Patent Publication Number: US-7593010-B2

Title: Software-implemented transform and lighting module and pipeline for graphics rendering on embedded platforms using a fixed-point normalized homogenous coordinate system

Description:
TECHNICAL FIELD 
   The present invention relates in general to graphics rendering systems and more particularly to a software-implemented transform and lighting module and pipeline designed and optimized for embedded platforms (such as mobile computing devices) that is based on fixed-point operations including a variable-length fixed point representation for numbers and a normalized homogenous coordinates system for vector operations. 
   BACKGROUND OF THE INVENTION 
   The importance of three-dimensional (3D) enabled embedded platforms has become increasingly important due to users&#39; expectations of multimedia-rich environments in products ranging from DVD players, set-top boxes, Web pads and mobile computing device (including handheld computing devices) to navigational equipment and medical instrumentation. As users continue to expect equal or nearly equal graphics quality on embedded devices as on their desktop systems, applications designed to run on embedded platforms continue to converge with their desktop equivalents. Thus, the need for 3D graphics rendering is vital in today&#39;s embedded systems. 
   One of the more popular 3D rendering standards available today is Direct3D by Microsoft® Corporation. Direct 3D is an application-programming interface (API) for manipulating and displaying 3D objects. Direct3D provide programmers and developers with a way to develop 3D applications that can utilize whatever graphics acceleration hardware is installed on the system. Direct3D does an excellent job in supporting efficient rendering in desktop applications. These desktop systems typically have powerful central processing units (CPUs), math coprocessors, and graphics processing units (GPUs). 
   Typical graphic rendering standards (such as Direct3D) designed for desktop systems use floating-point operations for the transform and lighting process. In embedded systems, however, the CPUs may not be powerful enough to support floating-point operations and there is typically no coprocessor or GPU for accelerating the floating-point operations. Thus, software-implemented transform and lighting is important and useful for use on embedded platforms. 
   Current software-implemented transform and lighting (T&amp;L) pipelines are based on floating-point operations. These pipelines assume that powerful graphics hardware and processors are available. However, these current T&amp;L processing pipelines based on floating-point operations are based on execution by a GPU (such as is available on a desktop system) instead of only a CPU (as typically is available in an embedded platform). Moreover, floating-point software routines are notoriously slow, expensive, require large amounts of memory, and have a large code size. Therefore, there exists a need for a software-implemented T&amp;L pipeline that is optimized for operation on an embedded platform and does not require powerful hardware and processing power. There is also so need for a software-implemented T&amp;L pipeline that is fast, efficient, requires little memory and has a small code size such that it is ideal for embedded platforms. 
   SUMMARY OF THE INVENTION 
   The invention disclosed herein includes a transform and lighting module and method (or pipeline) that is optimized for use on embedded platforms (such as mobile computing devices). The transform and lighting module and pipeline include of number of features that are designed specifically to increase efficiency and performance on embedded devices. Specifically, current graphics standards are designed for desktop systems having math coprocessors and graphics processing units (GPU) in addition to a central processing unit (CPU). In addition, the CPUs of these desktop systems typically run several operations (or streamlines) at once and are highly parallelized. However, embedded devices often lack coprocessors and GPUs, and have CPUs that run a single streamline and a not designed for parallel operations. 
   The transform and lighting module and pipeline includes a single streamline branched architecture that allows efficient processing on a CPU of an embedded device and saves computational time. This architecture is facilitated by use of a vertex cache that stores vertices as needed to avoid duplication in processing of the vertices. For example, if some input vertices have already been lit, then those vertices are sent to the vertex cache to avoid the lighting processing and thereby avoid duplicate processing. The vertex cache is implemented in software and not hardware. This alleviates the need for additional hardware that is at a premium in an embedded device. 
   The transform and lighting module and pipeline also improve efficiency by use of a culling module. The culling module is positioned before the lighting module to examine each vertex and determine whether to cull or keep the vertex. Culling is performed using back face culling and view frustum culling. By discarding the vertices that do not need to be lit or are not needed, the culling module decreases the number of vertices being processed by the lighting module. Since the lighting module is the most computationally intensive module of the transform and lighting module, this improves processing efficiency. 
   In general, the transform and lighting module inputs data in 3D, processes the data, and outputs the data in 2D screen coordinates. The transform and lighting module includes the transformation module that converts the input rendering data from model space to clip space. The vertex cache, which is software implemented, is used to store the vertices as needed. The culling module examines each vertex to determine whether it should be culled. If so, then it is discarded. Otherwise, the processing continues on the vertex. 
   Culling is performed in at least one of two ways. First, a back face culling technique examines each of the vertices to determines whether a back face of a triangle is formed. If so, then the vertex is culled. A second technique involved determining whether a vertex is outside of one view frustum clip plane. If so, then the vertex is culled. 
   The transform and lighting module also includes a lighting module, for computing color from the vertices, and a transformation generation and transformation module, for computing clip space coordinates. The transform and lighting module also includes view frustum clipping module that involves interpolation of the color and the texture of the vertices. The view frustum clipping module is designed for normalized homogenous coordinate system (NHCS) fixed-point operations and processes the clip space coordinates. The view frustum clipping module works with both vertex transformed by the texture generation and transformation module and on vertex transformed by an application outside of the transform and lighting module. 
   The transform and lighting pipeline includes inputting rendering data containing vertices. The rendering data is in model space. The rendering data then is transformed from model space into clip space. Each of the vertices then are examined prior to lighting to determine whether to cull the vertex. Culled vertices are discarded and the remainder is kept. The vertices are stored in a vertex cache as needed. The vertex cache is software-implemented and facilitates a single streamline branched architecture. This architecture avoids duplication in processing of the vertices. View frustum clipping is performed on clip space coordinates after lighting and texture generation and transformation of their vertices. Preferably, the clip space coordinates are NHCS clip space coordinates. The output of the transform and lighting pipeline are 2D screen coordinates. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention can be further understood by reference to the following description and attached drawings that illustrate aspects of the invention. Other features and advantages will be apparent from the following detailed description of the invention, taken in conjunction with the accompanying drawings, which illustrate, by way of example, the principles of the present invention. 
     Referring now to the drawings in which like reference numbers represent corresponding parts throughout: 
       FIG. 1  is a block diagram illustrating a general overview of the NHCS graphics rendering system including a transform and lighting module that resides on the driver module. 
       FIG. 2  illustrates an example of a suitable computing system environment in which the transform and lighting module and method may be implemented. 
       FIG. 3  is a block diagram illustrating the details of an exemplary implementation of the NHCS graphics rendering system and shown in  FIG. 1 . 
       FIG. 4  is a detailed block diagram illustrating the details of the transform and lighting module shown in  FIG. 3 . 
       FIG. 5  is a general flow diagram illustrating the operation of the single streamline branched pipeline (or method) of the transform and lighting module shown in  FIG. 4 . 
       FIG. 6  is a detailed flow diagram illustrating the operation of the culling module shown in  FIG. 4 . 
       FIG. 7  is a working example of the transform and lighting module and pipeline as is shown for illustrative purposes only. 
       FIG. 8  illustrates an exemplary implementation of a buffer to store culling planes. 
       FIGS. 9A and 9B  illustrate an exemplary implementation of normalized vectors in a D3DM Phong Model. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   In the following description of the invention, reference is made to the accompanying drawings, which form a part thereof, and in which is shown by way of illustration a specific example whereby the invention may be practiced. It is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention. 
   I. General Overview 
   Typical graphic rendering standards (such as Direct3D) designed for desktop systems use floating-point operations for the transform and lighting process. In embedded systems, however, the CPUs may not be powerful enough to support floating-point operations and there is typically no coprocessor or GPU for accelerating the floating-point operations. Thus, software-implemented transform and lighting is important and useful for use on embedded platforms. 
   The transform and lighting module and pipeline disclosed herein is optimized for use on embedded platforms (such as mobile computing devices). The transform and lighting module and pipeline is designed specifically to increase efficiency and performance on embedded devices and does not require powerful processors or graphics hardware. The transform and lighting module and pipeline includes a single streamline branched architecture that allows efficient processing on a CPU of an embedded device and saves computational time. This architecture is facilitated by use of a vertex cache that stores vertices as needed to avoid duplication in processing of the vertices. The transform and lighting (T&amp;L) module and pipeline differ from existing T&amp;L techniques as follows. First, the T&amp;L module and pipeline is a single pipeline, while existing techniques are multiple pipeline. Second, the T&amp;L module and pipeline reduces the lighting processing by culling vertices before lighting, while existing T&amp;L techniques transform and light all vertices. Third, the T&amp;L module and pipeline includes a software-implemented vertex cache that is located in the T&amp;L module, while existing T&amp;L techniques may include a vertex cache between the T&amp;L layer and the rasterizer. These differences make the T&amp;L module and pipeline disclosed herein more efficient and practical for use on embedded devices. 
     FIG. 1  is a block diagram illustrating a general overview of a NHCS graphics rendering system  100  including a transform and lighting module disclosed herein. The system  100  typically resides on a computing device  110 , such as a mobile computing device. In general, the system  100  inputs raw rendering data  120 , processes the data  120  and outputs processed rendering data  130  suitable for rendering by a rendering engine (not shown). The raw rendering data  120  typically is in a floating-point format. 
   As shown in  FIG. 1 , the NHCS graphics rendering system  100  includes a task module  140 , an application program interface (API) module  150 , and a driver module  160 . The task module  140  inputs the raw rendering data  120  in a floating-point format and converts the data  120  into a desired fixed-point format. In some embodiments, the task module  140  is capable of converting the data  120  in a floating-point format into either a traditional fixed-point format or a preferred NHCS fixed-point format. The converted data then is sent to the API module  150 . The API module  150  creates buffers for storing the converted data. In addition, the API module  150  prepares a command buffer for the driver module  160 . The driver module  160  contains the mathematical operation and graphics functions to prepare the data for rendering. The data is in a fixed-point format (preferably a NHCS fixed-point format) and the mathematical operation and graphics functions are specially created to process the fixed-point data. As explained in detail below, the transform and lighting module is contained in the driver module  160 . The output is the processed rendering data  130  that is ready to be rendered by a rendering engine. 
   II. Exemplary Operating Environment 
   The transform and lighting module and method disclosed herein and contained within the NHCS graphics rendering system  100  is designed to operate in a computing environment. The following discussion is intended to provide a brief, general description of a suitable computing environment in which the transform and lighting module and method may be implemented. 
     FIG. 2  illustrates an example of a suitable computing system environment  200  in which the transform and lighting module and method may be implemented. The computing system environment  200  is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention. Neither should the computing environment  200  be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in the exemplary operating environment  200 . 
   The transform and lighting module and method are operational with numerous other general purpose or special purpose computing system environments or configurations. Examples of well known computing systems, environments, and/or configurations that may be suitable for use with the transform and lighting module and method include, but are not limited to, personal computers, server computers, hand-held, laptop or mobile computer or communications devices such as cell phones and PDA&#39;s, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, distributed computing environments that include any of the above systems or devices, and the like. 
   The transform and lighting module and method may be described in the general context of computer-executable instructions, such as program modules, being executed by a computer. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote computer storage media including memory storage devices. With reference to  FIG. 2 , an exemplary system for implementing the transform and lighting module and method contained on the NHCS graphics rendering system  100  includes a general-purpose computing device in the form of a computer  210  (the computer  210  is an example of the computing device  110  shown in  FIG. 1 ). 
   Components of the computer  210  may include, but are not limited to, a processing unit  220 , a system memory  230 , and a system bus  221  that couples various system components including the system memory to the processing unit  220 . The system bus  221  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. By way of example, and not limitation, such architectures include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus. 
   The computer  210  typically includes a variety of computer readable media. Computer readable media can be any available media that can be accessed by the computer  210  and includes both volatile and nonvolatile media, removable and non-removable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes volatile and nonvolatile removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. 
   Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by the computer  210 . Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. 
   Note that the term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of any of the above should also be included within the scope of computer readable media. 
   The system memory  230  includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM)  231  and random access memory (RAM)  232 . A basic input/output system  233  (BIOS), containing the basic routines that help to transfer information between elements within the computer  210 , such as during start-up, is typically stored in ROM  231 . RAM  232  typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit  220 . By way of example, and not limitation,  FIG. 2  illustrates operating system  234 , application programs  235 , other program modules  236 , and program data  237 . 
   The computer  210  may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only,  FIG. 2  illustrates a hard disk drive  241  that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive  251  that reads from or writes to a removable, nonvolatile magnetic disk  252 , and an optical disk drive  255  that reads from or writes to a removable, nonvolatile optical disk  256  such as a CD ROM or other optical media. 
   Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The hard disk drive  241  is typically connected to the system bus  221  through a non-removable memory interface such as interface  240 , and magnetic disk drive  251  and optical disk drive  255  are typically connected to the system bus  221  by a removable memory interface, such as interface  250 . 
   The drives and their associated computer storage media discussed above and illustrated in  FIG. 2 , provide storage of computer readable instructions, data structures, program modules and other data for the computer  210 . In  FIG. 2 , for example, hard disk drive  241  is illustrated as storing operating system  244 , application programs  245 , other program modules  246 , and program data  247 . Note that these components can either be the same as or different from operating system  234 , application programs  235 , other program modules  236 , and program data  237 . Operating system  244 , application programs  245 , other program modules  246 , and program data  247  are given different numbers here to illustrate that, at a minimum, they are different copies. A user may enter commands and information into the computer  210  through input devices such as a keyboard  262  and pointing device  261 , commonly referred to as a mouse, trackball or touch pad. 
   Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner, radio receiver, or a television or broadcast video receiver, or the like. These and other input devices are often connected to the processing unit  220  through a user input interface  260  that is coupled to the system bus  221 , but may be connected by other interface and bus structures, such as, for example, a parallel port, game port or a universal serial bus (USB). A monitor  291  or other type of display device is also connected to the system bus  221  via an interface, such as a video interface  290 . In addition to the monitor, computers may also include other peripheral output devices such as speakers  297  and printer  296 , which may be connected through an output peripheral interface  295 . 
   The computer  210  may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer  280 . The remote computer  280  may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer  210 , although only a memory storage device  281  has been illustrated in  FIG. 2 . The logical connections depicted in  FIG. 2  include a local area network (LAN)  271  and a wide area network (WAN)  273 , but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet. 
