =ΔR M −ΔR M *ΔX MFRAC =10  (8) 
       
     
     As in the example of FIG. 3, the graphics processor determines the x-coordinate of pixel  605  by rounding X M  down to the nearest integer. Thus, (x, y)=(2, 2) for pixel  605 . From equation (7) the red color intensity for initial pixel  605  is R MEC =45. Because X DIR 0, the graphics engine renders the remaining pixels of the initial scan line  2  in the positive direction. Since the initial scan line width is W M =2.0, the graphics engine renders pixels  606  and  607  to complete scan line  2 . The graphics processor determines the red color values for pixels  606  and  607  by adding ΔR O  to each of the preceding pixels. Thus, pixel  606  has a red color value of 65, and pixel  607  has a red color value of 85. 
     After the initial scan line is completed, the graphics processor computes the next main slope pixel x-coordinate as 2.75−0.25=2.50, drawing pixel  610  at (x, y)=(2, 3) as in the previous example. The red color value for pixel  610  may be determined by adding R MEC =10 to the red value for the previous main slope pixel  605 . Thus, pixel  610  has color  55 . The remaining pixels values and red intensities along scan line  3  are determined likewise, ΔR O =20 governing the color gradient between pixels. Hence, pixels  611  and  612  have red color intensities 75 and 95, respectively. The graphics processor continues to compute the pixel coordinates and red color values in this manner, rendering each of the pixels along scan lines  4  and  5 . Accordingly, main slope pixels are assigned red color values of 65 (pixel  615 ) and 75 (pixel  620 ). Remaining pixels along scan line  4  have red colors 85 (pixel  616 ) and 105 (pixel  617 ), while remaining pixels along scan line  5  have red colors 95 (pixel  621 ) and 115 (pixel  622 ). 
     Since the x-coordinate of the main slope pixel  620  of line  5  is 2.00, the x-coordinate of main slope pixel  625  of line  6  is 2.00+ΔX M =1.75. The graphics processor again rounds the x-coordinate down to (x, y)=(1, 5) to draw pixel  625 . As with previous main slope pixels, the graphics processor determines a preliminary red color value by adding ΔR MEC  to the red value of the previous main slope pixel  620 . Thus, pixel  625  has color 75+ΔR MEC =85. Because subtracting 0.25 from 2.00 to compute the x-coordinate requires a borrow across the radix point of 2.00, however, the graphics processor must perform an ortho-adjust to pixel  625  by subtracting ΔR O  from the preliminary red color value of pixel  625 . Thus, pixel  625  has a final red color intensity of 75+ΔR MEC −ΔR O =65. The graphics processor draws remaining pixels  626 ,  627 , and  628  along scan line  6  as in previous examples, calculating each red color value by adding ΔR O  to the red color value of the preceding pixel. Thus the color values along scan line  6  become 85 (pixel  626 ), 105 (pixel  627 ), and 125 (pixel  628 ). 
     It is immediately apparent that polygon  400  has a smooth red color gradient throughout scan lines  2  through  6 , in contrast with polygon  300 . As in the example of FIG. 3, the color difference between each consecutive pixel on a given scan line equals ΔR O , or  20 . In contrast to the example of FIG. 3, the red color gradient along any pixel column equals a constant value of 10 in polygon  400 . For instance, the pixels along column  4  have red color intensities 85 (pixel  607 ), 95 (pixel  612 ), 105 (pixel  617 ), 115 (pixel  622 ), and 125 (pixel  628 ). Pixels in columns  2 ,  3 , and  4  also exhibit a smooth, even red color gradient. 
     Continuing to render the final scan line  7 , the graphics processor draws main slope pixel  630  by calculating from the previous main slope x-coordinate a new x-coordinate of 2.5. Rounding down, the graphics processor draws new main slope pixel  630  at (x, y)=(2, 7). The graphics processor determines the red color value for pixel  630  by adding ΔR M  to the red color value of pixel  625 . Thus, pixel  630  receives red color 65+ΔR M =75. The graphics processor draws remaining pixels along scan line  7 , determining each new color value by adding ΔR O  to each previous pixel color. Thus, pixel  631  has red color 95, pixel  632  has red color 115, and pixel  633  has red color 135. Again, the pixels along scan line  7  differ in red color intensity from the pixels of scan line  6  by a constant value of 10. Hence, polygon  400  appears to have a smooth red color gradient throughout. 
     While the drawing technique above used red color as an example, the present invention applies substantially the same technique to calculate any pixel characteristic, including green or blue color, red, green, or blue specular lighting, depth, texture, fog, alpha blending, or any other pixel characteristic. 
     Along with register file  415 , the present invention can implemented entirely within the polygon engine  410 , shown with greater detail in FIG.  7 . The polygon engine  410  preferably comprises an execution engine  700 , an ortho correction engine  705 , a draw engine  710 , an ortho adjust engine  770 , and a plurality of interpolators  715 . The interpolators include a red interpolator  715   a , a green interpolator  715   b , a blue interpolator  715   c , a specular red interpolator  715   d , a specular green interpolator  715   e , and a specular blue interpolator  715   f . Also included are interpolators for U, V, and W texture-mapping components, including a U interpolator  715   g , a V interpolator  715   h , and a W interpolator  715   j . Finally, a preferred embodiment of graphics processor  410  includes a depth interpolator  715   k  and an X interpolator  720 . Alternatively, the graphics processor  410  may include interpolators  715  (not shown) with substantially identical connections for other visual pixel characteristics such as fog and alpha blending. Because interpolators  715   a ,  715   b ,  715   c ,  715   d ,  715   e ,  715   f ,  715   g ,  715   h ,  715   j , and  715   k  represent substantially similar components, they will be referred to collectively as interpolators  715 . 
     The host interface  405  couples to the execution engine  700  and register file  415  via HEIF bus signals which transmit the polygon parameters and standard control signals from the host computer (not shown) to the register file  415 . The execution engine  700  further couples to the register file  415  through address and control signals, which are used by the graphics processor  450  to read from and write to the register file  415 . The execution engine  700  also transmits a request signal to the ortho correction engine  705  which indicates that the register file  415  holds uncorrected polygon parameters. The ortho correction engine  705  sends ortho_control signals to the register file  415  to request uncorrected polygon parameters such as main slope and delta main parameters. The register file  415  transmits these uncorrected parameters to the ortho correction engine  705  via register_data signals. In a similar manner, the ortho correction engine  705  transmits corrected polygon parameters to be stored into the register file  415  via corrected_data signals. 
