Abstract:
A method and apparatus for efficiently rasterizing graphics is provided. The method is intended to be used in combination with a frame buffer that provides fast tile-based addressing. Within this environment, frame buffer memory locations are organized into a tile hierarchy. For this hierarchy, smaller low-level tiles combine to form larger mid-level tiles. Mid-level tiles combine to form high-level tiles. The tile hierarchy may be expanded to include more levels, or collapsed to included fewer levels. A graphics primitive is rasterized by selecting an starting vertex. The low-level tile that includes the starting vertex is then rasterized. The remaining low-level tiles that are included in the same mid-level tile as the starting vertex are then rasterized. Rasterization continues with the mid-level tiles that are included in the same high-level tile as the starting vertex. These mid-level tiles are rasterized by rasterizing their component low-level tiles. The rasterization process proceeds bottom-up completing at each lower level before completing at higher levels. In this way, the present invention provides a method for rasterizing graphics primitives that accesses memory tiles in an orderly fashion. This reduces page misses within the frame buffer and enhances graphics performance.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application is a continuation of U.S. patent application Ser. No. 09/145,516 filed Sep. 2, 1998 now U.S. Pat. No. 6,611,272 which claims benefit of 60/091,599 filed Jul. 2, 1998, the entirety of which is incorporated herein by reference. 
   RELATED APPLICATIONS 
   The following application claims the benefit of U.S. Provisional Application Ser. No. 60/091,599 entitled “Method and Apparatus for Rasterizing in a Hierarchical Tile Order” by Zahid S. Hussain and Timothy J. Millet, filed Jul. 2, 1998, the disclosure of which is incorporated in this document by reference. 

   FIELD OF THE INVENTION 
   The present invention relates generally to systems for computer graphics. More specifically, the present invention includes a method and apparatus for efficiently rasterizing graphics primitives. 
   BACKGROUND OF THE INVENTION 
   Computer systems (and related devices) typically create three-dimensional images using a sequence of stages known as a graphics pipeline. During early pipeline stages, images are modeled using a mosaic-like approach where each image is composed of a collection of individual points, lines and polygons. These points, lines and polygons are known as primitives and a single image may require thousands, or even millions, of primitives. Each primitive is defined in terms of its shape and location as well as other attributes, such as color and texture. 
   The primitives used in early pipeline stages are transformed, during a rasterization stage, into collections of pixel values. The rasterization stage is often performed by a specialized graphics processor (in low-end systems, rasterization may be performed directly by the host processor) and the resulting pixel values are stored in a device known as a frame buffer. A frame buffer is a memory that includes a series of randomly accessible memory locations. Each memory location in the frame buffer defines a corresponding pixel included in an output device where the image will ultimately be displayed. To define its corresponding pixel, each memory location includes a series of bits. Typically, these bits are divided into separate portions defining red, blue and green intensities. Each memory location may also include depth information to help determine pixel ownership between overlapping primitives. 
   During the rasterization stage, the graphics processor renders each primitive into the frame buffer. The graphics processor accomplishes this task by determining which frame buffer memory locations are included within the bounds of each primitive. The included memory locations are then initialized to reflect the attributes of the primitive, including color and texture. 
   The rasterization stage is followed by a display stage where a display controller transforms the pixel values stored in the frame buffer into signals that drive the output device being used. The display controller accomplishes this task by scanning the memory locations included in the frame buffer. The red, blue and green portions of each location are converted into appropriate output signals and sent to the output device. 
   The throughput of a graphics pipeline is highly dependent on frame buffer performance. This follows because the frame buffer functions as a middleman between the rasterization stage and the display stage. As a result, the frame buffer becomes the focus of repeated memory accesses by both the graphics processor and the display controller. The number of these accesses may be quite large. The frame buffer must be able to sustain a high rate of these accesses if it is to avoid becoming a performance bottleneck. 
   Frame buffers are typically fabricated using arrays of dynamic random access memory (DRAM) components. Compared to other technologies, such as static random access memories (SRAMs), DRAM components represents a better trade off between performance and cost. At the same time, achieving acceptable frame buffer performance may be far more complicated when DRAM components are used. The complexity involved in DRAM use stems from the addressing scheme used by these components. For this scheme, memory locations are addressed using a combination of a row address and a column address. Row and column addresses are supplied in sequence—row address first, column address second. Depending on the specific type of DRAM components used, this two-step addressing scheme may be too time consuming to sustain the memory access rate required for frame buffer use. 
