Patent Publication Number: US-4930092-A

Title: Polygon display apparatus and method

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to the field of computer graphics and more particularly to the field of displaying three dimensional (3-D) objects in two-dimensional (2-D) form. 
     2. Description of the Prior Art 
     Computer graphics systems for displaying 3-D objects in 2-D form, such as on a video screen, must first define the surface or shape of the object to be displayed. The surface is defined in terms of each of the three coordinates (x, y, and z) of many points (or vertices) that are used to define the surface. Equations are used to specify how the x coordinate, for example, varies with variations of the parameters &#34;s&#34; and &#34;t&#34;, for example. As the complexity of the objects to be displayed increases, it is still necessary to minimize both (1) the time required to process the surface equations so as to obtain the values of the x, y, and z coordinates of the vertices that define the surface of the object, and (2) the amount of data that must be stored as the x, y, and z coordinates are computed. 
     In the past, as the values of the x, y, and z coordinates of the vertices have been obtained, selected vertices have been organized into polygons formed by from 3 to &#34;n&#34; vertices, where &#34;n&#34; is selected according to the complexity of the surface to be displayed. Some of those systems separately identified each polygon. For example, in the Thesis &#34;A Computer Graphics Program for the Generation of Half-Tone Images with Shadows&#34;, by K. C. Kelley, published in 1970, (NTIS AD-698 124), for each polygon there was provided a block of information including plane coefficients, reflectivity coefficients, the number of vertices of the polygon and the coordinates of the first through the &#34;n th  &#34; vertices. 
     Similarly, the Thesis &#34;Applications of B-Spline Approximation to Geometric Problems of Computer-Aided Design&#34;, by Richard Riesenfeld, Computer Science, University of Utah, UTEC-CSc-73-126, Mar., 1973, (&#34;Riesenfeld&#34;) dealt with individual Bezier polygons (that were open or closed) to define curves that are to be rendered on a graphics display. 
     In a different approach to managing the data corresponding to the x, y, and z coordinates of the vertices that define an object, Foley and Van Damm, in their text &#34;Fundamentals of Interactive Computer Graphics&#34;, (&#34;Foley&#34;), refer to an object defined by a mesh formed by polygons. The vertices were said to be stored in a vertex list and a polygon was said to be defined by a list of pointers into the vertex list. Thus, a polygon made up of vertices 3, 5, 7, and 10 in the vertex list would be represented as P=(3, 5, 7, 10). Foley refers to the storage of each vertex just once, such that considerable space is saved, which appears to indicate that the values of all of the vertices that define the object are obtained at once and stored. 
     Computer graphics systems for displaying 3-D objects in 2-D form are also now expected to show the object with hidden lines removed and to shade the surface of the object as if a light source were shining on the object. For example, the above-noted Kelley Thesis provided a capability of locating the light source at a point away from the observer. 
     To provide the hidden line and shading capabilities, the normal to the surface of a polygon (the &#34;polygon normal&#34;) has been determined. As reported in the Foley text at pages 582 and 583, Gouraud shading, or intensity interpolation shading, uses such polygon (or surface) normals to obtain the vertex (or point) normal by averaging the polygon normals of all polygons that share a given vertex. The intensity at the given vertex is obtained by using the vertex normal with a shading model. The limitations of such averaging of polygon normals are discussed in the text Principles of Interactive Computer Graphics, by W. M. Newman and R. F. Sproull, McGraw-Hill, 1973, (&#34;Newman&#34;). Such limitations include inaccuracy of the point normal, the increased time required to obtain the point normal due to the large number of surrounding polygons that share the given vertex, and the need to store additional vertex data that is required due to the increased number of surrounding polygons. 
     With respect to shading capabilities, the intensity at a given point or vertex must be determined. This is done by first obtaining the point normal and then using intensity models to obtain the intensity at the point as a function of the point normal. Point normals have been obtained by taking the cross product of the partial derivatives of the given vertex with respect to parameters such as &#34;s&#34; and &#34;t&#34;. Thus, the partial derivatives  dx  / ds  and  dx  / dt  were used. Having such cross-product, the values of the parameters and other variables in the cross-product equation were then substituted into such equation. Experience indicated that it required prohibitive amounts of computation time to recompute all of the terms of the partial derivative equation with respect to each of the coordinates x, y, and z for each value of the parameters s and t simply to obtain the value of the partial derivatives with respect to x, y, and z. To avoid such extensive computations, the averaging of polygon normals discussed by Newman has been used. 
     In the dissertation, &#34;A Subdivision Algorithm for Computer Display of Curved Surfaces&#34;, by Edwin Catmull, Dec., 1974, UTEC-CSc-74-133, Department of Computer Science, University of Utah.) (&#34;Catmull&#34;) it was also recognized that to produce high quality computer-generated images of surfaces and curved solid objects on a raster-scan output device, it was necessary to both accurately represent the surfaces and to control shading and texture. Catmull discussed the difficulties in using polygons to approximate curved surfaces. Catmull referred to a report by Henri Gouraud, &#34;Computer Display of Curved Surfaces&#34;, Department of Computer Science, University of Utah, UTEC-CSc-71-113, Jun., 1971, who associated a scalar light intensity value with each vertex of a polygon. Catmull also referred to a report by Bui Tuong-Phong, entitled &#34;Illumination for Computer-Generated Images&#34;, Department of Computer Science, University of Utah, UTEC-CSc-71-113, Jun. 1971. Phong interpolated the entire surface normal vector between vertices and edges instead of the scalar intensity values that Gouraud used. This yielded a normal at every display point, (or vertex) which was used to calculate intensity. Catmull noted that such normal was not the mathematically correct one. As understood, inaccuracy results from the amount of interpolation between points that are calculated exactly. 
     Catmull also discussed using surface normals for intensity calculations (page 34) and referred to Phong who calculated intensity using the surface normal and the light sources. This was complicated because the equation of the normal to a bi-cubic patch is a fifth degree polynomial. This method requires more information in order to obtain the tangents and, of course, it requires the extra computation involved in taking a cross product at every point. 
     In a thesis by James Henry Clark entitled, &#34;3-D Design of Free-Form B-Spline Surfaces&#34;, Department of Computer Science, University of Utah, UTEC-CSc-74-120, Sept., 1974, (&#34;Clark&#34;), page 27 refers to computation of an equation (2.15) for each point of a surface. The equation (2.15) is for S as a function of s and t. The point is made that to perform computations of 16 times for each value of s and t simply to compute one vertex requires too many computations. On page 28, to reduce the computations a table look-up is suggested based on pre-computing the equation using values of s and t. Then, at initialization time, the surface is completely evaluated and stored in tabular form. 
     SUMMARY OF THE INVENTION 
     In contrast to the prior art that computed and stored all vertices that define the surface of the object, so that vertices could be later organized into polygons, the preferred embodiment of the present invention determines the values of a relatively few vertices and efficiently groups them into polygons before determining the value of additional vertices to achieve the following objectives. 
     Polygon display apparatus and method according to the present invention reduce the time required to compute the vertices of the polygons that represent objects to be displayed and the vertex (or point) normals associated with such vertices. 
     The present invention also significantly reduces the requirements for storing data that represents the values of polygon vertices by computing such vertices on a row by row basis, where such rows are selected so as to substantially reduce the time required to obtain values for the vertex coordinates and the vertex normals. 
     With these objects in mind, the polygon display apparatus and method of the present invention use an equation having two parameters to define the coordinates of a vertex that represents a point on the surface of an object to be displayed. At first, one of the parameters (e.g. &#34;t&#34;) is kept constant to enable factoring of the equation to reduce the time required to obtain the values of a row of vertices as the other parameter (e.g. &#34;s&#34;) is varied. A second row of vertices is obtained in the same manner after adjusting the value at which the first parameter is held constant. Similar factoring is performed in each of the partial derivative equations for the partial derivative of the coordinates with respect to &#34;s&#34; and &#34;t&#34;, e.g.,  dx  / ds  and  dx  / dt  for the x coordinate of the vertices. This factoring is based on the fact that &#34;t&#34; is no longer a variable with respect to each row, such that the number of logic operations necessary to obtain the value of the partial derivatives for the vertex is substantially reduced. 