     The ortho correction engine  705  additionally couples to the draw engine  710  via request signal to indicate that the current polygon parameters have been corrected in the register file  415 . The draw engine transmits an acknowledge signal to the execution engine  700  and the ortho correction engine  705  to indicate that pixel rendering will commence. The draw engine further couples to the register file  415  via register_data signals to receive polygon parameters X M , ΔX M , W M , ΔW M 1, ΔW M 2, main_count, and opp_count. The draw engine transmits load_main, inc_main, and inc_ortho signals to the interpolators  715  and  720  to synchronize the interpolator calculations which will be described below in more detail with respect to FIGS. 8A,  8 B, and  8 C. The interpolators further receive error-corrected starting main slope values, delta main values, and delta ortho values from the register file  415 . Similarly, the X interpolator receives a starting main slope x-coordinate X M  and a delta X main value ΔX M  from the register file  415 . 
     Controlled by the draw engine  710  via load_main, inc_main, and inc_ortho signals, the X interpolator sends an x-coordinate to the pixel pipe  420  via an xpos signal and transmits an xstep signal to the ortho adjust engine  770 . The ortho adjust engine  770  further receives ΔX MFRAC  and x-direction X DIR  (not shown), which, in conjunction with xstep, determines whether the interpolators should ortho-adjust the current main slope characteristic values. Accordingly, the ortho adjust unit  770  transmits either an add or a sub signal to the interpolators to indicate ortho addition or subtraction, respectively. 
     Still referring to FIG. 7, the interpolators  715  transmit pixel characteristics and coordinate values to combined in the pixel pipe  420  for pixel-rendering via Rout (red pixel value), Gout (green pixel value), Bout (blue pixel value), sRout (specular red pixel value), sGout (specular green pixel value), sBout (specular blue pixel value), Uout (U texture value), Vout (V texture value), Wout (W texture value), Zout (Z pixel coordinate), and xpos (X pixel coordinate) signals. The pixel pipe  420  combines these pixel values and couples to the memory controller  425  by standard methods. 
     Now referring to FIG. 8A, a preferred embodiment of an ortho correction engine  705  comprises a 20×1 multiplexer  800 , a 10×1 multiplexer  805 , a 2×1 multiplexer  810   a , a multiplier  815 , pipeline register s  820   a  and  820   b , and an adder/subtractor  825   a . The multiplexers  800 ,  805 , and  810   a  couple to the outputs of register file  415  to receive uncorrected polygon parameters. Multiplexers  800  and  805  may contain additional, unused input terminals. In an alternative embodiment, however, these unused inputs may be connected register file  415  to support other types of pixel characteristics in substantially the same way as are the used inputs of the present example. The 32×1 multiplexer  800  receives starting parameters via register_data signals consisting of R M  (red main), G M  (green main), B M  (blue main), sR M  (specular red main), sG M  (specular green main), sB M  (specular blue main), U M  (U texture main), V M  (V texture main), W M  (W texture main), and Z M  (Z-depth main) and also receives corresponding delta main parameters ΔR M  (delta red main), ΔG M  (delta green main), ΔB M  (delta blue main), ΔsR M  (delta specular red main), ΔsG M  (delta specular green main), ΔsB M  (delta specular blue main), ΔU M  (delta U main), ΔV M  (delta V main), ΔW M  (delta W main), and ΔZ M  (delta Z main). The 10×1 multiplexer  805  receives delta ortho parameters ΔR O  (delta red ortho), ΔG O  (delta green ortho), ΔB O  (delta blue ortho), ΔsR O  (delta specular red ortho), ΔsG O  (delta specular green ortho), ΔsB O  (delta specular blue ortho), ΔU O  (delta U ortho), ΔV O  (delta V ortho), ΔW O  (delta W ortho), and ΔZ O  (delta Z ortho). Finally, the 2×1 multiplexer  810   a  receives X MFRAC  and ΔX MFRAC  from register file  415 . 
     The state of multiplexers  800 ,  805 , and  810   a  are controlled by all or some of the select signals SEL[ 4 : 01 ]. All of the select signals, SEL[ 4 : 01 ], control the state of 20×1 multiplexer  800 , selecting one input signal to multiplexer  800  to appear at the output of multiplexer  800 . Similarly, select signals SEL[ 4 : 1 ] choose one input from 10×1 multiplexer  805  to appear on the output of multiplexer  805 . Finally, SEL[ 0 ] determines which input of 2×1 multiplexer  810   a  is switched to the output of multiplexer  810   a.    
     Multiplier  815  and adder/subtractor  825   a  implement the calculations of equations (1) and (2), above. Multiplexers  805  and  810   a  feed delta ortho operands and x-coordinate operands, respectively, to multiplier  815 . The product of multiplier  815  is then fed into pipeline register  820   b . Concurrently, 20×1 multiplexer  800  feeds either a starting parameter or a main parameter (as determined by SEL[ 4 : 0 ]) into pipeline register  820   a . The outputs of pipeline registers  820   a  and  820   b  provide operands to adder/subtractor  825   a , which also receives the binary signal X DIR  indicating the rendering direction, as explained previously. If X DIR =0, the adder/subtractor  825   a  adds the operands provided by pipeline register s  820   a  and  820   b , while, if X DIR =1, the adder/subtractor  825   a  subtract the operands provided by pipeline register s  820   a  and  820   b . The adder/subtractor  825   a  sends the corrected parameter values via a corrected_data signal to register file  415 . 
     It should be noted that the ortho correction engine  705  essentially calculates error-corrected parameters by adding an “error term” to each uncorrected parameter, where the absolute value of the error term is given by the product of multiplier  815  and the sign of the error term is given by rendering direction X DIR . Thus, the error term can be defined as sgn{X DIR }*ΔC O *X MFRAC , where sgn{X DIR }=−1 if X DIR =0 (positive), and sgn{X DIR }=+1 if X DIR =1 (negative). For instance, to calculate an error-corrected starting parameter C MEC , the ortho correction engine  705  adds either +ΔC O *X MFRAC  or −ΔC O  *X MFRAC  to C M , as given by equation (1). Accordingly, the ±ΔC O *X MFRAC  term constitutes the error term. Similarly, the ortho correction engine  705  adds the error term ±ΔC O  *ΔX MFRAC  to ΔC M  to calculate an error-corrected delta main parameter ΔC MEC . Note that the error terms as defined above always are directly proportional to either the fractional starting x-coordinate X MFRAC  or the fractional delta X-main parameter ΔX MFRAC  and may be either positive or negative, as determined by the rendering direction X DIR . 
     While the calculations of the ortho correction engine  705  would produce substantially the same calculations without pipeline registers  820   a  and  820   b  or with alternative pipeline register configurations, pipeline registers  820   a  and  820   b  are provided in the present embodiment to enable pipelining, a standard method of reducing calculation time in computer devices. For more information on pipelining, refer to  Computer Organization and Design  by Patterson and Hennessy (Morgan Kaufmann Publishers, Inc., 1994) and  VLSI Digital Signal Processors  by Madisetti (Butterworth-Heinmann, 1995). 