   Fortunately, many DRAM components also provide a faster page addressing mode. For this mode, a sequence of column addresses may be supplied to a DRAM component after the row address has been supplied. Accesses within a row require only a single address. The overall effect is that accessing a DRAM component is much faster when a series of accesses is confined to a single row. Accessing a location included in a new row, referred to as a page miss, is much slower. 
   For this reason, frame buffers are often designed to maximize consecutive accesses within DRAM rows and to minimize page misses. One way in which this is accomplished is to structure the frame buffer so that graphics primitives tend to map to a single DRAM row or a small number of DRAM rows. Memory tiling is an example of this type of frame buffer structuring. In frame buffers that use memory tiling, the memory locations included in a DRAM row map to a rectangular block of pixels. This contrasts with more typical frame buffer construction where DRAM rows map to lines of pixels. Memory tiling takes advantage of the fact that many primitives fit easily into blocks and that few fit easily into lines. In this ways memory tiling reduces page misses by increasing the chances that a given primitive will be included within single DRAM row or a small number of DRAM rows. 
   Another way to maximize consecutive accesses within DRAM rows and to minimize page misses is to position a cache memory between the graphics processor and the frame buffer. The cache memory collects accesses performed by the graphics processor and forwards them to the cache on a more efficient row-by-row basis. 
   Memory tiling and cache memories are both effective techniques for improving the performance of DRAM based frame buffers. Unfortunately, the rasterization technique used within most frame buffers does not fully exploit the full potential of memory tiling or cache memories used in combination with memory tiling. This follows because rasterization is typically performed on a line-by-line basis. When used in a tiled frame buffer, line-by-line rasterization effectively ignores the tiled structure of the frame buffer. As a result, a given rasterization may alternately access and re-access a given set of tiles. This results in an increased number of DRAM page misses and decreases the throughput of the frame buffer and graphics pipeline. As a result, there is a need for rasterization methods that more effectively exploit the full potential of memory tiling and cache memories used in combination with memory tiling. 
   SUMMARY OF THE INVENTION 
   An embodiment of the present invention includes a method and apparatus for efficiently rasterizing graphics primitives. In the following description, an embodiment of the present invention will be described within the context of a representative graphics pipeline. The graphics pipeline is a sequence of components included in a host computer system. This sequence of components ends with a frame buffer followed by a display controller. 
   The frame buffer is a random access memory device that includes a series of memory locations. The memory locations in the frame buffer correspond to pixels included in an output device, such as a monitor. Each memory location includes a series of bits with the number and distribution of bits being implementation dependent. For the purpose of description, it may be assumed that each memory location includes four eight bit bytes. Three of these bytes define red, blue and green intensities, respectively. The fourth byte, alpha, defines the pixel&#39;s coverage or transparencies. 
   The memory locations included in the frame buffer are preferably organized using a tiled addressing scheme. For this scheme, the memory locations included in the frame buffer are organized to correspond to rectangular tiles of pixels included in the output device. The number of pixels (and the number of frame buffer memory locations) included in a single tile may vary between different frame buffer implementations. In most cases, the tile size will be a power of two. This provides a convenient scheme where more significant address bits choose a specific tile and less significant address bits choose an offset within the specific tile. In cases where the frame buffer is fabricated using DRAM or DRAM-like memory components it is preferable for each tile to map to some portion of DRAM row. Thus, each DRAM row includes one or more memory tiles. 
   The display controller scans the memory locations included in the frame buffer. For each location scanned, the display controller converts the red, blue and green intensities into appropriate output signals. The display controller sends these output signals to the output device being used. The display controller continually repeats this scanning process. In this way, the contents of the frame buffer are continuously sent to the output device. 