     The selection of only two rows of vertex and point normal data substantially reduces the amount of data that must be stored as polygons are defined. For ease of reference, such reduction in such data to two rows is referred to as a &#34;minimum set&#34; of vertex data since two rows of data are required before polygons are defined, as discussed below. 
     Additionally, once the minimum set of point and point normal data has been obtained and stored for each of the two rows, that data for the two rows is then processed into polygons. In particular, polygons that define the surface of the object to be displayed are identified in a predetermined sequence using a minimum number of steps. Adjacent identified polygons share an edge and that edge is not displayed twice when all polygon outlines are displayed, thus avoiding the problems noted in the Foley text at page 509. Further, this minimizes the amount of row storage required to record the polygons because all of the polygons are not stored at one time for the entire object. Rather, a first polygon is selected by identifying a first group of three vertices from the two rows. Those three vertices are separately stored in polygon storage and then output as a group as a first polygon. Then, one additional vertex is selected from row storage and it is stored in polygon storage by replacing the first vertex in polygon storage, such that only vertices 2, 3, and 4 are in the polygon storage. Vertices 2, 3, and 4 are output as a group as the second polygon. This process is repeated until all of the vertices in the first two rows have been used to define successive polygons. 
     Then the values of the third row of polygon vertices and the respective point normals are obtained and written into row storage in substitution for the first row and the process of defining additional successive polygons is continued. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is an schematic drawing showing how parameters &#34;s&#34; and &#34;t&#34; vary to define vertices of an object to be displayed; 
     FIG. 2 is an enlarged portion of FIG. 1 showing how the vertices shown in FIG. 1 are organized to form polygons; 
     FIG. 3 is an enlarged portion of FIG. 2 showing selected polygons in a polygon row no. 1 formed from vertices of vertex rows 1 and 2; 
     FIGS. 4A and 4B, when joined at lines (a)-(e), form a schematic diagram of a processor that generates signals representing the vertices and polygons shown in FIGS. 1-3; and 
     FIG. 5 is a schematic diagram of a circuit for controlling the processor. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Introduction 
     Referring first to FIGS. 1, 2, and 4B in the preferred embodiment of the present invention, cubicorder surfaces are used to represent an object (not shown) to be displayed in a 3-D field of view. A standard 3-D world or object space coordinate system is used. The locations of the points (not shown) of such surfaces vary as a function of the parameters &#34;s&#34; and &#34;t&#34; that are shown in FIG. 1. The locations of vertices 9 represent the points and are expressed in terms of coordinates x, y, and z that define distances within the 3-D world or object space coordinate system. 
     According to the method of the present invention, the surfaces of the object are graphically displayed by controlling the values of electrical signals 10 (FIG. 4B) that represents the x, y, and z coordinates of the vertices 9 that define the points on the surface of the object. The method includes the steps of selecting an equation having at least two parameters (e.g. &#34;s&#34; and &#34;t&#34; shown in FIG. 1) for defining the x, y, and z coordinates of the vertices 9. A first of the parameters (e.g. &#34;t&#34;) is identified and the value of it is kept constant at a first amount, shown as 0 in FIG. 1. The equation is then factored to express the coordinates x, y, and z in terms of the first constant &#34;t&#34; and of a variable represented by a second one (e.g. &#34;s&#34;) of the parameters. The equation is sequentially solved by substituting a predetermined series of values for the second one of the parameters (&#34;s&#34;), shown as s=0 through s=1.0 in FIG. 1, to control the values of certain of the vertex electrical signals 10 (FIG. 4B) and to define the coordinates x, y, and z of a series 11 (FIG. 2) of the vertices V1, V3, V5, V7 . . . V21 (FIG. 2) that form a first vertex row 12 (row 0 in FIG. 1) of the points on the surface of the object. The factoring and solving steps are repeated using a second amount (t=0.2, FIG. 1) for the first parameter &#34;t&#34; and the same predetermined series, shown as s=0 through s=1.0 in FIG. 1, of values for the second one &#34;s&#34; of the parameters to control the values of certain other of the electrical signals 10 and to define at least the coordinates of a second series 13 (FIG. 2) of the vertices 9 that form a second vertex row 14 (row 1 in FIG. 1) of the points on the surface of the object. 
     In other aspects of the method of the present invention, the vertices V1, V3 . . . V21 of the first vertex row 12 and the vertices V2, V4 . . . V22 of the second vertex row 14 are organized into triangular polygons 15, identified as P1 through P100 in FIG. 1, by performing the further step of selecting the coordinates x, y, and z of a pair 16 formed by the vertices V1 and V3 at one end 17 of the first vertex row 12 and the coordinates x, y, and z of one vertex V2 from the corresponding end 17 of the second vertex row 14 to identify and store those electrical signals 10 that represent a first polygon P1 (FIG. 1) that defines certain of the points on the surface of the object to be displayed. The coordinates x, y, and z of the next vertex V4 (FIGS. 2 and 3) of the second vertex row 14 are selected and the coordinates x, y, and z of the first vertex V1 of the first vertex row 12 are discarded to generate and store electrical signals 10 that represent the next polygon P2 (FIG. 1) adjacent the last-formed polygon P1 for defining an additional point on the surface of the object. The selection of the coordinates x, y, and z of the next vertices 9 is continued by alternately and sequentially selecting the next vertex 9 from the first vertex row 12 and then from the second vertex row 14 and discarding a selected previous vertex 9 so that the additional electrical signals 10 are generated for representing a series 18 (FIG. 1) of polygons 15 formed between all of the vertices 9 of the first and second vertex rows 12 and 14, respectively. The series 18 corresponds to polygon row 1 shown in FIG. 1. 
     The factoring and solving steps are repeated using a third amount (&#34;t&#34;=0.4) for the first parameter &#34;t&#34; and the same predetermined series, shown as s=0 through s=1.0 in FIG. 1, of values for &#34;s&#34; to control the values of certain other of the electrical signals 10 and to define the coordinates x, y, and z of an additional series 19 of the vertices 9 that form an additional vertex row 2 of the points on the surface of the object. After the electrical signals 10 are obtained to define the vertices 9 of each new vertex row, the coordinates x, y, and z of the vertices 9 of the prior vertex row (e.g. 14) and the new vertex row (e.g. 19) are selected to generate electrical signals 10 that represent the polygons 15 defined by those two vertex rows 14 and 19. The vertex rows 14 and 19 (vertex row numbers 1 and 2 in FIG. 1) form a polygon row 20 (shown as polygon row number 2 in FIG. 1). 
     Referring to FIGS. 4A, 4B, and 5, apparatus 21 of the present invention generates the signals 10 for graphically displaying the object. A font RAM circuit 22 is provided for registering the vertex electrical signals 10 that represent the first and second vertex rows 12 and 14 of the vertex locations. In FIG. 5, a control circuit 23 is shown for selecting the vertex electrical signals 10 that represent the pair 16 of vertices V1 and V3 at the one end 17 of the first vertex row 12 and the V2 vertex 9 at the corresponding end 17 of the second vertex row 14 to provide the vertex electrical signals 10 that define the first or P1 polygon 15 for graphically displaying the object. 
     The control circuit 23 then selects from the RAM circuit 22 a next vertex electrical signal 24 representing the next or V4 vertex 9 of the second vertex row 14 and substitutes the next signal 24 for one of the previously selected signals 10 to define the next or P2 polygon 15 along the first polygon row 18 for representing the object. The control circuit 23 alternately selects the vertex electrical signals 10 representing the vertices 9 from the first vertex row 12 and then from the second vertex row 14 so that the series 18 of polygons 15 is formed between the first and second vertex rows 12 and 14, respectively. The control circuit 23 is rendered effective upon selection of the vertex electrical signals 10 representing all of the vertices 9 of the first and second vertex rows 12 and 14, respectively, for writing into the RAM circuit 22 the vertex electrical signals 10 representing vertices 9 in the third vertex row 19 in place of the first row 12 of vertices 9. The control circuit 23 then cycles to select vertex electrical signals 10 representing a pair 25 of vertices 9 (V2 and V4) at the one end 17 of the second vertex row 14 and the V24 vertex 9 at the corresponding end 17 of the third vertex row 19 to provide the vertex electrical signals 10 that define an additional P21 polygon 15 for graphically displaying the object. A register 26 (FIG. 4A) defines the number (shown as six in FIG. 1) of vertex rows 12, 14, etc. to be used in displaying the object. In response to forming the last or P100 polygon 15 from the last two vertex rows 27 and 28 of that number (six) of rows, the control circuit 23 stops. 
     SURFACE DEFINITION EQUATIONS 
     Considering how the cubic-order surfaces of the object to be displayed are defined, the parametric form for cubic-order surfaces is stated in equation (1) below. Equation (1) is the general form for Hermite, Bezier, and B-spline curves according to the Foley text, page 524. Equations for the y and z coordinates are equivalent. ##EQU1## where s=(0,1), t=(0,1). 
     Equation (1) can also be expressed as: 
     