   When used in a LAN networking environment, the computer  210  is connected to the LAN  271  through a network interface or adapter  270 . When used in a WAN networking environment, the computer  210  typically includes a modem  272  or other means for establishing communications over the WAN  273 , such as the Internet. The modem  272 , which may be internal or external, may be connected to the system bus  221  via the user input interface  260 , or other appropriate mechanism. In a networked environment, program modules depicted relative to the computer  210 , or portions thereof, may be stored in the remote memory storage device. By way of example, and not limitation,  FIG. 2  illustrates remote application programs  285  as residing on memory device  281 . It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used. 
   III. System Components 
     FIG. 3  is a block diagram illustrating the details of an exemplary implementation of the NHCS graphics rendering system  100  shown in  FIG. 1 . In this exemplary implementation, the NHCS graphics rendering system  100  is implemented in a Direct3D mobile environment. Microsoft® Corporation in Redmond, Wash., developed Direct3D (D3D) and it has become a rendering standard. Traditionally, D3D supports efficient rendering in desktop personal computer (PC) applications. These PCs typically have powerful CPUs and GPUs and can support intensive graphics rendering. For other embedded devices, such as mobile computing devices, D3D dos not fit because it needs powerful processing units. The NHCS graphics rendering system and method disclosed herein enables the use of D3D on mobile computing devices (D3DM). The NHCS graphics rendering system and method includes a software-based transform and lighting module having a fixed-point mathematical library and corresponding graphics functions. The mathematical library is optimized for use on mobile computing devices and makes efficient use of the limited resources available on mobile computing devices. 
   The basic structure of D3DM is that there is a “thin” API module and a “thick” driver module. In the thin API module, the interface is simple and straightforward. Thus, the API code provides integration with the operating system and hosting for the display driver, but does not provide any actual drawing code. In the thick driver module, most of the work is forwarded by the API module and performed in the driver module. Thus, the thick driver module includes drawing code, which may only be overridden by the display driver. 
   The design of D3DM is based on the fact that models can be described in terms of primitives. In turn, each primitive is described in terms of a plurality of vertexes (or vertices). A vertex is the point at which two lines meet. The vertex carries a great of information. For example, the vertex contains the 3-D coordinates and weight. In addition, there is color information, often specified in the form of a diffuse and a specular color. This color data is commonly coded in the “RGBA” format (for red, green, blue and alpha). The vertex also contains a normal, the vector that is orthogonal to its surface, and the texture coordinates that represent the texture and its position for the vertex. The vertex may have several texture coordinates in case more than one texture is applied to the vertex. Further, the vertex may contain texture fog as well as other information such as point size. Thus, the vertex, the smallest unit in a 3-D scene, contains a large amount of information. 
   D3DM loads this data for the vertex into vertex buffers. The data then is processed by the transform and lighting (T&amp;L) module where the output is pixel color values for a frame buffer. The transform and lighting module and method contain a mathematical library that is used by D3DM. The mathematical library is used to implement the transform and lighting. The D3DM drivers expose all of the features of the mobile computing device on which an application is running, thus achieving maximum drawing performance. 
   Referring to  FIG. 3 , the task module  140  includes a math library and translator  300 , an application  305 , and floating-point data  310 . In general, the task module  140  inputs the floating-point data  310  and converts the data  310  into a fixed-point format or a NHCS fixed-point format. The converted data then is sent to buffers created by the API module  150 . The math library and translator  300  converts the data  310  and performs preliminary mathematical operations on the converted data. In addition, the math library and translator  300  defines a specific data structure for the converted data. The preliminary mathematical operations and data structure definitions are discussed in detail below. 
   The API module  150  creates buffers for storing the converted data and preparing the data for the driver module  160 . The API module  150  includes an index buffer  315 , for storing indices, and a vertex buffer  320 , for storing vertex information. The index buffer holds a value for each vertex. The value is called an index. Indices are used to retrieve a vertex in the vertex buffer. Each index is an offset in the current vertex buffer of the data for this vertex. This allows for the sharing of vertex data between multiple vertices and avoids the duplicated storage of vertices when two neighboring triangles share vertices. The API module  150  also includes commands  325  that provide instructions for the rendering and texture  330  that provides texture information. The API module  150  includes a wrapper  335  that packages the commands  325  and provides convenience, compatibility and security for the commands  325 . This ensures the that the commands  325  are ready for the driver module  160 . A command buffer  340  stores the wrapper  335  prior to them being sent to the driver module  160 . 
   The driver module  160  prepares data for the raster. In addition, the driver module  160  prepares the data for use by a rendering engine. This means that the data is translated into the language of the computing device&#39;s graphics hardware and causes particular primitives to be drawn. The driver module  160  includes the transform and lighting (T&amp;L) module  345  and a rasterizer  350 . The T&amp;L module  345  includes all necessary mathematical operations and graphic functions in the NHCS fixed-point data format. The T&amp;L module  345  is discussed in detail below. The rasterizer prepares the rendering data to be sent to the raster. 
   IV. Transform and Lighting Module 
   In general, the transform and lighting module  345  is a single streamline branched pipeline that takes input data in 3D and processes the input data into output data containing 2D screen coordinates. The single streamline branched pipeline is particularly well-suited for less powerful CPUs, such as those CPUs contained on embedded devices. The single streamline branched pipeline architecture of the transform and lighting module  345  saves computational time and greatly increases efficiency. Transform and lighting processing in desktops and other systems having graphics processing unit (GPU) is quite different from the single streamline branched pipeline T&amp;L processing using a low-power CPU. With a GPU, T&amp;L processing is highly parallelized (i.e., multiple streamlines) and branches are not necessary. In fact, branches can stall the streamlines and slow down the processing and computation. 
     FIG. 4  is a detailed block diagram illustrating the details of the transform and lighting module  345  shown in  FIG. 3 . The transform and lighting module  345  receives input data  400  that is 3D data. An index buffer  405  stores indices of the input data  400  and a vertex buffer  410  stores vertex information of the input data  400 . A determination is made as to whether the vertex information of the input data  400  has been previously transformed. If the vertex information has been previously transformed, vertex information is sent to a vertex cache  415 . If the vertex information has not been previously transformed, a transformation module  420  performs the transformation on the vertex information. The transformation module  420  transforms the vertex information from module space into clip space. This transformation is performed using NHCS fixed-point matrix operations as detailed below. 
   The vertex cache  415  is an important part of the transform and lighting module  345 . The vertex cache  415  makes possible the single streamline branched pipeline architecture by providing a location for vertex information and other data to be stored while other module are processing. In addition, the vertex cache  415  provides storage for vertex information and data that has already been processed by a certain module. In at least one embodiment, Direct3D mobile (D3DM) is implemented in the NHCS graphics rendering device  100  and the vertex cache  415  is software implemented into D3DM. On the other hand, traditional Direct3D implementations use hardware to store vertex information. The software-implemented vertex cache  415  saves precious memory and processing power. 
   A culling module  425  is used to perform culling of the transformed vertex information prior to the information being processed by a lighting module  430 . The lighting module is the most computationally intensive module in the transform and lighting module  345 , and positioning the culling module  425  before the lighting module  430  eliminates unnecessary (and computationally expensive) processing by the lighting module  430 . This placement of the culling module  425  before the lighting module  430  reduces the number of vertex needing to be lit by the lighting module and saves both time and processing power. 
   The culling module  425  receives as input the transformed vertex information. If the vertex information was transformed as part of the input data  400 , this information was stored in the vertex cache  415  and is retrieved by the culling module  425  from the vertex cache  415 . On the other hand, if the transformation module  420  performed the transformation, the culling module  425  receives the transformed vertex information directly from the transformation module  420 . 
   As explained in detail below, some vertex may be discarded by the culling module  425  and some vertex may be retained. The vertexes that are retained are processed further by checking the vertex cache  415  to determine whether the vertexes have been previously lit. For vertexes that have been previously lit, they are passed directly to the view frustum clipping module  435 . If the vertex is unlit, the vertex is sent to the lighting module  430 . 
   The lighting module  430  computes color for vertex. As discussed below, there are several functions that the lighting module  430  can use to light the vertex. After the vertex is lit, they are sent to a texture generation and transformation module  440 . The texture generation and transformation module  440  computes texture coordinates and transforms the coordinates after generation. 
   It should be noted that the placement of the culling module  425  in the transform and lighting pipeline is important. In D3D, culling is performed after lighting and after texture generation and transformation. In the transform and lighting pipeline of disclosed herein, the culling module  425  is implemented in D3DM and is positioned before the lighting module  430  and before the texture generation and transformation module  440 . This placement saves processing power and computational time because the lighting module  430  is the most computationally intensive part of the transform and lighting module  345 . 
   The view frustum clipping module receives data either from the vertex cache  415  or directly from the texture generation and transformation module  440 . View frustum clipping is applied to vertex after lighting and texture generation and transformation because it involves interpolation of the color and texture coordinate. The view frustum clipping module  435  is designed for NHCS fixed-point operations and processes the clip space coordinates that have already been lit and had texture generation and transformation applied. The view frustum clipping module  435  works well with both vertex transformed in by the texture generation and transformation module  440  and vertex transformed by an application before entering the transform and lighting module  345 . 
   The data exiting the view frustum module  435  is divided into vertex information and indices. The vertex information is sent to another vertex buffer  445  and the indices are sent to another index buffer  450 . The output data of the transform and lighting module  345  is 2D data that can be rendered on a screen or monitor of an embedded device. 
   V. Operational Overview of Single Streamline Branched T&amp;L Pipeline 
   The transform and lighting module  345  includes a single streamline branched pipeline that is optimized for executing on CPUs of embedded platforms.  FIG. 5  is a general flow diagram illustrating the operation of the single streamline branched pipeline (or method) of the transform and lighting module  345  shown in  FIG. 4 . The pipeline begins by inputting rendering data in model space (box  500 ). The rendering data contain vertex (or a plurality of vertices). Next, the rendering data is transformed from model space into clip space (box  510 ). Each vertex then is examined to determine whether that vertex should be culled (box  520 ). The culling process is described in detail below with reference to the culling module  425 . 
   If a vertex is culled, then it is discarded. Otherwise, processing is continued on the vertexes that are not culled (box  530 ). Processing includes lighting as well as texture generation and transformation. Throughout the transform and lighting pipeline, a vertex cache is provided to store vertex as needed (box  540 ). The vertex cache facilitates a single streamline branched architecture of the pipeline. By providing a vertex cache to store data, the pipeline avoids processing duplication. For example, if input rendering data has already been lit, then the lit data can be stored in the vertex cache to avoid processing by the lighting module. 
   Next, view frustum clipping is performed on normalized homogenous coordinate system (NHCS) clip coordinates of the vertex after lighting and texture generation and transform (box  550 ). The NHCS fixed-point format allows computations and operations to be performed on the clip coordinates such that a range can be predicted. Any data outside of the range is truncated. This processing of the data in the NHCS fixed-point format allows more efficient use of valuable memory and processing power. The NHCS fixed-point format is described in more detail below. 
   VI. Operational Details and Working Example 
     FIG. 6  is a detailed flow diagram illustrating the operation of the culling module  425  shown in  FIG. 4 . The culling module  425  is used to perform culling of transformed vertex information before processing by the lighting module  430 . The placement of the culling module  425  is important. In the transform and lighting pipeline disclosed herein, the culling module is positioned before the lighting module  430 . Since the lighting module is the most computationally intensive module in the transform and lighting module  345 , and positioning the culling module  425  before the lighting module  430  increases processing efficiency by reducing the number of vertex processed by the lighting module. 
   Referring to  FIG. 6 , the culling process of the culling module  425  begins by inputting vertex information in clip space (box  600 ). Next, each of the vertex is examined (box  610 ). One of two types of culling examinations is performed. A first type of culling is a back face culling. Back face culling checks whether the vertex forms a back face of a triangle (box  620 ). If not, then the vertex being examined is kept (box  630 ). Otherwise, if the vertex forms the back face of a triangle, then the vertex is discarded (box  640 ). 
   A second type of culling is a view frustum culling. View frustum culling checks whether the vertex is outside of one view frustum clip plane (box  650 ). If not, then the vertex is kept (box  630 ). On the other hand, if the vertex is outside of one view frustum clip plane, then the vertex is discarded (box  640 ). 
   NHCS Fixed-Point Operations 
   The transform and lighting module and pipeline disclosed herein use a normalized homogenous coordinate system (NHCS) to perform some operations on the rendering data. NHCS is a high-resolution variation of fixed-point number representation. In general, fixed-point representation of numbers is a way to represent a floating-point number using integers. Briefly, representing a number in a floating-point representation means that the decimal does not remain in a fixed position. Instead, the decimal “floats” such that the decimal always appears immediately after the first digit. As discussed above, using a floating point representation on a mobile device may not be possible due to processor and other hardware limitations. 
   The alternative is to use fixed-point number representation that is executed using integer functions. On mobile, wireless and other embedded platforms, the CPU may not be powerful enough to support floating-point operations and there typically are no coprocessors for accelerating the floating-point operations. Another important issue is that most floating point software routines are quite slow. Fixed-point is a much faster way to handle calculations. 
   Fixed-point number representation is a way to speed up any program that uses floating point. Typically, some of the bits are use for a whole part of the number and some bits are used for a fractional part. For example, if there are 32 bits available, a 16.16 configuration means that there are 16 bits before the decimal (representing the whole part of the number) and 16 bits after the decimal (representing the fractional part of the number). In this case, the value 65535.999984741211 is the largest possible number for the 16.16 configuration. This is obtained by setting the decimal portion to all 1&#39;s (in binary). The value 65535 with 16 bits is obtained for the whole part of the number. If 65535 is divided by 65536, then the value 0.999984741211 is obtained for the fractional part. There are other variants such as 24.8 (24 bits before the decimal and 8 bits after) and 8.24 (8 bits before the decimal and 24 bits after). The configuration type depends on the amount of precision that an application needs. 
   In an exemplary embodiment of the optimized transform and lighting module and pipeline, Direct3D for mobile devices (D3DM) is used. In order to operate in the D3DM transform and lighting module and pipeline, floating point numbers need to be converted to NHCS fixed-point numbers. Preferably, the conversion is easy as possible (so that the range of the input vertices does not need to be known) while preserving the precision of the data. NHCS fixed-point number representation achieves these objectives. 
   NHCS is a type of vertex representation. NHCS can eliminate the annoying overflow, and provides a wider data space. For example, without NHCS, the model space vertex coordinates range from 2 −16 ˜2 15 , assuming that a 16-bit mantissa is used. On the other hand, if NHCS is used, the model space vertex coordinates range from 2 −31 ˜2 31 . By adopting NHCS it can be seen that both range and precision are greatly increased. 
   NHCS also makes the conversion from floating-point to fixed-point easy. It is not necessary to know the exact range of the input vertices. NHCS also eliminates the factitious overflow and takes advantage of the full storage of the buffer. Moreover, NHCS has the advantage of providing a wider data representation given the same precision. NHCS also preserves all transform and lighting operations and makes use of the “w” in homogeneous coordinate representation. 
   Working Example 
     FIG. 7  is a working example of the transform and lighting module and pipeline as is shown for illustrative purposes only. The transform and lighting module and pipeline shown in  FIG. 7  includes a transformation module  700 , a lighting module  704  and a texture generation and transformation module  708 . It should be noted that the modules  700 ,  704  and  708  are specific implementations of the modules  420 ,  430  and  440  shown in  FIG. 4 . Specifically, in  FIG. 7  the transform and lighting module is an implementation of the Direct3D mobile (D3DM) rendering standard for embedded devices. 
   The transformation module  700  inputs vertices (or vertex)  712 . In this working example, a flexible vertex format is supported. This means that the necessary components of the vertex can be selected. The following vertex structures are supported: 
   