     To calculate the expression of equation (1), or C MEC =C M ±ΔC O *X MFRAC , SEL[ 0 ] is set to 0 so that 2×1 multiplexer  810   a  transmits X MFRAC  to multiplier  815 . Concurrently, SEL[ 4 : 1 ] selects a starting main parameter C M , via 20×1 multiplexer  800 , and a delta ortho parameter ΔC O , via 10×1 multiplexer  805 . For example, to calculate a corrected value for the specular red starting parameter, or sR MEC =sR M ±ΔsR O * X MFRAC , SEL[ 4 ,  3 ,  2 ,  1 , 0 ]=[0, 0, 1, 1, 0] routes sR M  to pipeline register  820   a  routes ΔsR O  and X MFRAC  to the multiplier  815 . After processing its operands, the multiplier  815  feeds the product ΔsR O *X MFRAC  into pipeline register  820   b . During the subsequent cycle, pipeline register s  820   a  and  820   b  feed the operands sR M  and ΔsR O *X MFRAC , respectively, to adder/subtractor  825   a , which adds the operands if X DIR =1 or subtracts the operands if X DIR =0. The adder/subtractor  825   a  then feeds the resulting corrected specular red starting value sR MEC  into the register file  415 , overwriting the uncorrected specular red value sR M  with the error-corrected specular red value sR MEC . 
     To calculate the expression of equation (2), or ΔC MEC =ΔC M ±ΔC O *ΔX MFRAC , SEL[ 0 ] is set to 1 so that 2×1 multiplexer  810   a  transmits ΔX MFRAC  to multiplier  815 . Concurrently, SEL[ 4 : 1 ] selects a delta main parameter ΔC M , via 20×1 multiplexer  800 , and a delta ortho parameter ΔC O , via 10×1 multiplexer  805 . For example, to calculate a corrected value for the delta Z main parameter, or ΔZ MEC =ΔZ M ±ΔZ O *ΔX MFRAC , SEL[ 4 ,  3 ,  2 ,  1 ,  0 ]=[1, 0, 0, 1, 1] routes ΔZ M  to pipeline register  820   a  routes ΔZ O  and ΔX MFRAC  to the multiplier  815 . After processing its operands, the multiplier  815  feeds the product ΔZ O *ΔX MFRAC  into pipeline register  820   b . During the subsequent cycle, pipeline register s  820   a  and  820   b  feed the operands ΔZ M  and ΔZ O *ΔX MFRAC , respectively, to adder/subtractor  825   a , which adds the operands if X DIR =1 or subtracts the operands if X DIR =0. The adder/subtractor  825   a  then feeds the resulting corrected delta Z main value ΔZ MEC  into the register file  415 , overwriting the uncorrected delta Z main value ΔZ M  with the error-corrected specular red value ΔZ MEC . 
     Referring now to FIG. 8B, a preferred embodiment of an interpolator  715  includes a main slope interpolator  830  cascaded with an ortho interpolator  835 . The main slope interpolator  830  calculates the characteristic values for pixels along the polygon main slope. The ortho interpolator  835  either transmits the main slope characteristic value directly (if the current pixel is the main slope pixel) or adds the delta ortho parameter to the current characteristic value (if the current pixel is not a main slope pixel). Main slope interpolator  830  includes an adder  840   a , and adder/subtractor  825   b , 2×1 multiplexers  810   b  and  810   c , register  820   c , and OR gates  845   a  and  845   b . Adder  840   a  couples to the register file  415  to receive a delta main parameter, ΔC MEC , receiving ha second, feedback operand from the output of register  820   c . The output of adder  840   a  feeds the adder/subtractor  825   b , which receives a second operand, a delta ortho parameter ΔC O , from the register file  415 . The adder/subtractor  825   b  adds its two input operands if the add signal is asserted or subtracts its two input operands if the sub signal is asserted. The ortho adjust unit generates the add and sub signals, as will be described with respect to FIG.  8 C. 
     The adder  810   a  and adder/subtractor  825   b  feed 2×1 multiplexer  810   b , and OR gate  845   a  controls the state of multiplexer  810   b . If add is asserted or if sub is asserted, then the output of OR gate  845   a  is asserted, and multiplexer  810   b  selects the adder/subtractor  825   b  output. If neither add nor sub is asserted, then the OR gate  845   a  output is not asserted, and the multiplexer  810   b  selects the adder  840   a  output. Multiplexer  810   b  feeds the low (“0”) input of multiplexer  810   c , and the register file  415  feeds the high (“1”) input to multiplexer  810   c . The state of multiplexer  810   c  is controlled by the load main signal from the draw engine  710 . The output of multiplexer  810   c  couples to the input of register  820   c , and the output of OR gate  845   b  controls the state of register  820   c . OR gate  845   b  receives input from the load_main and inc_main signals. Thus, if the draw engine  710  asserts either load_main or inc_main, then the output of OR gate  845   b  becomes asserted, and the C MAIN  signal is set to the value at the input of register  820   c.    
     When the load_main signal is asserted, multiplexer  810   c  and register  820   c  change C MAIN  to C MEC , which is the starting characteristic value. After loading the starting characteristic parameter C MEC  is this manner, the load_main signal is deasserted. Subsequently, when neither the add signal nor the sub signal is asserted, the output of multiplexer  810   b  reflects the adder  840   a  output. Thus, the path from adder  840   a  through multiplexers  810   b  and  810   c  and register  820   c  cause the adder  840   a  to operate as an accumulator whenever inc_main is asserted, computing the next main slope value by adding ΔC MEC  to the current main slope value. To ortho adjust a main slope characteristic value as described with respect to the example of FIG. 6, the ortho adjust unit  770  causes the output of OR gate  845   a  to become asserted by asserting either the add or sub signal. When the output of OR gate  845   a  is asserted, multiplexer  810   b  selects the adder/subtractor  825   b  output. Because adder/subtractor  825  either adds delta ortho to or subtracts delta ortho from the current main slope pixel value (i.e., the output of multiplier  840   a ), the output of multiplexer  810   b  receives from adder/subtractor  825  the ortho-adjusted characteristic value, given by equation (4) or (5), respectively. In this manner, the main slope interpolator  830  computes the main slope characteristic values which provide input to the ortho interpolator  835  via the C MAIN  signal. 
     Still referring to FIG. 8B, ortho interpolator  835  includes a 2×1 multiplexer  810   d , a refer  820   d , and an adder  840   b . Multiplexer  810   d  selects C MAIN  if OR gate  845   b  is asserted or the output of adder  840   b  if OR gate  845   b  is not asserted. Multiplexer  810   d  generates the output signal C OUT , which both serves as the primary output of interpolator  715  and also provides a feedback input to register  820   d . Register  820   d  provides one operand for adder  840   b  and is controlled by the inc_ortho signal. Adder  840   b  receives a second operand ΔC O  from the register file  415 . After a main slope characteristic value C MAIN  is produced by the main slope interpolator  830 , inc_ortho is asserted for each pixel along the current scan line, so that ortho interpolator  835  calculates a characteristic value for each pixel. 
     Note that each interpolator  715   a ,  715   b ,  715   c ,  715   d ,  715   e ,  715   f ,  715   g ,  715   h ,  715   j , and  715   k  essentially operates according to the apparatus of FIG.  8 B. Thus, the input signals to interpolator  715 , C MEC , ΔC MEC , and ΔC O , correspond to the appropriate starting, delta main, and delta ortho inputs, respectively, of each interpolator  715   a ,  715   b ,  715   c ,  715   d ,  715   e ,  715   f ,  715   g ,  715   h ,  715   j , and  715   k . Accordingly, C OUT  of FIG. 8B corresponds to any of the outputs R OUT , G OUT , B OUT , sR OUT , sG OUT , sB OUT , U OUT , V OUT , W OUT , or Z OUT  of FIG.  7 . 