         x(s,t)=S*Dx*T.sup.t                                        (2) 
    
     where S=[s 3  s 2  s 1], T=[t 3  t 2  t 1], and T t  is the transpose of T. 
     Similarly, equations (1) and (2) can be expressed in terms of the &#34;y&#34; or &#34;z&#34; coordinates, in which cases the respective &#34;Dy&#34; and &#34;Dz&#34; terms are used instead of &#34;Dx&#34;. 
     Referring to equation (1), the terms &#34;D11&#34;, &#34;D12&#34;, &#34;D13&#34;, &#34;D14&#34;, &#34;D21&#34;, &#34;D22&#34;, &#34;D23&#34;, &#34;D24&#34;, &#34;D31&#34;, &#34;D32&#34;, &#34;D33&#34;, &#34;D34&#34;, &#34;D41&#34;, &#34;D42&#34;, &#34;D43&#34;, and &#34;D44&#34; represent coefficients that are defined by a coefficient matrix, there being one coefficient matrix for each coordinate x, y, and z. 
     For example, the x coefficient matrix is obtained as follows: 
     
         ______________________________________                                    
PROCESS 1                                                                 
Obtain x Coefficient Matrix                                               
Step No.    Step                                                          
______________________________________                                    
1.1         enter an input x coefficient matrix.                          
1.2         concatenate a magic matrix with the                           
            input x coefficient matrix.                                   
1.3         concatenate the resulting matrix with                         
            the transpose of the magic matrix.                            
1.4         get the x coefficient matrix.                                 
______________________________________                                    
 
    
     To perform Process 1, the matrix elements are arranged as follows: ##EQU2## Based on this arrangement, the input x coefficient matrix is: ##EQU3## where x(0,1) means the value of x at s=0, t=1, for example. 
     The magic matrix for the Hermite form of parametric cubic surfaces is defined on page 57 in the Catmull text as follows: ##EQU4## The Step 1.2 concatenation of the magic matrix and the input x coefficient matrix is a post-concatenation. With the input x coefficient matrix (4) referred to as B, and the resulting concatenated matrix referred to as C, the post-concatenation results in the following: ##EQU5## 
     In Step 1.3, the concatenated matrix C resulting from Step 1.2 and the transpose of the magic matrix (5) are concatenated. This is a post-concatenation. The transpose of the magic matrix is as follows: ##EQU6## 
     The resulting x coefficient matrix is referred to as &#34;D&#34; and is in the form of matrix (3), where: ##EQU7## 
     Process 1 is described in greater detail below in respect to Charts 1A through 1L. 
     Referring to the parametric form for cubic-order surfaces stated in equation (1) and the coefficient matrix elements (8), it is observed that &#34;s&#34; and &#34;t&#34;  are parameters that vary, whereas the elements of the x coefficient matrix (e.g. &#34;D11&#34;) represent a number (i.e., a constant). In the past, to obtain a value for the coordinate &#34;x&#34;, the particular &#34;s&#34; and &#34;t&#34; values were substituted for the s, s 2 , s 3 , etc. terms of equation (1) and calculated, or a look up table was used to obtain s 2 , s 3 , etc. for a given value of &#34;s&#34; and t 3  and t 2  for a given value of &#34;t&#34;. The various operations called for in equation (1) were then a function of the values of &#34;s&#34; and &#34;t&#34;. As noted above, when such values of &#34;s&#34; and &#34;t&#34; were substituted into equation (1), it took a prohibitive number of machine cycles to calculate all of the values of the coordinates x, y, and z of the various vertices 9. Further, such selection of &#34;s&#34; and &#34;t&#34; was unrelated to the eventual grouped relationship of certain vertices (e.g. V1, V2, and V3) into the polygons 15. 
     FACTORING SURFACE EQUATIONS 
     On the other hand, the preferred embodiment of the present invention substantially reduces the number of machine cycles required to compute the coordinates x, y, and z of each of the vertices 9 that represent the object. In particular, one of such parameters is held constant and the other of the parameters is varied. For each value of the other parameter, the x, y, and z coordinates of a given vertex 9 are obtained. Further, by holding the first parameter constant, equation (1) can be factored and reduced to a function of only one parameter. As a result, as the second parameter is varied, the number of machine cycles required to obtain the values of the coordinates of each vertex 9 is substantially reduced. The parameter held constant is then incremented by one increment and held constant while the other parameter is varied through its entire range. This results in obtaining values for the x, y, and z coordinates of only the two series 11 and 13 of vertices 9, which are stored and then organized into the form of the series 18 of polygons 15. Then a third vertex row 19 of vertices 9 is obtained and another series 20 of polygons 15 is formed. 
     Referring now in detail to FIG. 1, the selection of the increments by which the parameters &#34;s&#34; and &#34;t&#34; are varied (or incremented) may be appreciated. Each of the parameters &#34;s&#34; and &#34;t&#34; is incremented or stepped by amounts that indicate how finely the surface of the object to be displayed will be drawn. The surface of the object to be displayed is determined by a series of curves, generated by varying &#34;s&#34; and &#34;t&#34; from 0 to 1. The step sizes by which &#34;s&#34; and &#34;t&#34; vary determine how many curves will make up the surface. The smaller the step size, the more detailed the surface will appear. The smallest step size allowed for both &#34;s&#34; and &#34;t&#34; is 1. This means that 100/1, or 100, steps is the finest the surface may be divided. Since the values of &#34;s&#34; and &#34;t&#34; are used frequently, as well as their squared and cubed values, these values are precalculated and stored. 
     The &#34;s&#34; and &#34;t&#34; step sizes are input as two 16 bit integers with a value from 1 to 100. The &#34;s&#34; and &#34;t&#34; step sizes may be different. 
     The surface of the object to be displayed is generated in a grid-like manner. The &#34;s&#34; step size determines the number of columns used to generate the surface, while the &#34;t&#34; step size determines the number of rows used. The number of columns used is 1 plus (100/&#34;s&#34; step size). The number of rows is 1 plus (100/&#34;t&#34; step size). For example, FIG. 1 shows an &#34;s&#34; step size of 10. Since (100/10) is 10, there are 1+10=11 columns numbered 0 to 10. For data storage purposes, these columns have &#34;column addresses&#34;  of 0 to 10. Similarly, a &#34;t&#34; step size of 20 is shown. Since 100/20 is 5, there are 1+5=6, there are 6 vertex rows numbered 0 to 5, which are identified by the reference numbers 12, 14, 19, 27, and 28, respectively. Again, for data storage purposes, these vertex rows have vertex row addresses of 0 to 5. 
     As will be described below with respect to Process 2, a given vertex 9 is defined by substituting selected &#34;s&#34; and &#34;t&#34; values into the x coordinate equation and into the corresponding y and z coordinate equations. The &#34;s&#34; and &#34;t&#34; values that are used result in the given vertex 9 having a particular vertex row address and a particular column address. Thus, in the example shown in FIGS. 2 and 3, a first or V1 vertex 9 on the lower left end 17 of FIG. 2 has a vertex row address of &#34;0&#34; and a column address of &#34;0&#34; and is obtained using &#34;s&#34;=0 and &#34;t&#34;=0 (stated as &#34;(0,0)&#34;) in the x coordinate equation and in the corresponding y and z coordinate equations. The next V3 vertex 9 to the right has a 0 vertex row address and a 1 column address, and is obtained using &#34;s&#34;=0.1 and &#34;t&#34;=0, (stated as &#34; (0.1,0)&#34;). Considering a 1 vertex row address, the first or V2 (left hand most) vertex 9 has a 1 vertex row address and a 0 column address and is obtained using &#34;s&#34;=0 and &#34;t&#34;=0.2. 
     As described below with respect to Processes 3 and 4, the vertex data for a given (e.g. V1) vertex 9 is accessed, transformed and the point intensity calculated. This is done for the V2 vertex 9 and then for the V3 vertex 9. Then these three V1, V2, and V3 vertices 9 are combined to form the first polygon P1 (FIG. 1). 
     DEFINITION OF MULTIPLIERS FOR COORDINATE EQUATION (1) 
     With the &#34;s&#34; and &#34;t&#34; grid in mind, the effect of holding &#34;t&#34; constant while varying &#34;s&#34; may be appreciated by referring to equation (1) and to the coefficient matrix elements (8). When the parameter &#34;t&#34; is held constant while &#34;s&#34; is incremented, there is only one variable, namely &#34;s&#34;, in equation (1). This enables the factoring of equation (1) as follows: ##EQU8## Since &#34;t&#34; is constant for any vertex row address, the terms in parentheses in equation (9) are constant. To save on calculation time, those terms are calculated only once per row (and are stored) as follows: ##EQU9## The terms &#34;s3&#34;, &#34;s2&#34;, &#34;s1&#34;, and &#34;s0&#34; are referred to as &#34;multipliers&#34;. 
     Referring to the multiplier equations (10), it is observed that the multipliers s3, s2, s1 and s0 are constant for any given vertex row address since &#34;t&#34; is constant for any given vertex row address. Thus, equation (9) reduces to the following: 
     