     
       
         
             
           
             
                 
             
           
          
             
               typedef struct t_FVF { 
             
          
         
         
             
             
          
             
                 
               BOOL bFog;//Whether Fog component exists. Only for output vertex 
             
             
                 
               BOOL bDiff; //Whether Diffuse component exists. 
             
          
         
         
             
             
             
          
             
                 
               BOOL bSpec; 
               // Whether Diffuse component exists. 
             
             
                 
               BOOL bXYZ; 
               // Whether Coordinate component exists. 
             
          
         
         
             
             
          
             
                 
               BOOL bNorm; // Whether Normal component exists. 
             
          
         
         
             
             
             
          
             
                 
               int nTexNum; 
               // Number of textures. 
             
             
                 
               int nTexCoord; 
               // Number of texture coordinates. 
             
             
                 
               int nSize; 
               //total size of a vertex 
             
          
         
         
             
             
          
             
                 
               //offsets in a vertex 
             
          
         
         
             
             
             
          
             
                 
               int offFog; 
               // Offset of the Fog component. 
             
             
                 
               int offDiff; 
               // Offset of the Diffuse component. 
             
          
         
         
             
             
          
             
                 
               int offSpec; // Offset of the Specular component. 
             
          
         
         
             
             
             
          
             
                 
               int offXYZ; 
               // Offset of the Coordinates component. 
             
          
         
         
             
             
          
             
                 
               int offNorm; // Offset of the Normal component. 
             
          
         
         
             
             
             
          
             
                 
               int offTex; 
               // Offset of the Texture coordinate component. 
             
          
         
         
             
          
             
               } FVF; 
             
             
                 
             
          
         
       
     
   
   These structures describe whether a component is in the vertex, and gives its offset for memory access. 
   The transformation module  700  transforms a vertex from model space to clip space, with a NHCS fixed-point presentation. Before transformation, the pipeline will check a vertex cache  716  to see whether the vertex has been transformed before. If not, then the model space NHCS vertex will be transformed by a matrix, M wvp , to a NHCS clip space vertex. The matrices and functions discussed in this working example are discussed in detail below. Next, the transformed data is stored in the vertex cache  716 . Some necessary messages are also stored in the vertex cache  716  for later conversion of the NHCS clip space vertex to non-NHCS clip space vertex. In this working example, a delayed conversion of NHCS to non-NHCS was used for at least two reasons. First, the conversion needs division, and can be reduced by back face culling. Second, back face culling be performed on NHCS clip space vertex. 
   Although the pipeline outputs the clip space vertices, view space vertices are still useful for either lighting in view space or texture coordinate generation. There are two ways to do both the transform: (1) transform vertex from model space to view space by M wv , and then transform vertex from view space to clip space by M p ; and (2) transform vertex from model space to view space by M wv , and transform vertex from model space to clip space by M wvp . The second technique was used because when functions like view space lighting and texture coordinate generation were turned off, the bypassing of transform to view space does not affect the pipeline. 
   As shown in  FIG. 7 , the transformation module uses the function TransQuad_SFIX32( )  720 . This function transforms model space vertices contained in the input vertices  712  to clip space vertices in an NHCS fixed-point format. The function DivW_SFIX32( )  724  converts NHCS clip space vertices to non-NHCS clip space vertices. It should be noted that in this working example the culling is part of the transformation module  700 . In particular, the functions Backface_SFIX32( ) and View Frustum Cull  728  are used to cull the input vertices  712 . The vertex that are culled are discarded and the vertex that pass are sent to the lighting module  704  to be lit. A Calc Fog function  732  is used to perform fogging on the vertex information, if desired. 
   The lighting module  704  computes color according to lighting parameters for vertices of that are non-backface. An input vertex can be assigned with normal or with diffuse/specular color. However, one or the other must be assigned; both cannot be assigned 
   If a normal is inputted, the lighting is calculated for each vertex using a Phong model. The output contains diffuse color. A render state can be set to indicate whether specular color should be calculated or not. If diffuse/specular color is inputted, the vertex are simply copied to output vertices  736  when light is off. When light is on, the output specular color is set to zero in spite of the input specular color, and the output diffuse color is set to ambient color. 
   Lighting with normal can be calculated in model space or view space. Lighting in model space is faster but limited to rigid transform from model to world space. In addition, non-rigid transform form world space to view space causes incorrect lighting results. Lighting in view space is less limited. For example, a model-world matrix could be scaled asymmetrically to transform a sphere in model space into an ellipsoid in world space, and lit it correctly in view space, but incorrectly when lit in model space. This occurs because lighting in view space first transforms normal from model space to view space, and thus asymmetry is involved. 
   The lighting module  704  uses the following functions. The function SumNorm_SFIX32Quad( )  740  is used to obtain direction by subtracting light/view with vertices. The output is view direction and light direction. This information is input for the function Dot_SFIX16Triple( )  744 , which obtains a dot product of two normalized vectors. The dot product is set to a calculate RGB module  748  where RGB color values are computed. The function TransQuad_SFIX32( )  752  is used to transform from model space to view space using a NHCS fixed-point format. The output is a position in view space. The function TransNorm_SFIX16( )  756  is used to transform a normal from model space to view space. The output is a normal in view space. Other functions (not shown) included in the lighting module  704  are a Normalize_SFIX16Triple( ), which normalizes a SFIX16Triple, and a Power_UFIX16( ), which obtains a power for a specular component. 
   The texture generation and transformation module  708  computes texture coordinates and the coordinate transforms. In texture coordinate generation, view space normal, position or reflection can be used as texture coordinates. It should be noted that some of these coordinates may have already been calculated, and these results can be reused. For this purpose, three flags are set to check whether these items have been computed. These flags involve: 
   Flag of view space normal 
   Flag of position 
   Flag of view direction 
   Texture transform is conducted after texture coordinate generation. 
   The texture generation and transformation module  708  uses the following functions. For texture generation, four functions are used. First, TransQuad_SFIX32( )  760  is used to transform from model space to view space using a NHCS fixed-point format. Second, SubNorm_SFIX32Quad( )  764  is used to transform a normal from model space to view space. Third, TransNorm_SFIX16( )  768  is used to obtain direction by subtracting view with vertices. Fourth, CalcR_SFIX16Triple( ) is used to calculate reflection from the view and the normal. The texture coordinate transformation uses the function TransQuad_SFIX16( ) to transform the texture coordinates. 
   The vertex cache  716  contains the intermediate transform and lighting results for reducing unnecessary re-calculation while rendering a single frame. The vertex cache  716  is reset when rendering of a frame is completed. In this working example, the vertex cache  716  contains 32 vertices. It is defined as
         #define CACHESIZE 32       