     Turning now to FIG. 8C, X interpolator  720  consists essentially of two adders  840   c  and  840   d  which connect to register file  415 , 2×1 multiplexer  810   e , OR gate  845   d , and register  820   e . X interpolator  720  also includes logic gates for generating the xstep signal, including OR gates  845   c  and  870 , AND gates  850  and  860 , and register  820   f . Adder  840   c  preferably receives as one operand ΔX MINT  from the register file  415 , while adder  840   d  preferably receives as one operand ΔX MFRAC  from the register file  415 . The present embodiment preferably includes an x-coordinate having a 12-bit integer portion (ΔX MINT ) and a 16-bit fractional (ΔX MFRAC ), although alternative embodiments may include other bit-widths for different integer and fractional resolutions. Adder  840   c  further receives through its carry-in input a carry-out signal carry generated by adder  840   d . Another preferred embodiment of X interpolator  720  (not shown) includes a single adder, in contrast to cascaded adders  840   c  and  840   d , which accepts a first operand consisting of ΔX MINT :ΔX MFRAC . 
     The outputs of adders  840   c  and  840   d  are concatenated to feed a calculated x-coordinate to the low input (“0”) of multiplexer  810   e . The high input (“1”) to multiplexer  810   e  is provided by the starting x-coordinate X M  from the register file  415 . Multiplexer  810   e  accepts its high input (“1”) when load_main is asserted and accepts its low input when load_main is deasserted. The output of multiplexer  810   e  couples to register  820   e , the state of which is controlled by the output of OR gate  845   d , which receives inputs from inc_main and load_main. Thus, OR gate  845   d  is asserted if either inc_main or load_main is asserted, and is deasserted if neither inc_main or load_main is asserted. Register  820   e  provides the output signal xpos, which indicates the main slope x-coordinate of the current scan line. The most significant bits of xpos, which have a bit-width equivalent to that of ΔX MINT , provide the second operand for adder  840   c . Accordingly, adder  840   d  receives a second operand from the least significant bits of xpos, which have a bit-width equivalent to that of ΔX MFRAC . 
     To render a polygon, the draw engine  710  first asserts the load_main signal, switching multiplexer  810   e  to the high (“1”) input state and initializing the xpos signal to be equal to the starting x-coordinate X M . Following the initialization of xpos, load_main is deasserted, and the adders  840   c  and  840   d  add ΔX M  to xpos. Thereafter, pulsing the inc_main signal causes register  820   e  to produce an updated main slope x-coordinate, as calculated by the adders  840   c  and  840   d . 
     The logic circuitry used to generate the xstep output signal consists of OR gates  870  and  845   c , AND gates  850  and  860 , and register  820   f . OR gate  870  accepts each bit of ΔX MFRAC  as input. Accordingly, the output to OR gate  870  is asserted high if at least one of the input signals is asserted high and is asserted low otherwise. Thus, OR gate  870  is asserted low if ΔX M  is an integer and is asserted high if ΔX M  is not an integer. AND gate  860  includes one inverting input (−) which is driven by the carry signal. AND gate  860  receives two additional signals through noninverting inputs (+), including one input from OR gate  870  and another input from the sign bit of ΔX M , or ΔX MINT [ 27 ], in the register file  415 . AND gate  850  includes one inverting input and one non-inverting input. The inverting input of AND gate  850  receives the sign bit of ΔX M , or ΔX M [ 27 ], from register file  415  while the noninverting input receives the carry signal. The outputs of AND gates  850  and  860  drive OR gate  845   c , which feeds register  820   f . The output of register  820   f  comprises the xstep signal, which is asserted if either a borrow or carry condition occurs between adders  840   c  and  840   d . Since adders  840   c  and  840   d  accumulate the integer and fractional portions of the x-coordinate, respectively, the xstep signal indicates the need for an ortho-adjustment in interpolators  715 . 
     Still referring to FIG. 8C, a preferred embodiment of ortho adjust unit  770  includes XOR gate  855 , XNOR gate  865 , and AND gates  875   a  and  875   b . XOR gate  855  and XNOR gate  865  receive identical inputs X DIR  and ΔX M [ 27 ] (the sign bit of ΔX M ) and feed AND gates  875   a  and  875   b , respectively. AND gates  875   a  and  875   b  receive second inputs from the xstep signal, generated by the X-interpolator  720 . The resulting outputs of ortho adjust unit  770  are the sub (output of AND gate  875   a ) and add (output AND gate  875   b ) signals. Thus, asserting xstep causes either add or sub to become asserted, resulting in an ortho-adjustment to the pixel characteristic values calculated by the interpolators  715 . 
     A polygon engine may also be implemented through software instructions that are executed via a microprocessor, a programmable graphics processor or other programmable device, or any other device capable of interpreting software instructions. Accordingly, FIGS. 9A through 9G illustrate flowcharts of a preferred embodiment of software instructions that implement a graphics processor. Further, the Microfiche appendix contains instructions written in the C programming language to implement the embodiment of FIGS. 9A through 9G. The software program of FIGS. 9A through 9G is intended to operate in substantially the same manner as the hardware embodiment of FIGS. 6,  8 A,  8 B, and  8 C, and includes the error calculations of equations (1) through (6). The software embodiment of FIGS. 9A through 9G further define polygon parameters having bit widths equivalent to the bit widths of the corresponding parameters in the hardware embodiment, although alternative software embodiments may include polygon parameters comprising any number of bits. Further, the software embodiment is preferably capable of rendering both the upper and the lower polygon defining a random triangle or quadrangle. In addition, the software embodiment supports the pixel characteristics of fog and alpha blending. 
     Software execution in the present embodiment begins with the “start” block of the flowchart of FIG.  9 A. In block  900   a , the parameters defining a polygon, as listed in Table I, above, are initialized or accepted as program input. In addition, new variables are introduced to allow the x-coordinate and pixel characteristics to be rounded. As in the previous embodiment, the x-coordinate of a main slope pixel X M  is calculated by adding ΔX M  to the previous main slope x-coordinate. Because X M  is preferably a fractional-valued variable, an “X current” variable X cur  is introduced in block  900   a  to represent the rounded value of the current main slope pixel x-coordinate X M . 
     Since the characteristic values are rounded as well, a “C current” variable C cur  and a “C ortho” variable C ortho  for each pixel characteristic are also introduced in block  900   a . The “C ortho” variable C ortho  will be used to calculate the pixel characteristic values for pixels not on the main slope, allowing C M  to hold the characteristic value of the main slope pixel. The “C current” variable C cur  will be used to represent the rounded value of C ortho , so that the current, unrounded, value of C ortho  may be used to calculate a subsequent C ortho . Accordingly, each type of pixel characteristic is assigned a “C ortho” and a “C current” variable. Table II summarizes the variables defined and initialized in block  900   a  that are not listed in Table I. 
     