         x(s,t)=s.sup.3 (s3)+s.sup.2 (s2)+s(s1)+s0.                 (11) 
    
     Having equation (11), in order to obtain the value for the x coordinate of a particular vertex 9, it is simply necessary to insert into equation (11) the value of the multipliers s0, s1, s2, and s3 and the values of s 3 , s 2 , and s. Thus, the number of operations to determine the value of the x coordinate, for example, is reduced to 6, including 3 multiplies and 3 addition steps. 
     In the same manner as equation (11) was obtained, the following factored equations for the y and z coordinates of each vertex 9 are obtained and are as follows: 
     
         y(s,t)=s.sup.3 (s3)+s.sup.2 (s2)+s(s1)+s0                  (13) 
    
     
         z(s,t)=s.sup.3 (s3)+s.sup.2 (s2)+s(s1)+s0                  (14) 
    
     In equations (13) and (14), the elements D11, D12, D13, etc. in the multipliers s3, s2, s1, and s0 refer to the y or z coefficient matrices instead of to the x coefficient matrix. 
     DEFINITION OF MULTIPLIERS FOR POINT NORMAL VECTORS 
     The preferred embodiment of the present invention also substantially reduces the time necessary to obtain a point normal vector for each vertex 9. The point normal vector is used in obtaining a point intensity for each vertex 9. The point normal vector is the cross-product of the partial derivatives of the coordinates x, y, and z of the vertex 9 with respect to each variable (or parameter) &#34;s&#34; and &#34;t&#34;. The cross-product is as follows: ##EQU10## where: point normal x=(dz/ds*dy/dt)-(dy/ds*dz/dt) 
     point normal y=(dx/ds*dz/dt)-(dz/ds*dx/dt) 
     point normal z=(dy/ds*dx/dt)-(dx/ds*dy/dt) 
     Before the cross product can be obtained, the partial derivatives must be determined. To determine the partial derivatives, the point coordinate equation (1) is differentiated with respect to &#34;s&#34; and also with respect to &#34;t&#34;. The equations for the x coordinate of these partial derivatives are as follows: ##EQU11## where &#34;D&#34; is the x coefficient matrix. 
     These equations can be factored as follows: ##EQU12## The following terms of the factored equations (18) and (19) are calculated once per vertex row address, and the resulting constants or multipliers are stored at the following addresses: 
     
         ______________________________________                                    
Address of                                                                
Multiplier Term                                                           
______________________________________                                    
(dx/ds s2) 3*D11*t.sup.3 + 3*D12*t.sup.2 + 3*D13*t                        
           +3*D14                                                         
(dx/ds s1) 2*D21*t.sup.3 + 2*D22*t.sup.2 + 2*D23*t                        
                                (20)                                      
           + 2*D24                                                        
(dx/ds s0) D31*t.sup.3 + D32*t.sup.2 + D33*t + D34                        
(dx/dt s3) 3*D11*t.sup.2 + 2*D12*t + D13                                  
(dx/dt s2) 3*D31*t.sup.2 + 2*D22*t + D23                                  
                                (21)                                      
(dx/dt s1) 3*D31*t.sup.2 + 2*D32*t + D33                                  
(dx/dt s0) 3*D41*t.sup.2 + 2*D42*t + D43                                  
______________________________________                                    
 