   Each element in the vertex cache  716  is defined as: 
                                          typedef struct t_VertexCacheItem {                          UFIX8 flag;            BYTE* pVtx;            WORD idxDestVtx;            int shift;           SFIX32 w;            BYTE* pTnIVtx;            SFIX32Quad cpos;                         } VertexCacheItem;                        
Where,
         flag contains 6 bits culling flags, 1 bit for transformed and 1 bit for lit.   pVtx points to the original vertex in model space vertex buffer.   idxDestVtx is the index of the vertex in destination vertex buffer. It can be −1 if the vertex is clipping by view frustum.   shift is the shift bits for recover the real w in getting non-NHCS clip space coordinates.   w is the original w in model space vertices. This means that user-T&amp;L vertices are supported in this transform and lighting pipeline.   pTnIVix is the vertex after transform and lighting. It has a structure exactly the same as vertex in destination vertex buffer, and is allocated dynamically when destination vertex FVF changes.   cpos is the NHCS clip space coordinates. Note that there is also a clip space coordinate in pTnIVtx but it is non-NHCS. The reason for keep cpos in cache is that some vertices may failed in back face test, and lit is bypassed. Then cpos is used when the vertices enter with another triangle for back face test again.       
   When a SrcFixVertex is sent to the pipeline, the cid is first checked in SrcFixVertex to determine whether it is in the cache. If 0&lt;=cid&lt;CACHESIZE, then the pVtx is further checked in the cache item of cid, to determine if it equals the SrcFixVertex. If this check is passed, the vertex is already transformed and lit; an index in index buffer is all that needs to be added to form a new triangle. 
   There are no math functions in the vertex cache  716 . The entire transformed and lit vertices are save in the vertex cache  716 . Each vertex corresponds to an input vertex. It is for supporting view frustum clipping, if necessary. 
   View frustum clipping  780  also is performed in the transform and lighting pipeline. This clipping technique uses NHCS clip space coordinates. The clipping technique works equally well with vertices transformed by the texture generation and transformation module  708  as well as vertices transformed outside of the transform and lighting pipeline by an application. The pipeline outputs vertices  736  that are in 2D screen coordinates. 
   VII. Details of Data Structure and Mathematical Functions 
   The following are details of the data structure for the transform and lighting module and pipeline as well as mathematical functions used in the pipeline. 
   Data Structure for Transform &amp; Lighting 
   The data structure definition for the NHCS fixed-point format is shown in the following tables: 
   Basic Type 
                                              SFIX64:   signed 64-bit integer           UFIX64:   unsigned 64-bit integer           SFIX32:   signed 32-bit integer           UFIX32:   unsigned 32-bit integer           SFIX16:   signed 16-bit integer           UFIX16:   unsigned 16-bit integer           SFIX8:   signed 8-bit integer           UFIX8:   signed 8-bit integer                        
Structure Type
 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               SFIX64 
               SFIX64Quad[4] 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 4-element vector, and each element is a 64-bit signed integer. This vector can be either NHCS or non-NHCS. 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               SFIX64 
               SFIX64Triple[3] 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 3-element vector, and each element is a 64-bit signed integer. This vector can be either NHCS or non-NHCS. 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               SFIX32 
               SFIX32Quad[4] 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 4-element vector, and each element is a 32-bit signed integer. This vector can be either NHCS or non-NHCS. 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               SFIX32 
               SFIX32Triple[3] 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 3-element vector, and each element is a 32-bit signed integer. This vector can be either NHCS or non-NHCS. 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               SFIX16 
               SFIX16Quad[4] 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 4-element vector, and each element is a 16-bit signed integer. This vector can be either NHCS or non-NHCS. 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               SFIX16 
               SFIX16Triple[3] 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 3-element vector, and each element is a 16-bit signed integer. This vector can be either NHCS or non-NHCS. 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               UFIX8 
               UFIX8Quad[4] 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 4-element vector, and each element is an 8-bit unsigned integer. This vector is non-NHCS. This vector is used mainly for representing color RGBA components. 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               typedef 
               SFIX32Mat4x4 
               SFIX32[16]; 
             
             
                 
                 
             
          
         
       
     
   
   This data structure is used to store a 16-element matrix, which is 4 by 4. Each element of the matrix is a 32-bit unsigned integer. This matrix can be either NHCS or non-NHCS. 
   Default Mantissa Bits 
   The default mantissa bits listed here are for fixed-point data representation: 
                                                  #define DEFAULT_SFIX32   16   //default mantissa                   bits for 32-bit                   signed           #define ONE_SFIX32   30   //mantissa bits                   for 32-bit signed                   with (−1~1)           #define NORMAL_SFIX16   14   //normal mantissa                   bits for 16-bit                   signed           #define TEXTURE_SFIX16   12   //mantissa bits for                   16-bit texture                   coordinate           #define ONE_UFIX16   15   //mantissa bits for                   16-bit unsigned                   within (0~1)           #define COLOR_UFIX16   8   //color mantissa bits                   for 16-bit unsigned                        
Constant
 
   The constants listed here are for integer shifting during computation and conversion between different data formats: 
   
     
       
         
             
             
           
             
                 
             
           
          
             
               const SFIX32 
               SFIX32_1 = (SFIX32)1&lt;&lt;DEFAULT_SFIX32; 
             
             
               const SFIX32 
               ONE_SFIX32_1=(SFIX32)1&lt;&lt;ONE_SFIX32; 
             
             
               const SFIX16 
               NORMAL_SFIX16_1= 
             
             
                 
               (SFIX16)1&lt;&lt;NORMAL_SFIX16; 
             
             
               const int 
               POSTOTEX=ONE_SFIX32− TEXTURE_SFIX16; 
             
             
               const int 
               NORMTOTEX= NORMAL_SFIX16 − 
             
             
                 
               TEXTURE_SFIX16; 
             
             
                 
             
          
         
       
     
   
   The basic operations have the following data structure definition: 
   Type Convert 
   The following macros are conversion macros for converting between different data formats: 
                                  #define PosToTex(a)   ((SFIX16)((a)&gt;&gt;POSTOTEX))       #define NormToTex(a)   ((SFIX16)((a)&gt;&gt;NORMTOTEX))       #define FloatToSFIX32(a,n)   ((SFIX32)((a)*((SFIX32)1&lt;&lt;(n))) )       #define SFIX32ToFloat(a,n)   ((float)(a)/((SFIX32)1&lt;&lt;(n)))       #define FloatToSFIX16(a,n)   ((SFIX16)((a)*((SFIX16)1&lt;&lt;(n))))       #define FloatToUFIX16(a,n)   ((UFIX16)((a)*((UFIX16)1&lt;&lt;(n))))       #define SFIX16ToFloat(a,n)   ((float)(a)/((SFIX16)1&lt;&lt;(n)))       #define FloatToUFIX8(a)   ((UFIX8)((a)*255))                    
Operations
 
   The following macros are computation macros for computing between fixed-point data: 
   
     
       
         
             
             
           
             
                 
             
           
          
             
               #define Mul_SFIX32(a,b,n) 
               ( (SFIX32)(((SFIX64)(a)*(b))&gt;&gt;(n)) ) 
             
             
               #define Mul_UFIX32(a,b,n) 
               ( (UFIX32)(((UFIX64)(a)*(b))&gt;&gt;(n)) ) 
             
             
               #define Div_SFIX32(a,b,n) 
               ( (SFIX32)(((SFIX64)(a)&lt;&lt;(n))/(b)) ) 
             
             
               #define Mul_SFIX16(a,b,n) 
               ( (SFIX16)(((SFIX32)(a)*(b))&gt;&gt;(n)) ) 
             
             
               #define Mul_UFIX16(a,b,n) 
               ( (UFIX16)(((UFIX32)(a)*(b))&gt;&gt;(n)) ) 
             
             
               #define Mul_UFIX8(a,b,n) 
               ( ((UFIX16)(a)*(b))&gt;&gt;(n) ) 
             
             
                 
             
          
         
       
     
   
   The data structure definition for the different types of data are as follows: 
   Input Data 
                                   Name   Type   Mantissa bits                  Model space vertex   SFIX32Quad   NHCS       coordinates       Model space normal   SFIX16Triple   NORMAL_SFIX16       Model space texture   SFIX16   TEXTURE_SFIX16       coordinates       Model space   DWORD with       diffuse/specular color   A8R8G8B8       Vertex/Texture   SFIX32Mat4x4   DEFAULT_SFIX32       transform matrices       Light/view vectors for   SFIX32Quad   NHCS       lighting       Fog parameters   SFIX32   DEFAULT_SFIX32       Color in light/material   UFIX8Quad   0       Power in material   UFIX8   0                    
Output Data
 
                                               Name   Type   Mantissa bits                          Transformed vertex   SFIX32   ONE_SFIX32           coordinates (x, y, z)           Transformed vertex   SFIX32   DEFAULT_SFIX32           coordinates (w)           Color   DWORD with               A8R8G8B8           Texture coordinates   SFIX16   TEXTURE_SFIX16           Fog   SFIX32   DEFAULT_SFIX32                        
Intermediate data&#39;s type and mantissa bits are listed within each function.
 
   Details of each of the above data types is listed below. The reason why such data types and the mantissa bits were chosen are explained. 
   Lighting 
   Position/Direction 
   Light position or direction is taken as 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               Light position 
               SFIX32Quad, 
               NHCS 
             
             
                 
                 
             
          
         
       
     
   
   This representation provides the enough range and precision for lighting, and no extra cost exists comparing with the traditional representation such as non-NHCS. 
   Viewpoint 
   Viewpoint is represented as: 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               Viewpoint 
               SFIX32Quad, 
               NHCS 
             
             
                 
                 
             
          
         
       
     
   
   This representation provides enough range and precision for lighting, and no extra cost exists comparing with the traditional representation such as non-NHCS. 
   Lighting Color 
   Lighting color includes:
         Ambient.   Diffuse   Specular
 
Their representation is:
       

                                                  Lighting color   UFIX8Quad   No mantissa                        
This presentation is a natural expansion of color in D3D in A8R8G8B8 style.
 
Material Property
 
   Material color includes:
         Ambient.   Diffuse   Specular
 
Each of them is represented as:
       

                                                  Material color   UFIX8Quad   No mantissa                        
This presentation is a natural expansion of color in D3D In A8R8G8B8 style. The power component is represented as:
 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               Power component 
               UFIX8 
               No mantissa 
             
             
                 
                 
             
          
         
       
     
   
   In one embodiment of the NHCS graphics rendering system  100 , the power is assumed to be an integer from 0 to 127. 
   Normal 
   Normal is taken as: 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               Normal 
               SFIX16 
               NORMAL_SFIX16 mantissa 
             
             
                 
                 
             
          
         
       
     
   
   From empirical evidence, it is concluded that a 16-bit normal is enough for rendering a Microsoft® Windows CE® device window. In a preferred embodiment, the NORMAL_SFIX16 is equal to 14. Moreover, the 1 sign bit must be preserved and 1 additional bit should be preserved as integer part for normal coordinates like 1.0 or −1.0. 
   Texture Coordinate 
   Texture coordinate is represented as: 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               Texture 
               SFIX16 
               TEXTURE_SFIX16 mantissa 
             
             
                 
               coordinate 
             
             
                 
                 
             
          
         
       
     
   
   In a preferred embodiment, the TEXTURE_SFIX16 is equal to 12. Further, there is 1 bit for sign and 3 bits for an integer part. This provides supports for finite filing (−8˜8), and gives 4-bits sub-pixel resolution for a texture as large as (256×256). Note that there is a trade off between the tilting size and sub-pixel resolution. 
   Output Vertex Coordinate 
   The NHCS graphics rendering system  100  produces an output vertex suitable for a vertex shader. The representation is: 
                                                  x   SFIX32   ONE_SFIX32 mantissa           y   SFIX32   ONE_SFIX32 mantissa           z   SFIX32   ONE_SFIX32 mantissa           w   SFIX32   DEFAULT_SFIX32 mantissa                        
When a vertex is within a view frustum, the value for x, y will be within (−1, 1), and z in (0˜1). A vertex outside the view frustum will be clipped before output. That is why ONE_SFIX32 is given as 30 and does not suffer from overflow. The w component is not normalized in (−1˜1). A 16-bit fraction and a 15-bit integer is a good balance between the precision and range of w.
 
Matrices
 
   Prior to rendering, several matrices should be ready. All matrices are of the data structure SFIX32, with DEFAULT_SFIX32 bits mantissa. 
   Model Space to World Space 
   M w : Transform matrix from model space to world space. 
   Currently, a D3DM implementation assumes that the last column of this matrix is (0, 0, 0, 1) T . No error is returned, and if a user specifies a matrix with different last column texture coordinate and fog it will be incorrect. 
   World Space to View Space 
   M v : Transform matrix from world space to view space 
   Currently, a D3DM implementation assumes that the last column of this matrix is (0, 0, 0, 1) T . No error is returned, and if user specifies a matrix with different last column texture coordinate and fog it will be incorrect. 
   View Space to Clip Space 
   M p : Projection matrix from view space to clip space 
   Currently, a D3DM implementation assumes that the last column of this matrix is (0, 0, 1, 0) T  or (0, 0, a, 0) T . No error is returned. For correct fog, the last column should be (0, 0, 1, 0) T  to give a correct w value. This is called the W-friendly projection matrix. 
   Model Space to View Space 
   M wv : Matrix combination from model space to view space
 
M wv =M w M v  
 
   A D3DM implementation combines the matrices M w  and M v  and the last column of this matrix is (0, 0, 0, 1) T . No error is returned. If user specifies a matrix with different last column texture coordinate and fog it will be incorrect. 
   Model Space to Clip Space 
   M wvp : Matrix combination from model space to clip space
 
M wvp =M w M v M p  
 
A D3DM implementation combines the matrices M w , M v  and M p . The last column of this matrix is determined by the parameters of these matrices. No error is returned.
 