       
         
               
             
               
               
             
           
               
                 TABLE II 
               
             
             
               
                   
               
               
                 Additional program variables defined in block 900a. 
               
             
          
           
               
                 Variable 
                 Description 
               
               
                   
               
               
                 X cur   
                 current polygon x-coordinate 
               
               
                 A M   
                 Alpha main parameter 
               
               
                 A O   
                 Alpha ortho gradient parameter 
               
               
                 ΔA M   
                 delta Alpha main 
               
               
                 A ortho   
                 Alpha ortho variable-holds current, unrounded Alpha value 
               
               
                 A cur   
                 Alpha current variable-holds current, rounded Alpha value 
               
               
                 F M   
                 Fog main parameter 
               
               
                 F O   
                 fog ortho parameter 
               
               
                 ΔF M   
                 delta fog main 
               
               
                 F ortho   
                 fog ortho variable-holds current, unrounded fog value 
               
               
                 F cur   
                 fog current variable-holds current, rounded fog value 
               
               
                 R ortho   
                 red ortho variable-holds current, unrounded red value 
               
               
                 R cur   
                 red current variable-holds current, rounded red value 
               
               
                 G ortho   
                 green ortho variable-holds current, unrounded green value 
               
               
                 G cur   
                 green current variable-holds current, rounded green value 
               
               
                 B ortho   
                 blue ortho variable-holds current, unrounded blue value 
               
               
                 B cur   
                 blue current variable-holds current, rounded blue value 
               
               
                 sR ortho   
                 specular red ortho variable-holds current, unrounded, specular red value 
               
               
                 sR cur   
                 specular red current variable-holds current, rounded specular red value 
               
               
                 sG ortho   
                 specular green ortho variable-holds current, unrounded specular green value 
               
               
                 sG cur   
                 specular green current variable-holds current, rounded specular green value 
               
               
                 sB ortho   
                 specular blue ortho variable-holds current, unrounded specular blue value 
               
               
                 sB cur   
                 specular blue current variable-holds current, rounded specular blue value 
               
               
                 U ortho   
                 u-texel ortho variable-holds current, unrounded U texture value 
               
               
                 U cur   
                 u-texel current variable-holds current, rounded U texture value 
               
               
                 V ortho   
                 v-texel ortho variable-holds current, unrounded V texture value 
               
               
                 V cur   
                 v-texel current variable-holds current, rounded V texture value 
               
               
                 W ortho   
                 w-texel ortho variable-holds current, unrounded W texture value 
               
               
                 W cur   
                 w-texel current variable-holds current, rounded W texture value 
               
               
                 Z ortho   
                 Z ortho variable-holds current, unrounded z-coordinate 
               
               
                 Z cur   
                 Z current variable-holds current, rounded z-coordinate 
               
               
                   
               
             
          
         
       
     