    
     The method and apparatus 21 of the present invention obtain these multipliers as described in greater detail with respect to Process 2 below. 
     The equations (18) and (19) for the partial derivative coordinates can be expressed in their fully factored form in terms of the multipliers in (20) and (21) as follows: ##EQU13## 
     The partial derivative coordinates (equations (22) and (23)) are thus expressed in terms of the single variable (&#34;s&#34;). 
     The same factoring and calculation is performed for the coordinates y and z with their corresponding coefficient matrices to obtain equations equivalent to equations (22) and (23) for the partial derivatives of such coordinates with respect to &#34;s&#34; and &#34;t&#34;. 
     POINT NORMAL VECTORS 
     To enable the point normal to be obtained for a given vertex 9, a value of &#34;s&#34; is substituted into each such partial derivative equation (22) and (23) to obtain the values of the partial derivatives of x with respect to &#34;s&#34; and &#34;t&#34; for that value of &#34;s&#34;. &#34;s&#34; is incremented once for each vertex 9 that has a given vertex row address and the calculations are repeated to obtain all of the values of the partial derivatives of x with respect to &#34;s&#34; and &#34;t&#34; for that given vertex row address. 
     Similarly, a value of &#34;s&#34; is substituted into each such partial derivative equation (22) and (23) corresponding to the y coordinate to obtain the values of the partial derivatives of y with respect to &#34;s&#34; and &#34;t&#34; for that value of &#34;s&#34;. &#34;s&#34; is incremented once for each vertex 9 that has that given vertex row address and the calculations are repeated to obtain all of the values of the partial derivatives of y with respect to &#34;s&#34; and &#34;t&#34; for that given row address. 
     Similarly, a value of &#34;s&#34; is substituted into each such partial derivative equation (22) and (23) corresponding to the coordinate to obtain the values of the partial derivatives of z with respect to &#34;s&#34; and &#34;t&#34; for that value of &#34;s&#34;. &#34;s&#34; is incremented once for each vertex 9 that has that given vertex row address and the calculations are repeated to obtain all of the values of the partial derivatives of z with respect to &#34;s&#34; and &#34;t&#34; for that given vertex row address. 
     This results in values for the various partial derivatives of each coordinate x, y, and z with respect to &#34;s&#34; and &#34;t&#34;, which are necessary for computing the cross product vectors for the various vertices 9 having that given vertex row address. In the example shown, the cross product for the x coordinate of the point normal vector for the V1 vertex 9 is then obtained. Next, the cross product for the y coordinate of the point normal vector for the V1 vertex 9 is obtained, whereafter the cross product for the z coordinate of the point normal vector for the V1 vertex 9 is obtained. This process is repeated for all of the vertices V2 through Vn having that given vertex row address, where Vn in FIGS. 2 and 3 is 21 for vertex row address 0. 
     When the x, y, and z coordinates of the point normal vector have been obtained for each vertex 9 having that given vertex row address, the processing of the point normal vectors for that given vertex row address is complete. The other variable &#34;t&#34; is then incremented, &#34;s&#34; is set to &#34;0&#34; and then &#34;s&#34; is incremented to obtain values of the coordinates x, y, and z of the vertices 9 and of the corresponding point normals for the next vertex row address 1 that corresponds to the vertex row 14. 
     VERTEX COORDINATE AND POINT NORMAL VECTOR PROCESSING 
     In greater detail, the process of calculating values of the coordinates x, y, and z of the vertices 9 and the point normal vectors of the two row vertex addresses 0 and 1 corresponding to the vertex rows 12 and 14 is as follows. One coordinate of a vertex 9 and a partial derivative with respect to &#34;s&#34; and also with respect to &#34;t&#34; for the coordinate of that vertex 9 is calculated at a time. The x coordinate is described below, it being understood that the calculations of the y and z coordinates are similar. Equation (1) for generating the x coordinate of the point is as follows: ##EQU14## 
     As described above, equation (1) is factored for use with the minimum set of vertices 9 defined by vertex row addresses 0 and 1 and appears as equation (11) as follows: 
     
         x(s,t)=s.sup.3 (s3)+s.sup.2 (s2)+s(s1)+s0.                 (11) 
    
     Also having the partial derivative coordinate equations in factored form (equations (22) and (23) above), the generation of the vertices 9 for the first two vertex row addresses 0 and 1 and the point normal vectors corresponding to such vertices 9 proceeds by setting &#34;t&#34; to 0. Next, s3, s2, s1, and s0 are calculated and stored. Then the multipliers (dx/ds s2), (dx/ds s1), (dx/ds s0), (dx/dt s3), (dx/dt s2), (dx/dt s1), and (dx/dt s0) are computed and stored. Next &#34;s&#34; is set to 0. The x coordinate of the V1 vertex 9 (0,0) is calculated as follows: 
     
         vertex (s,t)x=(s3*s.sup.3)+(s2*s.sup.2)+(s1*s)+s0          (24) 
    
     The partial derivatives for the vertex V1 (0,0) are computed as follows: 
     
         dx/ds(s,t)=[(dx/ds s2)*s.sup.2 ]+[(dx/ds s1)*s]+(dx/ds s0) (25) 
    
     
         dx/dt(s,t)=[(dx/dt s3)*s.sup.3 ]+[(dx/dt s2)*s.sup.2 ]+[(dx/dt s1)*s]+(dx/dt s0)                                         (26) 
    
     To continue obtaining values for the x coordinate for the first vertex row address 0, &#34;s&#34; is incremented by &#34;s&#34; step size and the above computations for x point coordinate and x partial derivatives are repeated. This continues, with &#34;s&#34; being incremented each time by &#34;s&#34; step size until &#34;s&#34; is greater than one. Then &#34;s&#34; is set back to 0 and the process is repeated to obtain the y point coordinate and y partial derivatives. The same is then done for the z coordinate. 
     When the 0 vertex row address of vertices 9 and partial derivatives has been completed, the cross-product of the &#34;s&#34; and &#34;t&#34; partial derivatives is computed as described above and is stored as point normal vectors. &#34;t&#34; is then incremented by &#34;t&#34; step size, &#34;s&#34; is set back to 0, and the above steps are repeated for the vertices 9 and point normal vectors of the vertex row address 1 that corresponds to the vertex row 14. 
     The steps in Process 2 can thus be summarized as follows: 
     
         ______________________________________                                    
PROCESS 2                                                                 
Obtain Values for Vertex Coordinates x, y, and z                          
of Vertex Row Addresses 0 and 1 and                                       
Corresponding Point Normal Vectors                                        
Step #       Step                                                         
______________________________________                                    
2.0   For TCNTR = 0 and t step size do:                                   
2.0.1    for x, y, and z, do:                                             
       2.0.1.1                                                            
             calculate multipliers for par-                               
             tial derivatives with respect to                             
             &#34;s&#34; coordinates;                                             
       2.0.1.2                                                            
             calculate multipliers for par-                               
             tial derivatives with respect to                             
             &#34;t&#34; coordinates,                                             
       2.0.1.3                                                            
             calculate multiplies for point                               
             coordinates.                                                 
       2.0.1.4                                                            
             For column = 0 to 100 by &#34;s&#34;                                 
             step size, do:                                               
           2.0.1.4.1                                                      
                  calculate point coor-                                   
                  dinate,                                                 
           2.0.1.4.2                                                      
                  calculate partial de-                                   
                  rivative with respect                                   
                  to &#34;s&#34; coordinate,                                      
           2.0.1.4.3                                                      
                  calculate partial de-                                   
                  rivative with respect                                   
                  to &#34;t&#34; coordinates.                                     
           End for column = 0 to 100 by &#34;s&#34;                               
           step size.                                                     
         End for x, y, and z.                                             
2.0.2    For column = to 100 by &#34;s&#34; step size, do:                        
       2.0.2.1                                                            
             cross partial derivative coordi-                             
             nates to get point normal vec-                               
             tor.                                                         
End for column = 0 to 100 by &#34;s&#34; step size.                               
2.0.3    Toggle row F/F.                                                  
      End for TCNTR = 0 and &#34;t&#34; step size.                                
2.1   Set TCNTR to &#34;t&#34; step size times 2. TCNTR is                        
      the value of t that will be used to calculate                       
      the next vertex row address.                                        
______________________________________                                    
 