Mathematical Library
 
   The mathematical library includes mathematical operations and graphics functions. The mathematical library now will be discussed in detail. 
   Feature Division 
   The features of the mathematical library are divided into features that are supported by the rasterizer, resource management, and features supported by transform and lighting (T&amp;L). The mathematical library implements all features supported by T&amp;L. 
   Features Supported in the Rasterizer 
   The following features are features in the mathematical library that are supported by the rasterizer:
         Point, line list, line strip, tri list, tri strip and tri fan rendering   Point, wireframe, solid fill   Flat and Gouraud shading   Depth test with various compare mode and pixel rejection   Stencil compare and pixel rejection   Depth buffer-less rendering is supported as well   W buffer support   MipMap textures are supported (Interpolate)   8 stage multi-texture with D3D8 fixed function blending options   Point, linear, anisotropic, cubic and Gaussian cubic texture filtering   Alpha blending (with several bland modes)   Palletized textures   Perspective correct texturing (not on by default)   Color channel masking (COLORWRITEENABLE)   Dithering   Multisampling for FSAA   Texture address modes       

   Features Supported in Resource Management 
   Resources are objects that are resident in memory, such as textures, vertex buffers, index buffers and render surfaces. Resource management is the management of the various memory operations on these objects. These operations include allocation, copying, moving, locking for exclusive usage, unlock and de-allocation. The following features are features in the mathematical library that are supported in resource management:
         Swap chain creation and management for display   Depth/stencil buffer creation and management   Vertex buffer creation and management   Index buffer creation and management   Texture map creation and management   Many texture formats including DXT compressed texture   Scratch surface creation/management for texture upload   MipMap textures are supported (Build)   Dirty rectangular texture update mechanism   All buffers lockable (assuming driver support!)       

   Features Supported in T&amp;L 
   The following features are features in the mathematical library that are supported by in T&amp;L:
         Texture coordinate generation   View, projection and world transform matrices   Single transform matrix per texture coordinate set (8 sets max)   Up to 4 dimensions per texture coordinate set   Ambient/diffuse/specular lighting and materials   Directional and point lights   Back face culling   Fog (depth and table based)
 
Math Functions Indexed by Features
       

   In this section, the mathematical functions indexed by features are described. The functions cover transform, culling, lighting, culling, texture and other miscellaneous functions. In addition, the overflow and underflow (resolution loss) problems of these functions are discussed. 
   Transform Functions 
   
     
       
         
             
           
             
                 
             
             
               NHCS vector transform 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               int TransQuad_SFIX32(SFIX32Quad b, SFIX32Mat4x4 m, 
             
             
                 
               SFIX32Quad c) 
             
             
                 
               This function transforms a 32-bits NHCS vector b to 
             
             
                 
               another 32-bits NHCS vector c by matrix m. 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               b, 
             
             
                 
                 
               Input vector in SFIX32Quad in 
             
             
                 
                 
               NHCS format 
             
             
                 
                 
               m 
             
             
                 
                 
               Transform matrix in SFIX32Mat4x4 
             
             
                 
                 
               and DEFAULT_SFIX32 format. 
             
             
                 
                 
               c, 
             
             
                 
                 
               Output vector after transform 
             
             
                 
                 
               in SFIX32 format in NHCS 
             
             
                 
                 
               representation. 
             
             
                 
               Return 
               An integer indicates the shift 
             
             
                 
               value 
               bits in converting intermediate 64- 
             
             
                 
                 
               bits c to 32-bits NHCS c. 
             
             
                 
               Remarks 
               Overflow: 
             
             
                 
                 
               The maximum possible 
             
             
                 
                 
               intermediate value is: 4*(0x8000 
             
             
                 
                 
               0000*0x8000 0000) = 0x 1 
             
             
                 
                 
               0000 0000 0000 0000. This indicates 
             
             
                 
                 
               that a 64-bits intermediate 
             
             
                 
                 
               value will have overflow in the 
             
             
                 
                 
               intermediate data before NHCS. 
             
             
                 
                 
               Underflow: 
             
             
                 
                 
               Appears when truncated from 
             
             
                 
                 
               intermediate buffer. 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Matrix combination 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
          
             
               void MatMul4x4_SFIX32(SFIX32Mat4x4 m1, SFIX32Mat4x4 m2, 
             
             
               SFIX32Mat4x4 m3, UFIX8 n) 
             
             
               This function combines two 32-bits 4 × 4 matrices to 
             
             
               another 32-bits 4 × 4 matrix 
             
          
         
         
             
             
          
             
               Parameters 
               m1, m2 
             
             
                 
               Input matrices in SFIX32Mat4x4 
             
             
                 
               n 
             
             
                 
               Input shift bits for shifting the 
             
             
                 
               64-bits multiplication results to 
             
             
                 
               32-bits results. 
             
             
                 
               m3, 
             
             
                 
               Output combined matrix. 
             
             
               Return value 
               No return value 
             
             
               Remarks 
               Shift 
             
             
                 
               The matrices m1, m2, m3 can 
             
             
                 
               have different mantissa bits. 
             
             
                 
               Suppose m1 with a bits 
             
             
                 
               mantissa and m2 with b bits 
             
             
                 
               mantissa, to get a c-bits 
             
             
                 
               mantissa m3, we should set n = 
             
             
                 
               (a + b) − c 
             
             
                 
               Overflow: 
             
             
                 
               The maximum possible 
             
             
                 
               intermediate value is: 4*(0x8000 
             
             
                 
               0000*0x8000 0000) = 0x 1 0000 
             
             
                 
               0000 0000 0000. This indicates 
             
             
                 
               that a 64-bits intermediate 
             
             
                 
               value will have overflow 
             
             
                 
               in the intermediate data. 
             
             
                 
               When truncating the 64-bits 
             
             
                 
               intermediate result to 32- 
             
             
                 
               bits output, overflow is also 
             
             
                 
               possible. 
             
             
                 
               Underflow: 
             
             
                 
               Appears when truncated from 
             
             
                 
               intermediate buffer. 
             
             
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Non-NHCS vector transform 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               void TransQuad_SFIX16(SFIX16Quad b, SFIX32Mat4x4 m, 
             
             
                 
               SFIX16Quad c) 
             
             
                 
               This function transforms a 16-bits vector to a 
             
             
                 
               16-bits vector. 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               b, 
             
             
                 
                 
               Input vector in SFIX16Quad 
             
             
                 
                 
               with TEXTURE_SFIX16 bits 
             
             
                 
                 
               mantissa. 
             
             
                 
                 
               m 
             
             
                 
                 
               Transform matrix in 
             
             
                 
                 
               SFIX32Mat4x4 and 
             
             
                 
                 
               DEFAULT_SFIX32 format. 
             
             
                 
                 
               c 
             
             
                 
                 
               Output vector after transform 
             
             
                 
                 
               in SFIX16 format with 
             
             
                 
                 
               TEXTURE_SFIX16 bits mantissa. 
             
             
                 
               Return Value 
               No return value. 
             
             
                 
               Remarks 
               Overflow: 
             
             
                 
                 
               Appears when go out range 
             
             
                 
                 
               of TEXTURE_SFIX16 mantissa. 
             
             
                 
                 
               Underflow: 
             
             
                 
                 
               Appears when go out range 
             
             
                 
                 
               of TEXTURE_SFIX16 mantissa. 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
             
           
             
                 
                 
             
           
          
             
                 
               void TransNorm_SFIX16(SFIX16Triple b,SFIX32Mat4x4 m, 
             
             
                 
               SFIX16Triple c) 
             
             
                 
               This function transforms a 16-bit normal to a 
             
             
                 
               16-bits normal. 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               b 
             
             
                 
                 
               Input vector in SFIX16Triple 
             
             
                 
                 
               with NORAML_SFIX16 bits 
             
             
                 
                 
               mantissa. 
             
             
                 
                 
               m 
             
             
                 
                 
               Transform matrix in 
             
             
                 
                 
               SFIX32Mat4x4 and 
             
             
                 
                 
               DEFAULT_SFIX32 format. 
             
             
                 
                 
               c 
             
             
                 
                 
               Output vector after transform, 
             
             
                 
                 
               it is in SFIX16 format with 
             
             
                 
                 
               NORMAL_SFIX16 bits mantissa, 
             
             
                 
                 
               normalized. 
             
             
                 
               Return value 
               No return value. 
             
             
                 
               Remarks 
               Matrix 
             
             
                 
                 
               For transform normal, only the 
             
             
                 
                 
               upper 3 × 3 part of m is used. 
             
             
                 
                 
               Normalization: 
             
             
                 
                 
               The output is normalized by 
             
             
                 
                 
               Normalize_SFIX16Triple( ) 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               NHCS to non-NHCS convert 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               Void DivWW_SFIX32(SFIX32 w, int shift, SFIX32Quad c, 
             
             
                 
               SFIX32Quad cc) 
             
             
                 
               This function transforms a NHCS vertex to clip space 
             
             
                 
               non-NHCS vertex. 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               w 
             
             
                 
                 
               Input w to be divided from the 
             
             
                 
                 
               NHCS vertex, SFIX32. It is the 
             
             
                 
                 
               b[3] in TransQuad_SFIX32( ). 
             
             
                 
                 
               shift 
             
             
                 
                 
               Input shifted bits return 
             
             
                 
                 
               from TransQuad_SFIX32( ). For 
             
             
                 
                 
               calculating the correct w 
             
             
                 
                 
               c 
             
             
                 
                 
               Input vertex after 
             
             
                 
                 
               TransQuad_SFIX32( ), NHCS 
             
             
                 
                 
               cc 
             
             
                 
                 
               Output vertex with non-NHCS 
             
             
                 
                 
               SFIX32 format. cc[0]~cc[2] has 
             
             
                 
                 
               ONE_SFIX32 bits mantissa, and 
             
             
                 
                 
               cc[3] has DEFAULT_SFIX32 
             
             
                 
                 
               bits mantissa. 
             
             
                 
               Return value 
               No Return value 
             
             
                 
               Remarks 
               This function is related 
             
             
                 
                 
               to TransQuad_SFIX32( ). 
             
             
                 
                 
               With this function we get the 
             
             
                 
                 
               actual clip space vertex from 
             
             
                 
                 
               NHCS clip space vertex for 
             
             
                 
                 
               finally converting to float 
             
             
                 
                 
               point vertex and output to 
             
             
                 
                 
               vertex shader. 
             
             
                 
                 
             
          
         
       
     
   
                                          Void DivW_SFIX32(SFIX32 w, int shift, SFIX32Quad c,           SFIX32Quad cc)           This function transforms a NHCS vertex to clip space           non-NHCS vert x.                             Parameters   w               Input w to be divided from the               NHCS vertex, SFIX32. It is the               b[3] in TransQuad_SFIX32( ).               shift               Input shifted bits return               from TransQuad_SFIX32( ). For               calculating the correct w               c               Input vertex after               TransQuad_SFIX32( ), NHCS               cc               Output vertex with               DEFAULT_SFIX32 format.           Return value   No Return value           Remarks   This function is related               to TransQuad_SFIX32( ).               This function is used in               texture coordinate generation               from view space position,               so the precision and range is               different from               DivWW_SFIX32 above.                        
Culling Functions
 
   
     
       
         
             
           
             
                 
             
             
               Backface testing 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               BOOL Backface_SFIX32(SFIX32* a, SFIX32* b, SFIX32* c, 
             
             
                 
               BOOL bCCW) 
             
             
                 
               This function checks if the triangle (a, b, c) is a 
             
             
                 
               back face. 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a, b, c 
             
             
                 
                 
               3 sequential vertex 
             
             
                 
                 
               of an triangle, they 
             
             
                 
                 
               are in SFIX32Quad 
             
             
                 
                 
               with NHCS representation 
             
             
                 
                 
               bCCW 
             
             
                 
                 
               Face orientation, TRUE 
             
             
                 
                 
               for CCW, FALSE for CW 
             
             
                 
               Return value 
               BOOL, TRUE for back face, 
             
             
                 
                 
               FALSE for non-back face. 
             
             
                 
               Remarks 
               There is a sequential 
             
             
                 
                 
               multiplication of 3 operands. 
             
             
                 
                 
               NHCS is used to compress 
             
             
                 
                 
               the operand from 32-bits to 
             
             
                 
                 
               16-bits since we only 
             
             
                 
                 
               need the sign. 
             
             
                 
                 
             
          
         
       
     
   
   View Frustum Culling 
   View frustum culling removes the triangles whose vertices are outside of one view frustum plane. View frustum involves 6 planes:
         Left plane.   Right plane.   Top plane.   Bottom plane.   Near plane   Far plane.       