   The graphics processor rasterizes graphics primitives into the frame buffer. To accomplish this task, the graphics processor determines which frame buffer memory locations are included within the bounds of each primitive. The included memory locations are then initialized to reflect the attributes of the primitive, including color and texture. During rasterization, the graphics processor uses a hierarchy of memory tiles. Within this hierarchy, smaller tiles are grouped into larger tiles. These larger tiles may be grouped, in turn, into still larger tiles. For a representative embodiment of the present invention, the tile hierarchy includes three levels. The lowest level of the hierarchy is made up of four pixel by four pixel low-level tiles. These four-by-four tiles are grouped into eight-by-eight mid-level tiles and the eight-by-eight tiles are grouped into sixteen-by-sixteen high-level tiles. 
   The graphics processor begins the process of rasterizing a primitive by selecting one of the primitive&#39;s vertices as a starting vertex. The graphics processor then rasterizes the low-level tile that includes the starting vertex. When rasterization of the first low-level tile is complete, the graphics processor moves left-to-right, top-to-bottom through the remaining low-level tiles that are included in same mid-level tile as the first low-level tile. The graphics processor rasterizes each of these low-level tiles that include pixels within the primitive. When the last of these low-level tiles has been rasterized, the graphics processor has completely rasterized the first mid-level tile. 
   When rasterization of the first mid-level tile is complete, the graphics processor moves left-to-right, top-to-bottom through the remaining mid-level tiles that are included in same high-level tile as the first mid-level tile. The graphics processor rasterizes each of these mid-level tiles that include pixels within the primitive by repeating the method used to rasterize the first mid-level tile (i.e., by rasterizing their component low-level tiles). When the last of these mid-level tiles has been rasterized, the graphics processor has completely rasterized the first high-level tile. 
   When rasterization of the first high-level tile is complete, the graphics processor moves left-to-right, top-to-bottom through the remaining high-level tiles that span the primitive. The graphics processor rasterizes each of these high-level tiles by repeating the method used to rasterize the first high-level tile (i.e., by rasterizing their component low-level tiles which are rasterized, in turn, by rasterizing their component low-level tiles). When the last of these high-level tiles has been rasterized, the graphics processor has completely rasterized the primitive. 
   Effectively, the primitive is rasterized in a bottom-up fashion. The graphics processor rasterizes low-level tiles, mid-level tiles and high-level tiles, completing rasterization at each level before moving up the hierarchy. The use of the tile hierarchy increases the temporal locality of accesses within a given memory tile. Increasing temporal locality reduces between tile access. For frame buffers that support fast tile-based access, this enhances graphics throughput. The increased temporal locality of accesses within a given memory tile may also enhance cache memory performance. This is particularly true in cases where cache memory/frame buffer interaction is performed on a tile-by-tile basis. 
   Advantages of the invention will be set forth, in part, in the description that follows and, in part, will be understood by those skilled in the art from the description herein. The advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the appended claims and equivalents. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The accompanying drawings, that are incorporated in and constitute a part of this specification, illustrate several embodiments of the invention and, together with the description, serve to explain the principles of the invention. 
       FIG. 1  is a block diagram of a host computer system shown as an exemplary environment for an embodiment of the present invention. 
       FIG. 2  is a block diagram of a frame buffer in accordance with an embodiment of the present invention. 
       FIG. 3  is a block diagram of a memory tile in accordance with an embodiment of the present invention. 
       FIG. 4  is a block diagram of an exemplary graphics primitive overlaying a frame buffer to further describe an embodiment of the present invention. 
       FIG. 5  is a block diagram showing the value of an edge function computed for each of the memory locations in a low-level tile. 
       FIG. 6  is a block diagram of a rasterization apparatus in accordance with an embodiment of the present invention. 
       FIG. 7  is a block diagram of a edge evaluator in accordance with an embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Reference will now be made in detail to preferred embodiments of the invention, examples of which are illustrated in the accompanying drawings. Wherever convenient, the same reference numbers will be used throughout the drawings to refer to the same or like parts. 
   Environment 
   In  FIG. 1 , a host computer system  100  is shown as a representative environment for the present invention. Structurally, host computer system  100  includes a host processor, or host processors, of which host processors  102   a  through  102   d  are representative. Host processors  102  represent a wide range of commercially available or proprietary types. Host computer system  100  may include either more or fewer host processors  102  than the four shown for the representative environment of host computer system  100 . 