     After the variables of Table I and Table II are initialized in block  900   a , the program proceeds to block  904   a  to error-correct the C M  and ΔC M  variables for each pixel characteristic, according to equations (1) and (2), above. A software algorithm implementing block  904   a  is presented in greater detail below with respect to FIGS. 9B and 9C, and it will be assumed for the remainder of the example of FIG. 9A that C M  and ΔC M  represent error-corrected “C” main and “delta C main” parameters. Proceeding to block  908   a , local integer variables count and j are introduced to track the position of loops within the algorithm of FIG.  9 A. The variable count represents the rendering position with respect to the height of the current polygon and is initially defined as count=main_count+1. The variable count is decremented by one after each scan line, so that when count=0, the current polygon (i.e., either the upper or lower polygon) has been completely rendered. The value of j identifies whether the upper or lower polygon is being rendered and is initialized to j=0. The value of j is incremented after a polygon, either the upper or lower polygon, is rendered. Thus, when j=2, both the upper and lower polygon have been rendered. In the next block  912   a , variable width_count is defined as width_count=W M  and will be used in a manner similar to that of loop variables count and j, to count the number of pixels drawn on each scan line. 
     Moving to block  916   a , the program writes the x- and y-coordinates, as defined by X cur  and Y M  to the pixel pipe. Any suitable algorithm may be used to implement the pixel pipe. In the next block  920   a , the program writes the characteristic values to the pixel pipe, as defined by the C cur  variables, R cur , G cur , B cur , sR cur , sG cur , sB cur , U cur , V cur , W cur , F cur , A cur , and Z cur . 
     Following block  920   a , program execution proceeds to block  924   a , where, if X DIR =1, the program moves to block  932   a  to decrement X cur  by one. If X DIR =0in block  924   a , then the program increments X, by one in block  928   a . Blocks  928   a  and  932   a  both feed block  936   a , in which the interpolators for all pixel characteristics are updated. Block  936   a  is described in greater detail with respect to FIGS. 9D and 9E. From block  936   a , the program moves to block  940   a  where width is decremented by one to mark the completion of the rendering of one pixel. 
     Since the variable width is decremented after every pixel is rendered, program execution next moves to block  944   a  to determine if width=0, which would indicate the end of a scan line. If width≠0, then the scan line has not been fully rendered, and program execution branches to block  916   a  to render the next pixel of the current scan line. If width=0, indicating that the final pixel of the scan line has been rendered, then the program proceeds to block  948   a . In block  948   a , xstep is calculated according to the same rules outlined with respect to FIG. 8C, above. Program execution next moves to block  952   a , where X M  is incremented by ΔX M , and Y M  is incremented by one. Next moving to block  956   a , program execution branches according to the value of j. If j=0 in block  956   a , indicating that the upper polygon is being rendered, the program proceeds to block  960   a  and increments W M  by ΔW M  for the next scan line. If j=1 in block  956   a , indicating that the lower polygon is being rendered, the program proceeds to block  964   a  and increments W M  by ΔW O  for the next scan line. Blocks  960   a  and  964   a  both lead to block  968   a , in which the characteristic interpolators invoke equations (3) through (6) to compute the characteristic values for the main slope pixels of the next scan line. FIGS. 9F and 9G describe detailed procedures for updating the interpolators according to block  968   a.    
     Subsequent to block  968   a , program execution proceeds to block  972   a , where count is decremented by one to mark the completion of a scan line. Decision block  976   a  follows block  972   a  and branches according to the state of count. Specifically, if count≠0 in block  976   a , indicating that the final scan line of the current polygon has not been rendered, program execution returns to block  912   a  to begin rendering the next scan line. If count =0 in block  976   a , indicating that the final scan line of the current polygon has been rendered, program execution proceeds to block  980   a , where count is set equal to opp_count, and j is incremented by one. Following block  980   a , decision block  984   a  branches according to the value of j. If j=1, indicating that the upper polygon has just been rendered, program execution branches to block  912   a  to begin rendering the lower polygon. If j=2 in block  976   a , then the lower polygon rendering is complete, and program execution halts, terminating at the “end” block. 
     FIG. 9B represents a preferred embodiment of an error-correction algorithm implementing equations (1) and (2), such the algorithm of block  904   a  in FIG.  9 A. Although the steps of FIG. 9B are generally suitable for correcting parameters for any pixel characteristic, the Z-parameters preferably have longer bit-widths than do the other parameters and are preferably corrected according to the steps of FIG. 9C, as described below. Accordingly, FIG. 9B illustrates the steps for correcting the starting and delta main parameters for pixel characteristic “C,” which may represent any pixel characteristic other than Z. The program of FIG. 9B is preferably invoked separately for each pixel characteristic. 
     Execution of the algorithm of FIG. 9B begins at the “start” block and proceeds immediately to block  900   b . In block  900   b , the fractional portion of X M , or X MFRAC , is multiplied by ΔC O . The product of block  900   b  is preferably represented by a two&#39;s complement number. Moving next to block  904   b , if X DIR =0, then the product of block  900   b  is used to calculate an updated C M  in block  912   b  as C M −X MFRAC *ΔC O , as in equation (1). If X DIR =1 in block  904   b , then the product of block  900   b  is used to calculate an updated C M  in block  908   b  as C M +X MFRAC * ΔC O . 
     Blocks  912   b  and  908   b  both lead to block  916   b , where the most significant bits of C M  are then truncated such that C M  lies between −512 (or −2 9 ) and +512 (or +2 9 ). Specifically, −2 9 ≦C M  ≦+2 9 −2 −16 . In a preferred embodiment, C M  has a 16-bit fractional portion but is represented by a 32-bit signed integer variable. Thus, C M  is treated as a fixed-point number which is truncated in step  916   b  by keeping only a sufficient number of the least significant bits to retain the fractional portion (16 bits), the integer portion (9 bits), and the sign bit (1 bit), a total of 26 bits. “Truncation” as in step  916   b  therefore requires sign-extending C M  from the sign bit (or the 26 th  least significant bit) of C M . To sign-extend C M  in this manner, all bits to the left of the sign bit are set to the value of the sign bit. For a more detailed description of sign-extension and two&#39;s complement binary notation, refer to  Computer Organization and Design  by Patterson and Hennessy (Morgan Kaufmann Publishers, Inc., 1994). 