   A UFIX8 is set to hold 6 flags for culling.  FIG. 8  illustrates an exemplary implementation of a buffer to store culling planes. In particular,  FIG. 8  shows an UFIX8 format buffer to store the culling planes. View frustum culling is performed in clip space. If it is assumed that b is a NHCS coordinate in the clip space, the algorithm is: 
   
     
       
         
             
             
           
             
                 
                 
             
           
          
             
                 
               SFIX32Quad b; // NHCS clip space coordinates 
             
             
                 
               UFIX8 f=0; 
             
             
                 
               if (b[0]&lt;−b[3]) 
             
          
         
         
             
             
          
             
                 
               f |= 0x01; 
             
          
         
         
             
             
          
             
                 
               else if (b[0]&gt; b[3]) 
             
          
         
         
             
             
          
             
                 
               f |= 0x02; 
             
          
         
         
             
             
          
             
                 
               if (b[1]&lt;− b[3]) 
             
          
         
         
             
             
          
             
                 
               f |= 0x04; 
             
          
         
         
             
             
          
             
                 
               else if (b[1]&gt; b[3]) 
             
          
         
         
             
             
          
             
                 
               f |= 0x08; 
             
          
         
         
             
             
          
             
                 
               if (b[2]&lt;0) 
             
          
         
         
             
             
          
             
                 
               f |= 0x10; 
             
          
         
         
             
             
          
             
                 
               else if (b[2]&gt; b[3]) 
             
          
         
         
             
             
          
             
                 
               f |= 0x20; 
             
             
                 
                 
             
          
         
       
     
   
   If three flags for each vertex are obtained, an “AND” operation can be used to test whether the flags are outside of the same plane. 
   The flag is also useful in the vertex cache, and the 2 unused bits will indicate:
         Transformed status (indicates whether a vertex has been transformed)   Lit status (indicates whether a vertex has been lit).
 
Lighting Functions
       

   The direct3D for mobile supports both directional light and point light. The lighting model used is the Phong model for vertices. Lighting is done in model space. A material should be assigned to the object, and the ambient, diffuse, specular, power property is denoted as M Ambient , M Diffuse , M Specular  and M Power  respectively. In D3D, M Ambient , M Diffuse , M Specular  are defined as (r, g, b, a), and each component is a float within [0˜1]. 
   Each component only need be represented as: 
   
     
       
         
             
             
             
             
           
             
                 
                 
             
           
          
             
                 
               Lighting component 
               UFIX8 
               8 bits mantissa 
             
             
                 
                 
             
          
         
       
     
   
   The color of lighting is noted as L Ambient , L Diffuse  and L Specular . Given normalized vectors N, L and V, which represent vertex normal, vertex-light direction and vertex-view direction respectively, the color of a vertex can be calculated as:
 
 C=L   Ambient   M   Ambient   +L   Diffuse   M   Diffuse ( N·L )+ L   Specular   M   Specular ( N·H ) M     Power    
 
     FIGS. 9A and 9B  illustrate an exemplary implementation of normalized vectors in a D3DM Phong Model. As shown in  FIG. 9A , L is the vector from vertex to light, and N is the vertex normal. R is the reflection direction of light, which is symmetric to L by N. As shown in  FIG. 9B , V is the vector from vertex to view point, and H is the half vector of L+V. 
   All the vectors are transformed to the same space for “dot product” computation, and are normalized for calculation. In this implementation, the model space for saving the transformation of each vertex normal to view space was chosen. However, this choice also brings problems if the model transform contains shears and scaling. Although lighting in model space is discussed here, it is easy to extend the discussion to other spaces. Both lit in model space and lit in view space are supported in the rendering pipeline of the NHCS graphics rendering system. 
   
     
       
         
             
           
             
                 
             
             
               Invert Length of a Normal 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
          
             
               SFIX32 TripleInvLen(SFIX16Triple a) 
             
             
               This function gives the invert length of a SFIX16Triple, which is useful in 
             
             
               normalize 
             
          
         
         
             
             
          
             
               Parameters 
               a 
             
             
                 
               Un-normalized input 
             
             
                 
               in SFIX16 in NHCS 
             
             
               Return value 
               Invert length in SFIX32 
             
             
               Remarks 
               Assume a is a n-bits mantissa, 
             
             
                 
               the result is of 42 − n bits 
             
             
                 
               mantissa. It does not matter 
             
             
                 
               if 42 − n &gt; 32, because the 
             
             
                 
               calculation does not use n 
             
             
                 
               explicitly. 
             
             
                 
               Newton&#39;s iteration method 
             
             
                 
               is used here for solving the 
             
             
                 
               invert square root, using 
             
             
                 
               a 256-item lookup table. 
             
             
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               NHCS Vector Normalization 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
          
             
               Void Normalize_SFIX16Triple(SFIX16Triple a, SFIX16Triple b) 
             
             
               This function normalizes a NHCS SFIX16Triple. 
             
          
         
         
             
             
          
             
               Parameters 
               a 
             
             
                 
               Un-normalized input 
             
             
                 
               in SFIX16 in NHCS 
             
             
                 
               b 
             
             
                 
               Normalized output in SFIX16 
             
             
                 
               format with NORMAL_SFIX16 
             
             
                 
               mantissa 
             
             
               Return value 
               No return value 
             
             
               Remarks 
               We use SFIX32 to hold the 
             
             
                 
               intermediate TripleInvLen ( ) 
             
             
                 
               result to prevent overflow 
             
             
                 
               and keep precision. 
             
             
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Negative Normalization of NHCS Vector 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
          
             
               Void NagNormalize_SFIX16Triple(SFIX16Triple a, SFIX16Triple b) 
             
             
               This function gives a negative result to Normalize_SFIX16Triple 
             
          
         
         
             
             
          
             
               Parameters 
               A 
             
             
                 
               Un-normalized input 
             
             
                 
               in SFIX16 in NHCS 
             
             
                 
               b 
             
             
                 
               Normalized output in SFIX16 
             
             
                 
               format with NORMAL_SFIX16 
             
             
                 
               mantissa 
             
             
               Return value 
               No return value 
             
             
               Remarks 
               We use SFIX32 to hold the 
             
             
                 
               intermediate TripleInvLen ( ) 
             
             
                 
               result to prevent overflow. 
             
             
                 
               It is used in normalization 
             
             
                 
               of directional light. Gives a 
             
             
                 
               normal L from vertex to 
             
             
                 
               lighting source. 
             
             
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Subtraction of Two NHCS Vectors 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
          
             
               Void SubNorm_SFIX32Quad (SFIX32Quad a, SFIX32Quad b, 
             
             
               SFIX16Triple c) 
             
             
               This function calculates normal from subtraction of two NHCS vectors. 
             
          
         
         
             
             
          
             
               Parameters 
               a, b 
             
             
                 
               Input vectors in SFIX32 
             
             
                 
               with NHCS format 
             
             
                 
               c 
             
             
                 
               Normalized (a − b) in SFIX16 
             
             
                 
               with NORMAL_SFIX16 bits 
             
             
                 
               mantissa 
             
             
               Return value 
               No return value 
             
             
               Remarks 
               It is used in normalization 
             
             
                 
               of view direction V and light 
             
             
                 
               direction L when using point 
             
             
                 
               light. 
             
             
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Dot Production of Two Normalized Vectors 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               UFIX16 Dot_SFIX16Triple(SFIX16Triple a, 
             
             
                 
               SFIX16Triple b) 
             
             
                 
               This function returns the dot product of two 
             
             
                 
               normalized vector 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a, b 
             
             
                 
                 
               Normalized input in SFIX16 
             
             
                 
                 
               with DEFAULT_SFIX16 bits 
             
             
                 
                 
               mantissa. 
             
             
                 
               Return value 
               Dot product with ONE_UFIX16 
             
             
                 
                 
               bits mantissa 
             
             
                 
               Remarks 
               If the two vectors are normalized, 
             
             
                 
                 
               there will no overflow at 
             
             
                 
                 
               all because the result will 
             
             
                 
                 
               be within (0~1). 
             
             
                 
                 
               Value that less than 0 is clamped to 0. 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Power 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               UFIX16 Power_UFIX16 (UFIX16 a, UFIX8 n) 
             
             
                 
               This function returns the power(a, n) 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a 
             
             
                 
                 
               Power base with ONE_UFIX16 
             
             
                 
                 
               bits mantissa. 
             
             
                 
                 
               n 
             
             
                 
                 
               Power exponential within 0~127 
             
             
                 
               Return value 
               Power value with UFIX16 format 
             
             
                 
               Remarks 
               We use the efficient digit 
             
             
                 
                 
               of n to determine how much 
             
             
                 
                 
               multiply we need. 
             
             
                 
                 
               In rendering pipeline the n 
             
             
                 
                 
               can be fixed. We use static 
             
             
                 
                 
               variables to store the n and 
             
             
                 
                 
               its efficient digit. If n is the 
             
             
                 
                 
               same in the consequential calling, 
             
             
                 
                 
               the efficient digit will be 
             
             
                 
                 
               same as previous one instead of 
             
             
                 
                 
               calculated again. 
             
             
                 
                 
             
          
         
       
     
   
   Half Vector 
   The half vector is used to approximate the actual cos θ=(V·R) by cos ψ=(N·H) for calculating the specular component. H can be calculated by the normalized L and V: 
   
     
       
         
           H 
           = 
           
             
               L 
               + 
               V 
             
             
               
                 L 
                 + 
                 V 
               
               | 
             
           
         
       
     
   
   L and V are represented by SFIX16Triple with NORMAL_SFIX16 bits mantissa. To avoid overflow and keep precision, they are first added together as a SFIX32Triple. Next, the half vector H is made in NHCS SFIX16Triple, and H then is normalized. 
   Texture Coordinate Generation 
   Texture coordinate generation uses view space normal/position/reflection to generate the texture coordinates in each vertex. View space normal and position is available after lighting in view space. However, reflection vectors need to be calculated here. 
                           Reflection Vector from Normal and View                                    Void CalcR_SFIX16Triple(SFIX16Triple norm, SFIX16Triple view,       SFIX16Triple reflect)       This function calculates reflection vector from normal       and view                     Parameters   Norm           normalized normal in           SFIX16, NORMAL_SFIX16           view           normalized view direction           in SFIX16, NORMAL_SFIX16           reflect           Normalized output in SFIX16           format with NORMAL_SFIX16           mantissa       Return value   No return value       Remarks   R = 2(N · V)N − V                    
NHCS Clip Space Coordinates Clipping Algorithm
 
   The model-view transform and view-projective transform can be combined into a 4×4 matrix P 4×4 : 
   
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           x 
                         
                         
                           y 
                         
                         
                           z 
                         
                         
                           1 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     P 
                     
                       4 
                       × 
                       4 
                     
                   
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               x 
                               p 
                             
                             
                               w 
                               p 
                             
                           
                           ⁢ 
                           
                             w 
                             p 
                           
                         
                       
                       
                         
                           
                             
                               y 
                               p 
                             
                             
                               w 
                               p 
                             
                           
                           ⁢ 
                           
                             w 
                             p 
                           
                         
                       
                       
                         
                           
                             
                               z 
                               p 
                             
                             
                               w 
                               p 
                             
                           
                           ⁢ 
                           
                             w 
                             p 
                           
                         
                       
                       
                         
                           w 
                           p 
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 1 
                 ) 
               
             
           
         
       
     
   
   The term 
               x   p       w   p       =     x   w           
is defined, and is similar to y, z. In fact, the term is the normalized screen space coordinates. This assumes the correct wp is obtained for each vertex. Multiplying (1) by
 
             (       1     w   p       ⁢     P     4   ×   4       -   1         )     ,         
yields:
 
   
     
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         
                           x 
                           
                             w 
                             p 
                           
                         
                       
                       
                         
                           y 
                           
                             w 
                             p 
                           
                         
                       
                       
                         
                           z 
                           
                             w 
                             p 
                           
                         
                       
                       
                         
                           1 
                           
                             w 
                             p 
                           
                         
                       
                     
                   
                   ) 
                 
                 = 
                 
                   
                     ( 
                     
                       
                         
                           
                             x 
                             w 
                           
                         
                         
                           
                             y 
                             w 
                           
                         
                         
                           
                             z 
                             w 
                           
                         
                         
                           1 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     P 
                     
                       4 
                       × 
                       4 
                     
                     
                       - 
                       1 
                     
                   
                 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   Equation (2) is a linear equation, which indicates that 1/wp can be linearly interpolated. Given three vertices and three texture coordinates: (x i  y i  z i  1) and (u i  v i  1) (i=1, 2, 3) for a triangle, there exists an affine transform which maps texture coordinates to object space, if the triangle is not degenerated:
 
( uv 1) A   3×4 =( xyz 1)  (3)
 