   Host processors  102  are connected to a sequence of components beginning with a memory request unit  104  followed by a memory controller  106 . Memory controller  106  is followed by a system memory  108 . Host processors  102  use this sequence of components to access memory locations included in system memory  108 . As part of these accesses, host processors  102  send virtual memory access requests to memory request unit  104 . Memory request unit  104  translates the requests into corresponding physical memory access requests. The physical memory access requests are then passed to memory controller  106 . Memory controller  106  then accesses system memory  108  to perform the requested operations. For the described embodiment, memory controller  106  and system memory  108  support a range of page types, including tiled and linear pages. Memory controller  106  and system memory  108  also support a range of page sizes for both tiled and linear pages. 
   Memory controller  106  also functions as an interface that allows other components to access system memory  108 . In  FIG. 1 , memory controller  106  provides this type of interface to graphics processor  110  and input/output controller  112 . Preferably, graphics processor  110  performs the majority of its processing using the memory included in system memory  108 . This avoids the delays that result if graphics primitives or data are moved from system memory  108  to graphics processor  110 . Input/output controller  112  functions as a channel allowing host computer system  100  to be connected to a wide range of input/output devices, such as disk drives, non-volatile storage systems, keyboards, modems, network adapters, and printers. 
   As mentioned, host computer system  100  is shown as a representative environment for the present invention. Additional details of this representative environment are discussed in U.S. Pat. No. 6,104,417, entitled “A Unified Memory Computer Architecture With Dynamic Graphics Memory Allocation” of Michael J. K. Nielsen and Zahid S. Hussain. It should be appreciated, however, that the present invention is equally applicable to a range of computer systems and related devices and is not limited to the representative environment of host computer system  100 . 
   Graphics processor  110  uses one or more frame buffers of the type shown in  FIG. 2  and generally designated  200 . Frame buffer  200  is a random access memory device and includes a series of memory locations of which memory locations  202   a ,  202   b  and  202   c  are representative. Each memory location  202  corresponds to a single pixel included in an output device, such a monitor or video display. Memory locations  202  are arranged into a series of rows and columns. For the specific embodiment shown in  FIG. 2 , 1024 rows and 1280 columns are included. This corresponds to a monitor having 1024 rows and 1280 columns of pixels. Each memory location  202  includes a series of bits with the number and distribution of bits being implementation dependent. For the purpose of description, it may be assumed that each memory location  202  includes four eight bit bytes. Three of these bytes define red, blue and green intensities, respectively. The fourth byte included in each memory location  202 , is referred to as alpha and defines the pixel&#39;s coverage or transparencies. 
   Frame buffer  200  is typically fabricated using an array of memory components. These components may be selected from appropriate DRAM types, including VRAM and SDRAM types. For the specific embodiment of host computer system  100 , frame buffer  200  is dynamically allocated within system memory  108 . In other architectures, frame buffer  200  may be included within other suitable locations, such as graphics processor  110 . 
   Frame buffer  200  preferably includes a series of memory tiles of which memory tiles  204   a  and  204   b  are representative. Each memory tile  204  includes a series of memory locations  202  arranged as a rectangle. The size of memory tiles  204  is largely implementation dependent. Thus, frame buffer  200  may be configured to include large or small memory tiles  204 . The dimensions of memory tiles  204  are also largely implementation dependent. Thus, frame buffer  200  may include tall or wide memory tiles  204 . Even more generally, some implementations may allow frame buffer  200  to include a mixture of memory tiles  204  having a range of sizes and dimensions. For the specific embodiment shown in  FIG. 2 , each memory tile  204  includes a total of two-hundred and fifty-six memory locations  202  arranged as a sixteen-by-sixteen square. 
   Frame buffer  200  preferably uses an addressing scheme where more significant address bits choose a specific memory tile  204  and less significant address bits choose a specific memory location  202  within the selected memory tile  204 . In cases where frame buffer  200  is fabricated using DRAM or DRAM-like memory components it is preferable for each memory tile  204  to map to some portion of DRAM row. Thus, each DRAM row includes one or more memory tiles  204 . This allows memory locations within a memory tile  204  to be accessed using a single DRAM row address. For DRAM components that provide some type of fast intra-row accessing mode (such as page mode access) this allows memory locations  202  included within a tile to be rapidly accessed in succession. 