    
     Having obtained the values for the vertex coordinates of x, y, and z for each vertex 9 of vertex rows 12 and 14 and having obtained the point normal vector for each such vertex 9, that data is further processed to form the triangular polygons 15 so that all of the vertices 9 in a polygon 15 will be coplanar (see Process 4 below). Before a group of vertices 9 is output as a given polygon 15, the data representing the values of the x, y, and z coordinates of each vertex 9 of the polygon 15 and the point normal vector for each such vertex 9 is processed further. Initially, this involves transformation of the vertex coordinates from the object space coordinate system. Also, the point intensity is obtained for each such vertex 9 using the previously calculated point normal vector (see Process 3 below). 
     POINT INTENSITY PROCESSING 
     In regard to obtaining the point intensity, it should be understood that all of the vertices 9 that are calculated in Process 2 are in what is referred to as the world or object space coordinate system. That is, the object space coordinate system is used to define the object. On the other hand, the sun or source of light is specified by the user in terms of a 3D screen coordinate system, since the screen coordinate system is the one used by the user. The sun always points to the origin of the screen coordinate system, but the user can specify where the sun is shining from in terms of the x, y, and z coordinates of the screen (not shown). Since the screen coordinate system is not the same as the object coordinate system, the sun coordinates must be transformed into object space so that both the object and the sun are represented in the same coordinate system in order to obtain the point intensity. An inverse matrix is used to effect this transformation of the &#34;sun screen space&#34; to the &#34;sun object space&#34;. The transformed sun in the object space coordinate system is referred to as the &#34;sun object space&#34; and has x, y, and z coordinates. The processing of the intensity of each vertex 9 uses a point intensity equation and results in specifying the point intensity at that particular vertex 9. 
     In particular, the vector for the sun screen space is referred to as the &#34;sun screen space vector&#34;. The sun screen space vector has x, y, and z coordinates and is input in IEEE floating point format as follows: 
     sun screen space x ms 
     sun screen space x ls 
     sun screen space y ms 
     sun screen space y ls 
     sun screen space z ms 
     sun screen space z ls. 
     The sun screen space vector is transformed to a sun vector in the object space coordinate system, referred to as the &#34;sun object space vector&#34;. This is done by obtaining the inverse of the transformation matrix. The inverse of a matrix is obtained by first representing the matrix as follows: ##EQU15## Once in this form, a series of elementary row transformations is performed. The legal elementary row transformations are to interchange any two rows, to add some number times one row to any other row, and to multiply any row by some number. These row transformations are performed until the matrices (27) are as follows: ##EQU16## The inverse matrix is the I matrix on the right of the matrices shown in (28). 
     The sun object space vector (x,y,z) is calculated as follows: ##EQU17## The sun object space distance squared that is used in the point intensity calculation is calculated as follows: ##EQU18## 
     The point intensity of each vertex 9 is computed from the sun object space vector and the point normal vector of the vertex 9. The point normal vectors are not unit vectors, therefore their distance must be found. Since calculating distance requires a square root calculation, to avoid taking square roots, the distance found is the point normal distance squared. The point normal distance squared is computed for a vertex 9 as follows: ##EQU19## Point intensity dot is calculated for a vertex 9 as follows: ##EQU20## Since the point normal distance is squared, the point intensity dot is also squared. Before squaring it however, its sign is saved. Now the point intensity can be calculated as follows: ##EQU21## where sun object space distance squared is that which was identified as equation (30). 
     TRANSFORMATION OF VERTEX COORDINATES 
     As indicated above, the values of all of the vertices 9 that are calculated in Process 2 are in the object space coordinate system. Since the vertices are to be displayed in the screen coordinate system, the x, y, and z coordinates for each vertex are transformed by a standard transformation matrix, as appropriate. This is referred to below in connection with Process 3 which both transforms (step 3.6) the x, y, and z coordinates of a given vertex 9 into the screen coordinate system and calculates (steps 3.1-3.5) the point intensity for each vertex 9 according to the method described above. 
     The above determination of point intensity and the transformation of the x, y, and z coordinates of a given vertex 9 is summarized as follows: 
     
         ______________________________________                                    
PROCESS 3                                                                 
Obtain Point Intensities and Transform Vertices                           
Step #   Step                                                             
______________________________________                                    
3.0      For each vertex, do:                                             
3.1      Calculate point normal distance squared,                         
3.2      Dot point normal vector with sun object space                    
         vector to get point intensity dot.                               
3.3      Square point intensity dot saving its sign.                      
3.4      Set point intensity to point intensity dot                       
         squared divided by point normal distance squared                 
         times sun object space distance squared.                         
3.5      Set ms bit of point intensity to its saved sign                  
         bit.                                                             
3.6      transform point using transformation matrix.                     
3.7      Output x, y and z and point intensity to storage                 
         for use in Process 4.                                            
______________________________________                                    
 
    
     SELECTING VERTICES TO FORM POLYGONS 
     Having described the transformation of each vertex 9 that has previously been calculated and having described the method of obtaining the point intensity for each such vertex 9, the following Process 4 is performed for selecting the particular vertices 9 (e.g. V1, V2, and V3) that form a given triangular polygon 15 (e.g. P1 in FIG. 1). As each such vertex 9 for a given triangular polygon 15 is selected in Process 4, the above transformation and calculation of point intensity is performed by Process 3 for each such vertex 9. The vertices 9 thus processed are output as a group. 
     The method of defining the triangular polygons is summarized as follows: 
     
         ______________________________________                                    
PROCESS 4                                                                 
Obtain Polygons For Polygon Row 0                                         
Step #        Step                                                        
______________________________________                                    
START:   4.1      Get 1st x, y, z, I and store.                           
         4.2      Get 2nd x, y, z, I and store.                           
         4.3      Get 3rd x, y, z, I and store.                           
         4.4      Output polygon made up of stored                        
                  4.1-4.3                                                 
LOOP:    4.5      If input is a command STOP.                             
         4.6      Get next x, y, z, I and store in                        
                  4.1.                                                    
         4.7      Output next polygon made up of                          
                  4.1-4.3                                                 
         4.8      If input is a command STOP.                             
         4.9      Get next x, y, z, I and store in                        
                  4.2.                                                    
         4.10     Output next polygon made up of                          
                  4.1-4.3.                                                
         4.11     If input is a command STOP.                             
         4.12     Get next x, y, z, I and store in                        
                  4.3.                                                    
         4.13     Output next polygon made up of                          
                  4.1-4.3.                                                
         4.14     Go to LOOP.                                             
______________________________________                                    
 