   Combining (3) and (1), both sides are divided by the wp, and thus: 
                     (           u     w   p             v     w   p             1     w   p             )     ⁢   B     =     (           x   w           y   w           z   w         1         )             (   4   )               
Where  B=A   3×4   P   4×4    
   Equation (4) indicates u/wp, v/wp can be interpolated linearly. For perspective-correct texture mapping, after linearly interpolating u/wp, v/wp and 1/wp, the correct texture coordinates can be computed for projective-correct texture mapping. 
   The algorithm for interpolating between two points is:
 
Input: point(x 1p y 1p z 1p w 1p )(x 2p y 2p z 2p w 2p ),
 
Clip plane  ax   w   +by   w   +cz   w   +d= 0
 
   The intersection point (x p  y p  z p  w p ) will satisfy: 
   
     
       
         
           { 
           
             
               
                 
                   
                     x 
                     w 
                   
                   = 
                   
                     
                       
                         x 
                         p 
                       
                       / 
                       
                         w 
                         p 
                       
                     
                     = 
                     
                       
                         
                           
                             x 
                             
                               1 
                               ⁢ 
                               p 
                             
                           
                           / 
                           
                             w 
                             
                               1 
                               ⁢ 
                               p 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 
                                   x 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                   
                                 
                                 / 
                                 
                                   w 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                   
                                 
                               
                               - 
                               
                                 
                                   x 
                                   
                                     1 
                                     ⁢ 
                                     p 
                                   
                                 
                                 / 
                                 
                                   w 
                                   
                                     1 
                                     ⁢ 
                                     p 
                                   
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           t 
                         
                       
                       = 
                       
                         
                           x 
                           
                             1 
                             ⁢ 
                             w 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 x 
                                 
                                   2 
                                   ⁢ 
                                   w 
                                 
                               
                               - 
                               
                                 x 
                                 
                                   1 
                                   ⁢ 
                                   w 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           t 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     y 
                     w 
                   
                   = 
                   
                     
                       
                         y 
                         p 
                       
                       / 
                       
                         w 
                         p 
                       
                     
                     = 
                     
                       
                         
                           
                             y 
                             
                               1 
                               ⁢ 
                               p 
                             
                           
                           / 
                           
                             w 
                             
                               1 
                               ⁢ 
                               p 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 
                                   y 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                   
                                 
                                 / 
                                 
                                   w 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                   
                                 
                               
                               - 
                               
                                 
                                   y 
                                   
                                     1 
                                     ⁢ 
                                     p 
                                   
                                 
                                 / 
                                 
                                   w 
                                   
                                     1 
                                     ⁢ 
                                     p 
                                   
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           t 
                         
                       
                       = 
                       
                         
                           y 
                           
                             1 
                             ⁢ 
                             w 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 y 
                                 
                                   2 
                                   ⁢ 
                                   w 
                                 
                               
                               - 
                               
                                 y 
                                 
                                   1 
                                   ⁢ 
                                   w 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           t 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     z 
                     w 
                   
                   = 
                   
                     
                       
                         z 
                         p 
                       
                       / 
                       
                         w 
                         p 
                       
                     
                     = 
                     
                       
                         
                           
                             z 
                             
                               1 
                               ⁢ 
                               p 
                             
                           
                           / 
                           
                             w 
                             
                               1 
                               ⁢ 
                               p 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 
                                   z 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                   
                                 
                                 / 
                                 
                                   w 
                                   
                                     2 
                                     ⁢ 
                                     p 
                                   
                                 
                               
                               - 
                               
                                 
                                   z 
                                   
                                     1 
                                     ⁢ 
                                     p 
                                   
                                 
                                 / 
                                 
                                   w 
                                   
                                     1 
                                     ⁢ 
                                     p 
                                   
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           t 
                         
                       
                       = 
                       
                         
                           z 
                           
                             1 
                             ⁢ 
                             w 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 z 
                                 
                                   2 
                                   ⁢ 
                                   w 
                                 
                               
                               - 
                               
                                 z 
                                 
                                   1 
                                   ⁢ 
                                   w 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           t 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     1 
                     / 
                     
                       w 
                       p 
                     
                   
                   = 
                   
                     
                       1 
                       / 
                       
                         w 
                         
                           1 
                           ⁢ 
                           p 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             1 
                             / 
                             
                               w 
                               
                                 2 
                                 ⁢ 
                                 p 
                               
                             
                           
                           - 
                           
                             1 
                             / 
                             
                               w 
                               
                                 1 
                                 ⁢ 
                                 p 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       t 
                     
                   
                 
               
             
           
             
         
       
     
   
   Take into clip plane, yields: 
               ax     1   ⁢   w       +     by     1   ⁢   w       +     cz     1   ⁢   w       +   d   +       (       a   ⁢     (       x     2   ⁢   w       -     x     1   ⁢   w         )       +     b   ⁡     (       y     2   ⁢   w       -     y     1   ⁢   w         )       +     c   ⁡     (       z     2   ⁢   w       -     z     1   ⁢   w         )         )     ⁢   t       =   0               t   =       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p       +     dw     1   ⁢   p         )               w     1   ⁢   p       ⁡     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )                 
Then:
 
                   1   /     w   p       =       ⁢       1   /     w     1   ⁢   p         +       (       1   /     w     2   ⁢   p         -     1   /     w     1   ⁢   p           )     ⁢   t                   =       ⁢       1     w     1   ⁢   p         +         -     (       w     1   ⁢   p       -     w     2   ⁢   p         )       ⁢     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p       +     dw     1   ⁢   p         )                 w     1   ⁢   p       (         w     1   ⁢   p       ⁢     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -                     w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )       )                           =       ⁢                 w     1   ⁢   p       ⁡     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )       -                 (       w     1   ⁢   p       -     w     2   ⁢   p         )     ⁢     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p       +     dw     1   ⁢   p         )                 w     1   ⁢   p       ⁡     (         w     1   ⁢   p       ⁡     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )         )                     =       ⁢         a   ⁡     (       x     2   ⁢   p       -     x     1   ⁢   p         )       +     b   ⁡     (       y     2   ⁢   p       -     y     1   ⁢   p         )       +     c   ⁡     (       z     2   ⁢   p       -     z     1   ⁢   p         )       +     d   ⁡     (       w     2   ⁢   p       -     w     1   ⁢   p         )               w     1   ⁢   p       ⁡     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )                             And   ,     
     ⁢       x   w     =               b   ⁢     (         x     1   ⁢   p       ⁢     y     2   ⁢   p         -       x     2   ⁢   p       ⁢     y     1   ⁢   p           )       +     c   ⁢     (         x     1   ⁢   p       ⁢     z     2   ⁢   p         -       x     2   ⁢   p       ⁢     z     1   ⁢   p           )       +               d   ⁡     (         x     1   ⁢   p       ⁢     w     2   ⁢   p         -       x     2   ⁢   p       ⁢     w     1   ⁢   p           )                   w     1   ⁢   p       ⁡     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )                           y   w     =               a   ⁡     (         y     1   ⁢   p       ⁢     x     2   ⁢   p         -       y     2   ⁢   p       ⁢     x     1   ⁢   p           )       +     c   ⁡     (         y     1   ⁢   p       ⁢     z     2   ⁢   p         -       y     2   ⁢   p       ⁢     z     1   ⁢   p           )       +               d   ⁡     (         y     1   ⁢   p       ⁢     w     2   ⁢   p         -       y     2   ⁢   p       ⁢     w     1   ⁢   p           )                   w     1   ⁢   p       ⁡     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )                         z   w     =               a   ⁡     (         z     1   ⁢   p       ⁢     x     2   ⁢   p         -       z     2   ⁢   p       ⁢     x     1   ⁢   p           )       +     b   ⁡     (         z     1   ⁢   p       ⁢     y     2   ⁢   p         -       z     2   ⁢   p       ⁢     y     1   ⁢   p           )       +               d   ⁡     (         z     1   ⁢   p       ⁢     w     2   ⁢   p         -       z     2   ⁢   p       ⁢     w     1   ⁢   p           )                   w     1   ⁢   p       ⁡     (       ax     2   ⁢   p       +     by     2   ⁢   p       +     cz     2   ⁢   p         )       -       w     2   ⁢   p       ⁡     (       ax     1   ⁢   p       +     by     1   ⁢   p       +     cz     1   ⁢   p         )                 
After NHCS transform, gives:
 ( x   np   ,y   np   ,z   np   ,w   np )= c   w   c   v   c   p   w   nm ( x   p   ,y   p   ,z   p   ,w   p ) 
which gives:
 
   
     
       
         
           
             ( 
             
               
                 x 
                 
                   1 
                   ⁢ 
                   p 
                 
               
               , 
               
                 y 
                 
                   1 
                   ⁢ 
                   p 
                 
               
               , 
               
                 z 
                 
                   1 
                   ⁢ 
                   p 
                 
               
               , 
               
                 w 
                 
                   1 
                   ⁢ 
                   p 
                 
               
             
             ) 
           
           = 
           
             
               1 
               
                 
                   c 
                   1 
                 
                 ⁢ 
                 
                   w 
                   
                     1 
                     ⁢ 
                     nm 
                   
                 
               
             
             ⁢ 
             
               ( 
               
                 
                   x 
                   
                     1 
                     ⁢ 
                     np 
                   
                 
                 , 
                 
                   y 
                   
                     1 
                     ⁢ 
                     np 
                   
                 
                 , 
                 
                   z 
                   
                     1 
                     ⁢ 
                     np 
                   
                 
                 , 
                 
                   w 
                   
                     1 
                     ⁢ 
                     np 
                   
                 
               
               ) 
             
           
         
       
     
     
       
         
           
             ( 
             
               
                 x 
                 
                   2 
                   ⁢ 
                   p 
                 
               
               , 
               
                 y 
                 
                   2 
                   ⁢ 
                   p 
                 
               
               , 
               
                 z 
                 
                   2 
                   ⁢ 
                   p 
                 
               
               , 
               
                 w 
                 
                   2 
                   ⁢ 
                   p 
                 
               
             
             ) 
           
           = 
           
             
               1 
               
                 
                   c 
                   2 
                 
                 ⁢ 
                 
                   w 
                   
                     2 
                     ⁢ 
                     nm 
                   
                 
               
             
             ⁢ 
             
               ( 
               
                 
                   x 
                   
                     2 
                     ⁢ 
                     np 
                   
                 
                 , 
                 
                   y 
                   
                     2 
                     ⁢ 
                     np 
                   
                 
                 , 
                 
                   z 
                   
                     2 
                     ⁢ 
                     np 
                   
                 
                 , 
                 
                   w 
                   
                     2 
                     ⁢ 
                     np 
                   
                 
               
               ) 
             
           
         
       
     
   
   Thus, the final representation of (x w ,y w ,z w ,1/w p ) becomes: 
             l   /     w   p       =                 c   1     ⁢       w     1   ⁢   nm       ⁡     (       ax     2   ⁢   np       +     by     2   ⁢   np       +     cz     2   ⁢   np       +     dw     2   ⁢   np         )         -                 c   2     ⁢       w     2   ⁢   nm       ⁡     (       ax     1   ⁢   np       +     by     1   ⁢   np       +     cz     1   ⁢   np       +     dw     1   ⁢   np         )                     w     1   ⁢   np       ⁡     (       ax     2   ⁢   np       +     by     2   ⁢   np       +     cz     2   ⁢   np         )       -       w     2   ⁢   np       ⁡     (       ax     1   ⁢   np       +     by     1   ⁢   np       +     cz     1   ⁢   np         )                         x   w     =               b   ⁢     (         x     1   ⁢   np       ⁢     y     2   ⁢   np         -       x     2   ⁢   np       ⁢     y     1   ⁢   np           )       +     c   ⁢     (         x     1   ⁢   np       ⁢     z     2   ⁢   np         -       x     2   ⁢   np       ⁢     z     1   ⁢   np           )       +               d   ⁡     (         x     1   ⁢   np       ⁢     w     2   ⁢   np         -       x     2   ⁢   np       ⁢     w     1   ⁢   np           )                   w     1   ⁢   np       ⁡     (       ax     2   ⁢   np       +     by     2   ⁢   np       +     cz     2   ⁢   np         )       -       w     2   ⁢   np       ⁡     (       ax     1   ⁢   np       +     by     1   ⁢   np       +     cz     1   ⁢   np         )                         y   w     =               a   ⁡     (         y     1   ⁢   np       ⁢     x     2   ⁢   np         -       y     2   ⁢   np       ⁢     x     1   ⁢   np           )       +     c   ⁡     (         y     1   ⁢   np       ⁢     z     2   ⁢   np         -       y     2   ⁢   np       ⁢     z     1   ⁢   np           )       +               d   ⁡     (         y     1   ⁢   np       ⁢     w     2   ⁢   np         -       y     2   ⁢   np       ⁢     w     1   ⁢   np           )                   w     1   ⁢   np       ⁡     (       ax     2   ⁢   np       +     by     2   ⁢   np       +     cz     2   ⁢   np         )       -       w     2   ⁢   np       ⁡     (       ax     1   ⁢   np       +     by     1   ⁢   np       +     cz     1   ⁢   np         )                         z   w     =               a   ⁡     (         z     1   ⁢   np       ⁢     x     2   ⁢   np         -       z     2   ⁢   np       ⁢     x     1   ⁢   np           )       +     b   ⁡     (         z     1   ⁢   np       ⁢     y     2   ⁢   np         -       z     2   ⁢   np       ⁢     y     1   ⁢   np           )       +               d   ⁡     (         z     1   ⁢   np       ⁢     w     2   ⁢   np         -       z     2   ⁢   np       ⁢     w     1   ⁢   np           )                   w     1   ⁢   np       ⁡     (       ax     2   ⁢   np       +     by     2   ⁢   np       +     cz     2   ⁢   np         )       -       w     2   ⁢   np       ⁡     (       ax     1   ⁢   np       +     by     1   ⁢   np       +     cz     1   ⁢   np         )                 
And the representation of (x p , y p , z p , w p )
 