   Tile Hierarchy 
   Within frame buffer  200 , memory tiles  204  represent the highest level in a tile hierarchy. Other levels of this hierarchy are shown more clearly in  FIG. 3  where a memory tile  204  is shown to include four mid-level tiles  300   a  through  300   d . In turn, each mid-level tile  300  includes four low-level tiles  302   a  through  302   d . The overall result is that a three level hierarchy is formed. Within this hierarchy four-by-four low-level tiles  302  are grouped into eight-by-eight mid-level tiles  300  and eight-by-eight mid-level tiles  300  are grouped into sixteen-by-sixteen memory tiles  204 . Other hierarchies, including more or fewer levels, are equally possible. 
   Rasterization Method 
   An embodiment of the present invention provides a method for efficiently rasterizing graphics primitives. The rasterization method is intended to work in combination with a wide range of graphics primitive types, including points, lines and polygons. 
   Graphics processor, 110  begins the process of rasterizing a primitive by selecting one of the primitive&#39;s vertices as a starting vertex. Graphics processor  110  then rasterizes the low-level tile  302  that includes the starting vertex. When rasterization of the first low-level tile  302  is complete, graphics processor  110  moves left-to-right, top-to-bottom through the remaining low-level tiles  302  that are included in same mid-level tile  300  as the first low-level tile  302 . Graphics processor  110  rasterizes each of these low-level tiles  302  that include pixels within the primitive. When the last of these low-level tiles  302  has been rasterized, graphics processor  110  has completely rasterized the first mid-level tile  300 . 
   When rasterization of the first mid-level tile  300  is complete, graphics processor  110  moves left-to-right, top-to-bottom through the remaining mid-level tiles  300  that are included in same memory tile  204  as the first mid-level tile  300 . Graphics processor  110  rasterizes each of these mid-level tiles  300  that include pixels within the primitive by repeating the method used to rasterize the first mid-level tile  300  (i.e., by rasterizing their component low-level tiles  302 ). When the last of these mid-level tiles  300  has been rasterized, graphics processor  110  has completely rasterized the first memory tile. 
   When rasterization of the first memory tile  204  is complete, graphics processor  110  moves left-to-right, top-to-bottom through the remaining memory tiles  204  that span the primitive. Graphics processor  110  rasterizes each of these memory tiles  204  by repeating the method used to rasterize the first memory tile  204  (i.e., by rasterizing their component low-level tiles  302  which are rasterized, in turn, by rasterizing their component low-level tiles  302 ). When the last of these memory tiles  204  has been rasterized, graphics processor  110  has completely rasterized the primitive. 
   To better describe the rasterization method,  FIG. 4  shows an exemplary primitive  400  overlaying a portion of frame buffer  200 . Primitive  400  is a triangular polygon. This particular shape is chosen to be representative of primitives in general, with the understanding that the present invention is equally amenable to other primitive shapes and types. As shown in  FIG. 4 , primitive  400  is spanned by two memory tiles  204   a  and  204   b.    
   To begin rasterizing primitive  400 , graphics processor  110  selects a starting vertex from the vertices of primitive  400 . In general, the choice of vertex is somewhat arbitrary—meaning that the present invention may be adapted to initiate rasterization at any given point. To simplify the following description it is assumed however, that graphics processor  110  selects the upper left vertex of primitive  400  as the starting vertex. 
   After selecting the starting vertex, graphics processor  110  rasterizes the pixels in low-level tile  302  marked  1 . Rasterization starts at this location because low-level tile  302 - 1  includes the starting vertex. After rasterizing low-level tile  302 - 1 , graphics processor  110  moves left-to-right, top-to bottom within the mid-level tile  300  that includes the low-level tile  302 - 1 . Graphics processor  110  rasterizes each low-level tile  302  within this mid-level tile that includes pixels in primitive  400 . Specifically, graphics processor  110  moves right and rasterizes low-level tile  302 - 2 , and down to rasterize low-level tile  302 - 3 . 