    
     In Process 4, &#34;Get&#34; means get from Process 3 output. The &#34;1st&#34;, &#34;2nd&#34;, and &#34;3rd&#34; x, y, and z are the values of the x, y, and z coordinates of the respective V1, V2, and V3 vertices 9. &#34;I&#34; in Process 4 is the point intensity of each such given V1, V2 and V3 vertex 9. 
     Referring in particular to Process 4 and to FIGS. 1-3, triangular polygons 15 are processed across one polygon row at a time, e.g. across the polygon row 18, where the polygon row 18 is formed by the two vertex rows 12 and 14. For example, the first P1 polygon 15 processed for the polygon row 18 is made up of the V1, V2, and V3 vertices 9 obtained using respective &#34;s&#34;, &#34;t&#34; values of (0,0), (0.1,0), and (0,0.2). All other polygons 15 on the polygon row 18 only need one additional vertex 9 to define the next triangular polygon 15. In this example, the next or P2 polygon 15 processed is formed using respective &#34;s&#34;, &#34;t&#34; values of (0,0.2), (0.1,0), and (0.1,0.2) substituted into equation (11). The first or P2-1 vertex 9 of the P2 polygon 15 is the same as the third or P1-3 vertex 9 of the first or P1 polygon 15. The second or P1-2 and P2-2 vertices 9 of both of the P1 and P2 polygons 15 are the same V2 vertex 9. Therefore, it is only necessary to access one new vertex 9 having &#34;s&#34; and &#34;t&#34; values of (0.1,0.2) to obtain the third or P2-3 vertex 9 for the second or P2 polygon 15. 
     For the third or P3 polygon 15, the first or P3-1 vertex 9 is the V3 vertex 9, which was the P2-1 vertex 9 of the second or P2 polygon 15. The second or P3-2 vertex 9 using &#34;s&#34; and &#34;t&#34; values of (0.2,0) has to be accessed. The third or P3-3 vertex 9 is the V4 vertex 9, which was the third or P2-3 vertex 9 of the second or P2 polygon 15. 
     PROCESSING POLYGONS FOR A GIVEN POLYGON ROW 
     In greater detail and referring to FIGS. 2, 3 and 4B, steps 4.1-4.3 of the above method read intermediate vertex signals 30 (FIG. 4B) that represent the x, y, and z coordinates and the point intensity of the pair 25 of first and second V1 and V3 vertices 9 in the first vertex row 12 and of the first or V2 vertex 9 in the second vertex row 14. These intermediate vertex signals 30 are stored at locations identified as 4.1, 4.2, and 4.3, respectively. 
     Step 4.4 identifies these three intermediate vertex signals 30 as representing the first or P1 polygon 15. The signals 30 are serially read from the storage locations 4.1-4.3 and are output as the vertex signals 10 representing the P1 polygon 15. 
     The method then moves to a loop that includes step 4.5. If step 4.5 is not a &#34;command&#34;, step 4.6 is performed by reading the intermediate vertex signals 30 (FIG. 4B) that represent the coordinates x, y, and z and the point intensity of the next or V4 vertex 9, which is the next vertex 9 in the vertex row 14 diagonally opposite to the V1 vertex 9. The intermediate vertex signal 30 representing the V4 vertex 9 is read into storage location 4.1 in place of the intermediate vertex signal 30 representing the V1 vertex 9. At this time, storage locations 4.1-4.3 store signals 30 representing the second or P2 polygon 15. In step 4.7 those intermediate vertex signals 30 are identified as representing the second or P2 polygon 15. The intermediate vertex signals 30 are serially read from storage and are output as the vertex signals 10 representing the P2 polygon 15. 
     If the next input is not a &#34;command&#34; (step 4.8), the next step is performed by reading the next intermediate vertex signals 30 which represent the coordinates x, y, and z and the point intensity of the V5 vertex 9, which is the next vertex 9 in vertex row 12. The signals 30 for the V5 vertex 9 are read into storage location 4.2 in place of the signals 30 representing the V2 vertex 9. Storage locations 4.1-4.3 at this time store signals 30 representing the V3, V4, and V5 vertices 9 which represent the third or P3 polygon 15. The signals 30 are serially read from storage and are output as the vertex signals 10 representing the P3 polygon 15. 
     If the next input is not a &#34;command&#34; (step 4.11), step 4.12 is performed by reading the next intermediate vertex signal 30, which represent the coordinates x, y and z and the point intensity of the V6 vertex 9, which is the next vertex 9 in the vertex row 14. The intermediate vertex signals 30 for the V6 vertex 9 are read into storage location 4.3 in place of the intermediate vertex signals 30 for the V3 vertex 9. Storage locations 4.1-4.3 at this time store intermediate vertex signals 30 representing a fourth or P4 polygon 15. In step 4.13 those intermediate vertex signals 30 are serially read to identify them as representing the fourth or P4 polygon 15 and are output as the vertex signals 10 representing the P4 polygon 15. 
     Step 4.14 returns the method to LOOP, which starts at step 4.5. The cycle continues until both of the last (right hand) vertices 9 having the respective vertex rows 0 and 1 have been processed into polygons 15. In the example shown in FIG. 3, these are the V21 vertices 9 in the vertex row 12 and the V22 vertex 9 in the vertex row 14. The processing of the last or Pn polygon 15 (shown as P20) in the polygon row 18 causes a &#34;command&#34; to be input in the next one of the steps 4.5, 4.8, and 4.11, which stops the polygon processing. 
     TRANSITION TO NEXT VERTEX ROW AND NEXT POLYGON ROW 
     At this juncture, vertex signals 10 representing the P1-P20 polygons 15 defined by the first polygon row 18 have been output as a group, with each vertex 9 of the polygon 15 specified by x, y and z coordinates transformed into the screen space coordinate system. Each vertex 9 also has a point intensity value. The next step is to obtain the values of the x, y and z coordinates and the point normals for each vertex 9 in the third vertex row 19. This requires incrementing &#34;t&#34; by &#34;t&#34; step size and performing Process 2 again starting with step 2.0.1. Then, as described above, Processes 3 and 4 are performed so that vertex signals 10 representing the values of the transformed x, y and z coordinates for each vertex 9 in the polygon row 20 and the point intensity for such vertices 9 are output to define the polygons 15 of the polygon row 20. In the example shown in FIG. 2, this involves outputting the vertex signals 10 representing polygons P21 through P40. 
     Similarly, referring to FIG. 1 it can be understood that at the end of such processing of each polygon row, such as 20, the next vertex row, such as 31, and then the next polygon row, such as 32, are processed. This sequence is performed until the vertex signals 10 representing the polygons 15 of the fourth polygon row 33 and then the last polygon row 34 have been output. 
     DETAILED DESCRIPTION OF CONTROL APPARATUS 
     Referring now to FIGS. 4A, 4B, and 5 of the drawings, the methods of the preferred embodiment of the present invention may be performed by the preferred embodiment of the apparatus 21 of the present invention. The operation of the apparatus 21 is controlled by a control register 35 of the control circuit 23. The control register 35 is connected to a writable control store (WCS) 36 that stores various computer programs in the form of microcode. The control register 35 controls the selection of words or commands from the WCS 36 in conjunction with a next address register 37 and a this address register 38. 
     In response to a reset command, the address in the this address register 38 is reset to 0. This causes the WCS 36 to output an instruction signal 39 representing an instruction (&#34;instruction 0&#34;) that is at address 0 into the control register 35. At the next clock cycle, the signal 39 for the instruction 0 is sent to the apparatus 21 shown in FIGS. 4A and 4B. Also, a signal 40 representing the next address associated with the instruction 0 is shifted into the this address register 38. In the same cycle, a new this address signal 41 also addresses the WCS 36. At the next cycle, the WCS 36 outputs to the control register 35 an instruction signal 39 associated with the next address, such that on the next cycle the instruction signal 39 corresponding to the next address will be output to the apparatus 21. As the cycling continues, this operation continues by successively outputting the instruction signals 39 from the WCS 36 to the control register 35 to the apparatus 21. Also, to take into consideration events that should modify the next address, a branch logic circuit 42 is connected to enable modification of the next address signal 40 of each instruction based upon the occurrence of a condition, such as a flag or processor register reaching 0. The branch logic circuit 42 outputs a modify signal 43 that modifies the next address signal 40 that is transferred to the this address register 38. As a result, the WCS 36 outputs the instruction signal 39 that corresponds to the modified next address. An example of this is discussed later in connection with the processing of the P21 and P22 polygons 15 from the respective third and second vertex rows 19 and 14. In that case, the need for the modified addresses results from the fact that the data for the V24, V26, etc. vertices 9 of the vertex row 19 are written into vertex row 0 storage, such that the selection of the V24, V26 . . . V44 vertices 9 to form the P21, P22 . . . P40 polygons 15 has to be reversed. 
     In addition, a flag register 44 is provided for setting various flags for various conditions. The flag register 44 is connected to a flag store RAM 45 that stores flag signals 46 representing the flags for access during processing to indicate how certain operations are to be performed. 
     Also provided is a loop counter 47 of the counter circuit 26 that can be set to an initial value and then decremented each time an event occurs. The loop counter 47 outputs a signal 47&#39; when it decrements to 0. For example, it will input the signal 47&#39; to the branch logic circuit 42 which, if programmed to perform this function, will modify the next address signal 40 to access an appropriate next instruction. 
     A push/pop stack RAM 48 is used in conjunction with a subroutine that is called by the control register 35. The push/pop stack RAM 48 stores the particular next address that follows the subroutine. The stack RAM 48 outputs the next address signal 40 to the WCS 36 to cause it to output to the control register 35 the instruction signal 39 that corresponds to that next address. This is done because a particular subroutine may be called at different times during the various operations, such that there is no particular next address associated with a given subroutine. 
     Also, with respect to the next address following a particular subroutine, a conditional pop-logic circuit 49 is connected to both the push/pop stack RAM 48 and to the branch logic circuit 42. In this case, the use of the instruction at the next address in the push/pop stack RAM 48 is conditional upon an event happening, which event controls the conditional pop-logic circuit 49. If the event occurs, the conditional pop-logic circuit 49 outputs a signal 50 to instruct the push/pop stack RAM 48 to output its next address signal 40 to the WCS 36, which then outputs the instruction signal 39 corresponding to that next address. If the event does not occur, then the conditional pop-logic circuit 49 does not permit the next address to be read from the push/pop stack RAM 48. Instead, a second signal 51 from the conditional pop-logic circuit 49 is applied to the branch logic circuit 42 to modify the next address signal 40 that is transferred to the this address register 38. The instruction signal 39 corresponding to that modified next address is then output from the WCS 36 to the control register 35. 
     DETAILED DESCRIPTION OF COMPUTATION APPARATUS 
     Referring now in detail to FIGS. 4A and 4B, the portions of the apparatus 21 for arithmetic computation and data storage are shown. The operation of the apparatus 21 of the present invention to perform the method of the present invention is described in terms of the various Steps of Processes 1.0 through 4.0, where the computational operations required to perform a Step are described in the following Charts. Each such Chart has columns, in which a first column heading indicates the computational or machine cycle in which a particular operation is performed. This is indicated by the heading &#34;Cycle&#34;. In general, when data is input, 10 cycles occur before an output is obtained. The description of such Charts includes where data is stored, such as in a matrix Ram 52 (FIG. 4A). Similarly, data is stored in a translate RAM 53 and in a pipeline RAM section 56 of the RAM circuit 22. Data is also stored in a Reciprocal PROM 59 and in a y register 60 (FIG. 4A) associated with a processor 61. Data is also stored in the font RAM 22 and certain data from a data field (not shown) or from a FIFO register 63 is input directly to an F BUS 64, which can be connected to a Y BUS 65 and a C BUS 66. 
     The other column headings indicate what data is input to and output from certain other circuit elements shown in FIGS. 4A, 4B, and 5. Still referring to FIGS. 4A and 4B and to such Charts, respective A and B inputs 67 and 68 to a floating point multiplier 69 are labeled &#34;FPM A IN&#34; and &#34;FPM B IN&#34; and a C output 70 of the floating point multiplier 69 is labeled &#34;FPM C OUT&#34;. Similarly, a floating point adder 71 has an A input 72 identified as &#34;FPA A IN&#34; and a corresponding B input 73 is identified as &#34;FPA B IN&#34;. A C output 74 of the floating point adder 71 is identified as &#34;FPA C OUT&#34;. An A REG 1 register 75 is connected to the C output 74 of the floating point adder 71 and is referred to as &#34;A REG 1&#34; 75. An A REG 2 register 76 is connected to the A REG 1 75, an an A REG 3 register 77 is connected to the A REG 2 register 76. 
     The data section 55 of the matrix RAM 52 can store sixteen 4×4 matrices. In the present invention, four matrices are stored in the section 55, the x coefficient matrix, the y coefficient matrix, the z coefficient matrix and the transformation matrix. For each matrix, the data section 55 has a storage location for each ms and ls word of the sixteen elements of the given matrix. Thus, referring to the basic matrix arrangement in (3), there is an individual storage location for each ms and ls word of element 11, 12, 13, 14, etc. If new data, such as new element 11, is written into the location 11 of the matrix RAM 52 it will overwrite the previous data at location 11. Thus, before new data is written to a particular location, the current data must have been fully processed or written into a register. 
     The apparatus 21 also includes a PROM 78 connected to a sign generate circuit 79. An integer to floating point circuit 80 is also connected to the Y register 60. 
     A second FIFO register 81 (FIG. 4B) is provided for storing the vertex signals 10 before they are selected to define the polygons 15 and output at a terminal 82. 
     OPERATION 
     In the use of the present invention, a particular parametric form for cubic order surfaces is selected for defining the surface of an object to be displayed (see Equation (1)). The coefficient matrix (8) for the selected surfaces is obtained using Process 1. It will be recalled that Process Step 1.2 called for concatenating the magic matrix (5) with the input x coefficient matrix (4). 
     OPERATION--PROCESS STEP 1.2 CONCATENATE MAGIC MATRIX (5) WITH INPUT X COEFFICIENT MATRIX (4) 
     Referring to FIGS. 4A and 4B, Step 1.2 is performed by loading certain of the elements B of the input x coefficient matrix (4) into the matrix RAM 52. This is described with respect to the following Charts 1A through 1L. 
     