   
     
       
         
           
             w 
             p 
           
           = 
           
             
               
                 
                   
                     
                       
                         w 
                         
                           1 
                           ⁢ 
                           np 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             ax 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             by 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             cz 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                         
                         ) 
                       
                     
                     - 
                   
                 
               
               
                 
                   
                     
                       w 
                       
                         2 
                         ⁢ 
                         np 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           ax 
                           
                             1 
                             ⁢ 
                             np 
                           
                         
                         + 
                         
                           by 
                           
                             1 
                             ⁢ 
                             np 
                           
                         
                         + 
                         
                           cz 
                           
                             1 
                             ⁢ 
                             np 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         
                           w 
                           
                             1 
                             ⁢ 
                             nm 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ax 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               by 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               cz 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               dw 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                   
                 
               
               
                 
                   
                     
                       c 
                       2 
                     
                     ⁢ 
                     
                       
                         w 
                         
                           2 
                           ⁢ 
                           nm 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             ax 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             by 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             cz 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             dw 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             x 
             p 
           
           = 
           
             
               
                 
                   
                     
                       b 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               x 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               y 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           - 
                           
                             
                               x 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               y 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               x 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               z 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           - 
                           
                             
                               x 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               z 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                   
                 
               
               
                 
                   
                     d 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             x 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           ⁢ 
                           
                             w 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                         
                         - 
                         
                           
                             x 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                           ⁢ 
                           
                             w 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         
                           w 
                           
                             1 
                             ⁢ 
                             nm 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ax 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               by 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               cz 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               dw 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                   
                 
               
               
                 
                   
                     
                       c 
                       2 
                     
                     ⁢ 
                     
                       
                         w 
                         
                           2 
                           ⁢ 
                           nm 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             ax 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             by 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             cz 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             dw 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             y 
             p 
           
           = 
           
             
               
                 
                   
                     
                       a 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               y 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               x 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           - 
                           
                             
                               y 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               x 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               y 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               z 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           - 
                           
                             
                               y 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               z 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                   
                 
               
               
                 
                   
                     d 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             y 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           ⁢ 
                           
                             w 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                         
                         - 
                         
                           
                             y 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                           ⁢ 
                           
                             w 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         
                           w 
                           
                             1 
                             ⁢ 
                             nm 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ax 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               by 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               cz 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               dw 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                   
                 
               
               
                 
                   
                     
                       c 
                       2 
                     
                     ⁢ 
                     
                       
                         w 
                         
                           2 
                           ⁢ 
                           nm 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             ax 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             by 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             cz 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             dw 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             z 
             p 
           
           = 
           
             
               
                 
                   
                     
                       a 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               z 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               x 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           - 
                           
                             
                               z 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               x 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       b 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               z 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               y 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           - 
                           
                             
                               z 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             ⁢ 
                             
                               y 
                               
                                 1 
                                 ⁢ 
                                 np 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                   
                 
               
               
                 
                   
                     d 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             z 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           ⁢ 
                           
                             w 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                         
                         - 
                         
                           
                             z 
                             
                               2 
                               ⁢ 
                               np 
                             
                           
                           ⁢ 
                           
                             w 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         c 
                         1 
                       
                       ⁢ 
                       
                         
                           w 
                           
                             1 
                             ⁢ 
                             nm 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ax 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               by 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               cz 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                             + 
                             
                               dw 
                               
                                 2 
                                 ⁢ 
                                 np 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                   
                 
               
               
                 
                   
                     
                       c 
                       2 
                     
                     ⁢ 
                     
                       
                         w 
                         
                           2 
                           ⁢ 
                           nm 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             ax 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             by 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             cz 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                           + 
                           
                             dw 
                             
                               1 
                               ⁢ 
                               np 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
   
   In case the new intersection point will participate in further clipping, it can be written in NHCS form:
 
 x   np   =b ( x   1np   y   2np   −x   2np   y   1np )+ c ( x   1np   z   2np   −x   2np   z   1np )+ d ( x   1np   w   2np   −x   2np   w   1np )
 
 y   np   =a ( y   1np   x   2np   −y   2np   x   1np )+ c ( y   1np   z   2np   −y   2np   z   1np )+ d ( y   1np   w   2np   −y   2np   w   1np )
 
 z   np   =a ( z   1np   x   2np   −z   2np   x   1np )+ b ( z   1np   y   2np   −z   2np   y   1np )+ d ( z   1np   w   2np   −z   2np   w   1np )
 
 w   np   =w   1np ( ax   2np   +by   2np   +cz   2np )− w   2np ( ax   1np   +by   1np   +cz   1np )
 
And
 
 Cw=c   1   w   1nm ( ax   2np   +by   2np   +cz   2np   +dw   2np )− c   2   w   2nm ( ax   1np   +by   1np   +cz   1np   +dw   1np )
 
   Here, C is the shifted bits and w is the weight, and the interpolate parameter is: 
                   T   p     =         w   p     -     w     1   ⁢   p             w     2   ⁢   p       -     w     1   ⁢   p                       =         -     c   2       ⁢       w     2   ⁢   nm       ⁡     (       ax     1   ⁢   np       +     by     1   ⁢   np       +     cz     1   ⁢   np       +     dw     1   ⁢   np         )                     c   1     ⁢       w     1   ⁢   nm       ⁡     (       ax     2   ⁢   np       +     by     2   ⁢   np       +     cz     2   ⁢   np       +     dw     2   ⁢   np         )         -                 c   2     ⁢       w     2   ⁢   nm       ⁡     (       ax     1   ⁢   np       +     by     1   ⁢   np       +     cz     1   ⁢   np       +     dw     1   ⁢   np         )                             
Miscellaneous Functions
 
   There are some functions that have not been discussed in the previous sections. These functions include: (1) NHCS functions that perform NHCS conversion; and (2) EffiDigit functions that calculates efficient digit of an integer. These functions will now be discussed. 
   
     
       
         
             
           
             
                 
             
             
               Calculate Efficient Digits in UFIX8 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               UFIX8 EffiDigit_UFIX8(UFIX8 a) 
             
             
                 
               This function calculates efficient digits in an 
             
             
                 
               UFIX8 integer 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a 
             
             
                 
                 
               Input integer, unsigned 8- 
             
             
                 
                 
               bits integer in UFIX8 format 
             
             
                 
               Return value 
               Efficient digit of the integer, 
             
             
                 
                 
               which equals ceil(log 2 (abs(a)) in 
             
             
                 
                 
               UFIX8 format. 
             
             
                 
               Remarks 
               Using Bisearch algorithm 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Calculate Efficient Digits in SFIX32 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               UFIX8 EffiDigit_SFIX32(SFIX32 a) 
             
             
                 
               This function calculates efficient digits in an 
             
             
                 
               SFIX32 integer 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a 
             
             
                 
                 
               Input integer, signed 
             
             
                 
                 
               32-bits integer in SFIX32 format 
             
             
                 
               Return value 
               Efficient digit of the integer, 
             
             
                 
                 
               which equals ceil(log 2 (abs(a)) in 
             
             
                 
                 
               UFIX8 format. 
             
             
                 
               Remarks 
               Using Bisearch algorithm 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Calculate Efficient Digits in SFIX64 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               UFIX8 EffiDigit_SFIX64(SFIX64 a) 
             
             
                 
               This function calculates efficient digits in an 
             
             
                 
               SFIX64 integer 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a 
             
             
                 
                 
               Input integer, signed 64-bits 
             
             
                 
                 
               integer in SFIX64 format 
             
             
                 
               Return value 
               Efficient digit of the integer, 
             
             
                 
                 
               which equals ceil(log 2 (abs(a)) in 
             
             
                 
                 
               UFIX8 format. 
             
             
                 
               Remarks 
               Using Bisearch algorithm 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Conversion from SFIX64Quad to SFIX32Quad NHCS 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               int NHCS_SFIX64Quad (SFIX64Quad a, SFIX32Quad b) 
             
             
                 
               This functions convert from non-NHCS to NHCS 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a 
             
             
                 
                 
               Input integers, signed 64-bits 
             
             
                 
                 
               Quad, in SFIX64Quad format. 
             
             
                 
                 
               b 
             
             
                 
                 
               Output integers, signed 32-bits 
             
             
                 
                 
               Quad, in SFIX32Quad, NHCS 
             
             
                 
                 
               format. 
             
             
                 
               Return value 
               An integer records shift bits 
             
             
                 
                 
               from 64-bit non-NHCS to 32-bit 
             
             
                 
                 
               NHCS. 
             
             
                 
               Remarks 
               NHCS_SFIX64Quad is used 
             
             
                 
                 
               in transform. In transform, 
             
             
                 
                 
               we need not shift when 
             
             
                 
                 
               efficient digits of maximum 
             
             
                 
                 
               component are less than 
             
             
                 
                 
               storage bits. 
             
             
                 
                 
               In clip space has either 
             
             
                 
                 
               NHCS or non-NHCS, For 
             
             
                 
                 
               recovering the correct w, 
             
             
                 
                 
               it needs to record the shift 
             
             
                 
                 
               bits. 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Conversion from SFIX64Triple to SFIX16Triple NHCS 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               Void NHCS_SFIX64Triple(SFIX64Triple a, 
             
             
                 
               SFIX16Triple b) 
             
             
                 
               This functions perform NHCS conversion 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a 
             
             
                 
                 
               Input integers, signed 
             
             
                 
                 
               64-bits Triple, non-NHCS 
             
             
                 
                 
               b 
             
             
                 
                 
               output integers, signed 
             
             
                 
                 
               16-bits Triple, NHCS 
             
             
                 
               Return value 
               No return value 
             
             
                 
               Remarks 
               NHCS_SFIX64Triple is 
             
             
                 
                 
               used in lighting before 
             
             
                 
                 
               normalization. Either 
             
             
                 
                 
               efficient digit of maximum 
             
             
                 
                 
               component is less than 
             
             
                 
                 
               storage bits or not, we 
             
             
                 
                 
               need shift to preserve 
             
             
                 
                 
               precision. 
             
             
                 
                 
             
          
         
       
     
   
   
     
       
         
             
           
             
                 
             
             
               Conversion from SFIX64Triple to SFIX16Triple NHCS 
             
             
                 
             
           
          
             
                 
             
          
         
         
             
             
          
             
                 
               Void NHCS_SFIX64Triple(SFIX32Triple a, 
             
             
                 
               SFIX16Triple b) 
             
             
                 
               This functions perform NHCS conversion 
             
          
         
         
             
             
             
          
             
                 
               Parameters 
               a 
             
             
                 
                 
               Input integers, signed 
             
             
                 
                 
               32-bits Triple, non-NHCS 
             
             
                 
                 
               b 
             
             
                 
                 
               output integers, signed 
             
             
                 
                 
               16-bits Triple, NHCS 
             
             
                 
               Return value 
               No return value 
             
             
                 
               Remarks 
               NHCS_SFIX32Triple is 
             
             
                 
                 
               used in lighting before 
             
             
                 
                 
               normalization. Either 
             
             
                 
                 
               efficient digit of maximum 
             
             
                 
                 
               component is less than 
             
             
                 
                 
               storage bits or not, we 
             
             
                 
                 
               need shift to preserve 
             
             
                 
                 
               precision. 
             
             
                 
                 
             
          
         
       
     
   
   The foregoing description of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description of the invention, but rather by the claims appended hereto.