   At this point, graphics processor  110  has completely rasterized the first mid-level tile  300  (the final low-level tile  302  included within this mid-level tile  300  is completely outside of the boundaries of primitive  400 ). To continue the rasterization process, graphics processor  110  jumps to low-level tile  302 - 4  in the next mid-level tile  300 . Graphics processor  110  selects mid-level tiles  300  using the same left-to-right, top-to-bottom pattern used to traverse low level tiles  302 . After rasterizing low-level tile  302 - 4 , graphics processor  110  moves left-to-right, top-to-bottom within the mid-level tile  300  that includes the low-level tile  302 - 4 . Specifically, graphics processor  110  moves right and rasterizes low-level tile  302 - 5 , down and left to rasterize low-level tile  302 - 6 ,and right to rasterize low-level tile  302 - 7 . 
   At this point, graphics processor  110  has completely rasterized the first memory tile  204   a  (the remaining mid-level tiles  302  and their included low-level tiles  302  are completely outside of the boundaries of primitive  400 ). To continue the rasterization process, graphics processor  110  jumps to low-level tile  302 - 8  in the next memory tile  204   b . Graphics processor  110  selects memory tiles  204  using the same left-to-right, top-to-bottom pattern used to traverse mid-level tiles  300  and low level tiles  302 . After rasterizing low-level tile  302 - 8 , graphics processor  110  moves left-to-right, top-to-bottom within the mid-level tile  300  that includes the low-level tile  302 - 8 . Specifically, graphics processor  110  moves down and rasterizes low-level tile  302 - 9 . By rasterizing low-level tile  302 - 9 , graphics processor  110  completes rasterization of primitive  400 . 
   In the preceding description, graphics processor  110  selects memory tiles  204 , mid-level tiles  300  and low-level tiles  302  using a left-to-right, top-to-bottom traversal. In general, it should be appreciated that this particular pattern of traversal is only one of many possible patterns. In fact, the present invention may be adapted for use with any pattern that ensures that rasterization is completed at each lower level before proceeding to higher hierarchical levels. It should also be apparent that different patterns of traversal may be used at different hierarchical levels. Thus, graphics processor  110  may traverse memory tiles  204  using a first pattern of traversal, mid-level tiles  300  using a second pattern of traversal and low-level tiles  302  using a third pattern of traversal. 
   The preceding description also assumes that graphics processor  110  modifies the pattern of traversal to exclude memory tiles  204 , mid-level tiles  300  and low-level tiles  302  that fall entirely outside of a primitive being rasterized. To accomplish this modification, graphics processor  110  is preferably configured to include a lookahead mechanism. The lookahead mechanism determines, as the graphics processor  110  is rasterizing a given low-level tile  302 , which low-level tile should be rasterized next. The lookahead mechanism is preferably configured to ignore memory tiles  204 , mid-level tiles  300  and low-level tiles  302  that fall entirely outside of a primitive being rasterized. It should be appreciated however, that this type of mechanism, while preferable, is not required. Thus, graphics processor  110  may be configured to exhaustively traverse low-level tiles  302  within mid-level tiles  300  or mid-level tiles  300  within memory tiles  204 . 
   Graphics processor  110  uses the tile hierarchy to control the order in which low-level tiles  302  are selected during rasterization of graphics primitives. To maximize the efficiency of this ordering, graphics processor  110  is preferably configured to rasterize the sixteen memory locations  202  within a selected low-level tile  302  in a concurrent, or nearly concurrent fashion. For the described embodiment, graphics processor  110  achieves this concurrency by defining each edge of each primitive using a linear expression of the form: F(x,y)=Ax+By+C. Use of these equations means that all points on one side of an edge have F(x,y)≧0. All points on the other side of the same edge have F(x,y)≦0. To rasterize a low-level tile  302  for a given primitive, graphics processor  110  calculates each of the primitive&#39;s edge functions for each memory location  202  within the low-level tile  302 . For example, for a triangular primitive bounded by edges F(x,y), F′(x,y) and F″(x,y), graphics processor  110  would calculate each of these equations for each memory location  202  within the low-level tile  302  being rasterized. Graphics processor  110  determines that a memory location  202  is within a triangular primitive if an odd number of the primitive&#39;s edge functions are less than zero at the memory location  202 . 