     After block  916   b , C ortho  and C cur  are defined as C ortho =C cur =C M  in block  920   b . Because C cur  represents a pixel characteristic value appropriate for rendering, C cur  is next clipped, or saturated to lie between 0 and +255, as implemented with respect to blocks  924   b ,  928   b ,  932   b , and  936   b . Beginning with decision block  924   b , the program flow branches according to the value of C cur . If C cur &lt;0, then C cur  is set to zero in block  928   b . If C cur ≧0 in block  924   b , then the program moves to decision block  932   b , which further branches program flow according to the value of C cur . If C cur ≧256 in block  932   b , then the program proceeds to block  936   b , where C cur  is set equal to 255. Otherwise, the program moves from block  932   b  to block  940   b , without altering C cur . Program flow from block  928   b  or block  932   b  also moves to block  940   b . In block  940   b , the program calculates the product ΔX MFRAC * ΔC O , which is preferably represented by a two&#39;s complement number containing 40 significant bits, including the sign bit. Next continuing with block  944   b , the program branches according to the value of X DIR . Specifically, if X DIR =0, program execution continues to block  948   b , where the product of block  940   b  is used to calculate an updated ΔC M  as C M +ΔX MFRAC *ΔC O , as in equation (2). If X DIR =0 in block  944   b , then the product of block  940   b  is used to calculate ΔC M  in block  952   b  as C M -ΔX MFRAC *ΔC O . Blocks  952   b  and  948   b  represent the final stage of the algorithm of FIG.  9 B and terminate at the “end” block. 
     Now referring to FIG. 9C, a program for correcting Z M  and ΔZ M  begins at the. “start” block, proceeding immediately to block  900   b . In block  900   c , the fractional portion of X M , or X MFRAC , is multiplied by ΔZ O . The product of block  900   c  preferably represents a two&#39;s complement number containing 40 significant bits, including the sign bit. Moving next to block  904   c , if X DIR =0, then the product of block  900   c  is used to calculate an updated Z M  in block  912   c  as Z M −X MFRAC *ΔZ O , from equation (1). If X DIR =1 in block  904   c , then the product of block  900   c  is used to calculate an updated Z M  in block  908   c  as Z M +X MFRAC *ΔZ O , using equation (1). The error-corrected Z M , as calculated in either block  912   c  or  908   c , preferably is represented by a two&#39;s complement number containing 48 significant bits, including the sign bit as the most significant bit. Block  912   c  and block  908   c  both lead to block  916   c , in which the most significant bits of Z M  are truncated such that Z M  lies between −131,072 (or −2 17 ) and +131,072 (or +2 17 ). Specifically, −2 17 ≦Z M ≦+2 17 −2 −16 . In a preferred embodiment, Z M  has a 16-bit fractional portion and is represented by a 64-bit signed integer variable. Thus, Z M  is treated as a fixed-point number which is truncated in step  916   c  by keeping only a sufficient number of bits to retain the fractional portion (16 bits), the integer portion (17 bits), and the sign bit (1 bit), a total of 48 bits. “Truncation” as in step  916   c  therefore requires sign-extending Z M  from the 48 th  least significant bit. 
     Still referring to FIG. 9C, the program next moves to block  920   c , in which Z ortho  and Z cur  are defined as Z ortho =Z M  in block  920   c . Because Z cur  represents a pixel characteristic value appropriate for rendering, Z cur  is next clipped, or saturated to lie between 0 and +65,536, as implemented with respect to blocks  924   c ,  928   c ,  932   c ,  936   c , and  940   c . Beginning with decision block  924   c , the program flow branches according to the value of Z M . If Z M ≧65,536 (or +2 16 ), then Z cur  is set to +2 16 −2 −16  in block  940   c . If Z M &lt;65,536 in block  924   c , then the program moves to decision block  928   c , which further branches program flow according to the value of Z M . If Z M &lt;0 in block  928   c , then the program proceeds to block  936   c , in which Z cur  is set to zero. Otherwise, the program moves from block  928   c  to block  932   c , in which Z cur  is set equal to the 32 least significant bits of Z M , or Z MINT [ 15 : 0 ]:Z MFRAC . 
     Program flow from blocks  932   c ,  936   c , and  940   c  feed block  944   c . In block  944   c , the program calculates the product ΔX MFRAC *ΔZ O , which is preferably represented by a two&#39;s complement number containing 40 significant bits, including the sign bit. Next continuing with block  948   c , the program branches according to the value of X DIR . Specifically, if X DIR =0, program execution continues to block  952   c , where the product of block  944   c  is used to calculate an updated ΔZ MEC  as Z M +ΔX MFRAC *ΔZ O , using equation (2). If X DIR =0 in block  948   c , then the product of block  944   c  is used to calculate an updated ΔZ M  in block  956   c  as Z M −ΔX MFRAC *ΔZ O . Blocks  952   c  and  956   c  represent the final stage of the algorithm of FIG.  9 C and terminate at the “end” block. 
     FIG. 9D describes a preferred embodiment of an algorithm for updating a pixel characteristic interpolator to hold the characteristic value of the next pixel on the current scan line, such as in block  936   a  of FIG. 9A Although the steps of FIG. 9D are generally suitable for any pixel characteristic interpolator, the Z-interpolator preferably handles longer bit-widths than do the other interpolators and are preferably updated according to the steps of FIG. 9E, as described below. FIG. 9D illustrates the steps for updating the interpolator for pixel characteristic “C,” which may represent any pixel characteristic other than Z. The program of FIG. 9D is preferably invoked separately to update each pixel characteristic. 
     Now referring to FIG. 9D, program execution begins at the “start” block, proceeding immediately to block  900   d . In block  900   d , C ortho  is updated by adding ΔC O  to C ortho . Next moving to block  904   d , C ortho  is truncated such that C ortho  lies between −511 (or −2 9 ) and +512 (or +2 9 ). Specifically, −2 9 ≦C ortho ≦+2 9 −2 −16 . The truncation accounts for the fact that a computer implementing the steps of FIG. 9D will use a fixed-width accumulator. In a preferred embodiment, C ortho  has a 16-bit fractional portion but is represented by a 32-bit signed integer variable. Thus, C ortho  is treated as a fixed-point number which is truncated in step  904   d  by keeping only a sufficient number of bits to retain the fractional portion (16 bits), the integer portion (9 bits), and the sign bit (1 bit), a total of 26 bits. “Truncation” as in step  904   d  therefore requires sign-extending C ortho  from the 26 th  least significant bit. Proceeding from step  904   d , C cur  is set equal to C ortho . After block  904   d , C cur  is clipped, or saturated, to lie between 0 and +255, as implemented with respect to blocks  912   d ,  916   d ,  920   d , and  924   d.    
     Beginning with decision block  912   d , the program flow branches according to the value of C cur . If C cur &lt;0, then C cur  is set to zero in block  916   d , and program execution terminates at the “end” block. If C cur ≧0 in block  912   d , then the program moves to decision block  920   d , which further branches program flow according to the value of C cur . If C cur ≧256 in block  920   d , then the program proceeds to block  924   d , where C cur  is set equal to 255. Following block  924   d , the program terminates at the “end” block. If C cur &lt;256 in block  920   d , the program moves from block  920   d  to the “end” block, terminating execution without altering C cur . 
     Now referring to FIG. 9E, a program for updating the Z-interpolator begins at the “start” block, proceeding immediately to block  900   e . In block  900   e , Z ortho  is incremented by ΔZ O , the resulting sum preferably represented by a 48-bit signed integer. Next moving to block  904   e , Z ortho  is truncated such that Z ortho  lies between −131,072 (or −2 17 ) and +131,072 (or +2 17 ). Specifically, −2 17 ≦Z ortho ≦+2 17 −2 −16 . The truncation accounts for the fact that a computer implementing the steps of FIG. 9E will use a fixed-width accumulator. In a preferred embodiment, Z ortho  has a 16-bit fractional portion but is represented by a 64-bit signed integer variable. Thus, Z ortho  is treated as a fixed-point number which is truncated in step  904   e  by keeping only a sufficient number of bits to retain the fractional portion (16 bits), the integer portion (17 bits), and the sign bit (1 bit), a total of 34 bits. “Truncation” as in step  904   e  therefore requires sign-extending Z ortho  from the 34 th  least significant sign bit. Proceeding from step  904   e , the program branches in step  908   e  according to the value of Z ortho . If Z ortho ≧+65,536 (or +2 16 ), then the program proceeds to block  924   e , where Z cur  is “saturated down” to +2 16 −2 −16 . If Z ortho  &lt;+65,536 in block  908   e , then the program proceeds to block  912   e , branching again according to the value of Z ortho . In block  912   e , if Z ortho &lt;0, then Z cur  is “saturated up” to 0 in block  920   e . Otherwise, Z cur  is set equal to the 32 least significant bits of Z ortho , or Z ortho [15:0]:Z orthoFRAC , in block  916   e . Blocks  916   e ,  920   e , and  924   e  each terminate to the “end” block, completing the algorithm. 