   Graphics processor  110  preferably uses an additive process to evaluate edge functions for all of the memory locations  202  of a low-level tile  302  in a concurrent, or nearly concurrent, fashion. The additive process may be better understood by reference to  FIG. 5 .  FIG. 5  shows the values calculated by graphics process  110  for the memory locations  202  included in a low-level tile  302 . As shown, graphics processor  110  calculates the value F(x,y) for memory location  202   a  located at the lower, left hand corner of low-level tile  302 . Graphics processor  110  calculates the value F(x,y)+A for memory location  202   b  located one location to the right of memory location  202   a , F(x,y)+2A for memory location  202   c  located two locations to the right of memory location  202   a , and so on. Effectively, graphics processor  110  calculates edge functions for each memory location  202  to the right of memory location  202   a  by adding multiples of the constant A to the edge function calculated for memory location  202   a . In a similar fashion, graphics processor  110  calculates edge functions for each memory location  202  above memory location  202   a  by adding multiples of the constant B to the edge function calculated for memory location  202   a . Memory locations that are both to the right of, and above, memory location  202   a  have values calculated by adding appropriate multiples of A and B. The overall result is that graphics processor  110  need only calculate F(x,y), F′(x,y) and F″(x,y) once per low-level tile  302 . The calculated values are then extrapolated using a series of additions to all of the memory locations included in the low-level tile  302 . 
   Apparatus 
   The previously described methods are adaptable for use in a wide range of hardware and software environments. Typically, however, these methods are most efficient when they are fully or partially implemented within a specialized rendering apparatus. An apparatus of this type is shown in  FIG. 6  and generally designated  600 . 
   Rendering apparatus  600  includes a set of three edge evaluators  602   a  through  600   c . Each edge evaluator is connected by an input and control bus  604  to the remaining logic of graphics processor  110 . Each edge evaluator  602  is also connected to a respective adder tree  606   a  through  606   c . Adder trees  606  are connected, in turn, to an and gate  608 . The output of and gate  608  is connected to a fragment selection unit  610 . 
   Each edge evaluator  602  is configured to accept a set of parameters that characterize a linear equation of the form F(x,y)=Ax+By+C from graphics processor  110 . The parameters include an initial value for the equation and appropriate values for A and B. Graphics processor  110  sends these parameters to edge evaluators  602  using input and control bus  604 . Once initialized, edge evaluators  602  are configured to compute successive values for their associated edge equation. Edge evaluators  602  compute these values by adding A or B to their initial values as appropriate. 
   Before rasterizing a given primitive, graphics processor  110  computes initial values for each of the edge functions that describe the primitive. Graphics processor  110  computes these initial values using the x and y coordinates of the first memory location  204  within the initial low-level tile  302  that will be rasterized (i.e., the low-level tile that includes the starting vertex). Graphics processor  110  then initializes edge evaluators  602  to include the initial values and appropriate values for A and B. 
   Once initialization is complete, edge evaluators  602  output the value of their associated edge functions (i.e., the initial values computed for the first memory location  204  within the initial low-level tile  302  that will be rasterized). These output of each edge evaluator  602  is passed to a respective adder tree  606 . Each adder tree  606  performs a series of additions to create a set of sixteen output values. The output values are equivalent to the values shown in  FIG. 5 . In this way, each adder tree  606  re-computes the value it received from its associated edge evaluator for each x and y location within the low-level memory tile  302  being rasterized. 
   And gate  608  combines the three sets of sixteen values produced by the three adder trees  606 . The result is a single set of sixteen values. The single set of output values shows which memory locations  204  within the low-level tile  302  being rasterized are included within the primitive. The set of sixteen output values are passed to fragment selection unit  610 . 
   To continue the rasterization process, graphics processor  110  repeatedly directs edge evaluators  602  to reevaluate their output functions to reflect movement of the rasterization process to additional low-level tiles  302 . For each additional low-level tile  302 , adder trees  606  apply the reevaluated function to each of the memory locations  204  within the low-level tile  302  being rasterized. And gate  608  combines the values produced by adder trees  606  to produce unified sets of values showing the memory locations  204  that are included in the primitive being rasterized. 
   Details of edge evaluators  602  are better appreciated by reference to  FIG. 7 . In  FIG. 7 , it may, be seen that edge evaluator  602  includes A register  700  and B register  702 . These registers are used to store values for A and B, respectively. Edge evaluator  602  also includes X save registers  704  and Y save registers  706 . As will be described in more detail, these registers are used to store checkpointed output values of edge evaluator  602  at specific times during the rasterization process. X save registers  704  and Y save registers  706  are register sets. Each set includes one register for each level in the tile hierarchy being used. For the described embodiment, this means that there are three registers in both X save registers  704  and Y save registers  706 . Edge evaluator  602  also includes a current register  708 . Current register  708  it used to store the current value of the edge function associated with edge evaluator  602  (i.e., the current value of F(x,y)=Ax+By+C). 
   The outputs of A register  700  and B register  702  are connected to the data inputs of a step direction multiplexer  710 . The control input of step direction multiplexer  710  is connected to input and control bus  604 . This allows graphics processor  110  to select the output of step direction multiplexer  710  as either the output of A register  700  or B register  702 . The output of step direction multiplexer  710  is connected to a first input of an adder  712 . 
   The outputs of X save registers  704 , Y save registers  706  and current register  708  are connected to the data inputs of a current/restore multiplexer  714 . The control input of current/restore multiplexer  714  is connected to input and control bus  604 . This allows graphics processor  110  to select the output of current/restore multiplexer  714  as either the output of X save registers  704 , Y save registers  706  or current register  708 . The output of current/restore multiplexer  714  is connected to a second input of adder  712 . 
   The output of adder  712  is connected to a first data input of an initialization multiplexer  716 . The second data input of initialization multiplexer and the control input of data initialization multiplexer  716  are connected to input and control bus  604 . This allows graphics processor  110  to select the output of initialization multiplexer  716  as either the output of adder  712  or a value specified by graphics processor  110 . 
   The output of adder  712  is also connected to the inputs of X save registers  704  and Y save registers  706 . Write enable inputs for X save registers  704  and Y save registers  706  are connected to input and control bus  604 . This allows graphics processor  110  to selectively save the output of select the output of adder  712  in either X save registers  704  or Y save registers  706 . 
   The inputs of A register  700  and B register  702  are connected to input and control bus  604 . This allows graphics processor  110  to initialize A register  700  and B register  702  to include values for A and B, respectively. 
   To initialize edge evaluator  602 , graphics processor  110  computes an initial value for the edge function F(x,y)=Ax+By+C. As discussed, graphics processor  110  computes this initial value using the x and y coordinates of the first memory location  204  within the initial low-level tile  302  to be rasterized (i.e., the low-level tile that includes the starting vertex). Graphics processor  110  then uses input and control bus  604  to store the initial value in current register  708 . Graphics processor  110  also uses input and control bus  604  to store the values A and B in A register  700  and B register  702 , respectively. At the completion of initialization, the output of edge evaluator  602  is the initial value for the edge function computed by graphics processor  110 . 
   To continue the rasterization process, graphics processor  110  uses input and control bus  604  to cause step direction multiplexer  710  to select A register  700  or B register  702 . A register  700  is selected to cause edge evaluator  602  to reevaluate the initial value in current register  708  by adding A or B. The reevaluated value is stored in current register  708  and becomes the current output of edge detector  602 . Effectively, by selecting A register  700  or B register  702  and reevaluating the initial value, graphics processor  110  causes edge evaluator  602  to move the rasterization process one by low-level tile  302 . The movement may be left-to-right (when A register  700  is selected) or top-to-bottom (when B register  702  is selected) 
   CONCLUSION 
   The use of the tile hierarchy ensures that rasterization within a given memory tile  204  is completed before rasterization within another memory tile  204  is initiated. This increases the temporal locality of accesses within memory tiles  204  during the rasterization process. For frame buffers that support fast tile-based access, this enhances graphics throughput. The increased temporal locality of accesses within a given memory tile  204  may also enhance cache memory performance. This is particularly true in cases where cache memory/frame buffer interaction is performed on a tile-by-tile basis. In this way, the present invention provides an efficient method for rasterizing graphics primitives that fully exploits the use of memory tiling within frame buffers. 
   Other embodiments will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with a true scope of the invention being indicated by the following claims and equivalents.