     FIG. 9F describes a preferred embodiment of an algorithm used by a pixel characteristic interpolator to calculate the characteristic value of the main slope pixel of the next scan line, such as in block  968   a  of FIG.  9 A. Although the steps of FIG. 9F are generally suitable for any pixel characteristic interpolator, the Z-interpolator preferably handles longer bit-widths than do the other interpolators and are preferably updated for the next scan line according to the steps of FIG. 9G, as described below. FIG. 9F illustrates the steps for updating the interpolator for pixel characteristic “C,” which may represent any pixel characteristic other than Z. The program of FIG. 9F is preferably invoked separately for each pixel characteristic. 
     The “start” block of FIG. 9F begins the program execution by feeding into decision block  900   f . If xstep=1 in block  900   f , indicating the need for ortho-adjustment, then program execution proceeds to block  904   f , which branches according to the value of ΔX M [ 27 ]⊕X DIR , where⊕represents the logical XOR function. If xstep=1 in block  904   f , then the program moves to block  916   f . If ΔX M [ 27 ]⊕X DIR =1 in block  904   f , then the characteristic value of the next main slope pixel C M  is ortho-adjusted by subtracting ΔC O  from C M  in block  912   f . If ΔX M [ 27 ]⊕X DIR =1 in block  904   f , then C M  is ortho-adjusted by adding ΔC O  to C M  in block  912   f . Blocks  908   f  and  912   f  further feed block  916   f  In block  916   f , the next main slope characteristic value is calculated by adding ΔC M  to the current main slope characteristic value C M . Block  916   f  feeds block  920   f , in which C M  is truncated such that C M  lies between −512 (or −2 9 ) and +511 (or +2 9 ). Specifically, −2 9 ≦C M ≦+2 9 −2 −16 . In a preferred embodiment, C M  has a 16-bit fractional portion but is represented by a 32-bit signed integer variable. Thus, C M  is treated as a fixed-point number which is truncated in step  920   f  by keeping only a sufficient number of bits to retain the fractional portion (16 bits), the integer portion (9 bits), and the sign bit (1 bit), a total of 26 bits. 
     After block  920   f , C ortho  and C cur  are set to C ortho =C cur =C M  in block  924   f . Because C cur  represents a pixel characteristic value appropriate for rendering, C cur  is next clipped, or saturated to lie between 0 and +255, as implemented with respect to blocks  928   f ,  932   f ,  936   f , and  940   f . From block  924   f , the program moves to decision block  928   f , which branches the program flow according to the value of C cur . If C cur &lt;0, then C cur  is set to zero in block  932   f , and program execution terminates at the “end” block. If C cur ≧0 in block  928   f  then the program moves to decision block  936   f , which further branches program flow according to the value of C cur . If C cur ≧256 in block  936   f , then the program proceeds to block  940   f , where C cur  is set equal to 255. Following block  940   f , the program terminates at the “end” block. If C cur &lt;256 in block  936   f , the program moves from block  936   f  to the “end” block, terminating execution without altering C cur . 
     FIG. 9G illustrates a preferred embodiment of the algorithm used to update the Z-interpolator to calculate the main slope z-coordinate of the next scan line, as in block  968   a  of FIG.  9 A. The “start” block of FIG. 9G begins the program execution by feeding into decision block  900   g . If xstep=1 in block  900   g , then the program moves to block  908   g . If xstep=1 in block  900   g , indicating the need for ortho-adjustment, then program execution proceeds to block  904   g , which branches according to the value of ΔX M [ 27 ]⊕X DIR , where⊕represents the logical XOR function. If ΔX M [ 27 ]⊕X DIR =1 in block  904   g , then the characteristic value of the next main slope pixel Z M  is ortho-adjusted by subtracting ΔZ O  from Z M  in block  912   g . If ΔX M [ 27 ]⊕X DIR =1 in block  904   g , then Z M  is ortho-adjusted by adding ΔZ O  to Z M  in block  908   g . The result of blocks  912   g  and  908   g  preferably are 48-bit two&#39;s complement integers. Blocks  912   g  and  908   g  feed block  908   g . In block  916   g , the next main slope characteristic value is calculated by adding ΔZ M  to the current main slope characteristic value Z M . 
     Block  916   g  feeds block  920   g , in which Z M  is truncated to be between −131,072 (or −2 17 ) and +131,072 (or +2 17 ). Specifically, −2 17 ≦Z M ≦+2 17 −2 −16 . In a preferred embodiment, Z M  has a 16-bit fractional portion but is represented by a 64-bit signed integer variable. Thus, Z M  is treated as a fixed-point number which is truncated in step  920   g  by keeping only a sufficient number of bits to retain the fractional portion (16 bits), the integer portion (17 bits), and the sign bit (1 bit), a total of 34 bits. “Truncation” as in step  920   g  therefore requires sign-extending Z M  from the 34 th  least significant bit. 
     After block  920   g ; Z ortho  is set to Z ortho =Z M  in block  924   g . Because Z cur  represents a pixel characteristic value appropriate for rendering, Z cur  is next clipped, or saturated to lie between 0 and +65,536 (or +2 16 ). Taking into account the fractional portion, Z cur  is preferably saturated to 0≦Z cur ≦2 16 −2 −16 . Proceeding from block  924   g , program execution branches according to the value of Z M . If Z M ≧65,536, then Z cur  is set to 2 16 −2 −16  in block  944   g . If Z M &lt;65,536 in block  928   g , then the program moves to decision block  932   g , which branches according to the evaluation of Z M &lt;0. If Z M &lt;0, then Z cur  is set to 0 in block  940   g . Otherwise, Z cur  in block  936   g  is set equal to the 32 least significant bits of Z M . Blocks  936   g ,  940   g , and  944   g  each terminate to the “end” block, completing the algorithm of FIG.  9 G. 
     Hence, the present invention discloses a graphics processor capable of receiving polygon parameters from a display driver, correcting the polygon parameters to anticipate, or prevent, interpolation error, and then interpolate the polygon parameters to render the polygon. The graphics processor solely implements the error correction, allowing the software driver to focus only on calculating and transmitting the polygon parameters. Thus, the software driver need not deal with error correction calculations and can be designed to operate faster and more simply than before. Further, because the graphics processor corrects the polygon parameters before the polygon parameters are interpolated to render the polygon, the pixel characteristics are correct immediately after they are rendered, with no need for a subsequent error-correction calculation. Hence, the graphics processor implements error correction by actually preventing interpolation errors in the polygon prior to interpolation, as opposed to fixing existing interpolation errors. Because the graphics processor need not implement an extra error correction stage following polygon interpolation, the present invention is adapted to correct and interpolate polygons much more expediently than before. 
     These features as well as numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. It is intended that the following claims be interpreted to embrace all such variations and modifications.