         ______________________________________                                    
CHART 1A                                                                  
Process step 1.2                                                          
           FPM        FPM       FPM                                       
Cycle      A IN       B IN      C OUT                                     
______________________________________                                    
1          B14 ms     2 ms                                                
2          B14 ls     2 ls                                                
3          B13 ms     2 ms                                                
4          B13 ls     2 ls                                                
5          B12 ms     2 ms                                                
6          B12 ls     2 ls                                                
7          B11 ms     2 ms                                                
8          B11 ls     2 ls                                                
10                                                                        
11         B24 ms     -2 ms     2*B14 ms                                  
12         B24 ls     -2 ls     2*B14 ls                                  
13         B23 ms     -2 ms     2*B13 ms                                  
14         B23 ls     -2 ls     2*B13 ls                                  
15         B22 ms     -2 ms     2*B12 ms                                  
16         B22 ls     -2 ls     2*B12 ls                                  
17         B21 ms     -2 ms     2*B11 ms                                  
18         B21 ls     -2 ls     2*B11 ls                                  
19                                                                        
20                                                                        
21                              -2*B24 ms                                 
22                              -2*B24 ls                                 
23                              -2*B23 ms                                 
24                              -2*B23 ls                                 
25                              -2*B22 ms                                 
26                              -2*B22 ls                                 
27                              -2*B21 ms                                 
28                              -2*B21 ls                                 
29                                                                        
30                                                                        
31         B34 ms     -2 ms                                               
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         ______________________________________                                    
CHART 1B                                                                  
Process Step 1.2                                                          
         FPA        FPA        FPA                                        
Cycle    A IN       B IN       C OUT                                      
______________________________________                                    
2                                                                         
3                                                                         
4                                                                         
5                                                                         
6                                                                         
7                                                                         
8                                                                         
9                                                                         
10                                                                        
11       B44 ms     FPM OUT                                               
12       B44 ls     FPM OUT                                               
13       B43 ms     FPM OUT                                               
14       B43 ls     FPM OUT                                               
15       B42 ms     FPM OUT                                               
16       B42 ls     FPM OUT                                               
17       B41 ms     FPM OUT                                               
18       B41 ls     FPM OUT                                               
19                                                                        
20                                                                        
21       FPA OUT    FPM OUT    2*B14 + B44 ms                             
22       FPA OUT    FPM OUT    2*B14 + B44 ls                             
23       FPA OUT    FPM OUT    2*B13 + B43 ms                             
24       FPA OUT    FPM OUT    2*B13 + B43 ls                             
25       FPA OUT    FPM OUT    2*B12 + B42 ms                             
26       FPA OUT    FPM OUT    2*B12 + B42 ls                             
27       FPA OUT    FPM OUT    2*B11 + B41 ms                             
28       FPA OUT    FPM OUT    2*B11 + B41 ls                             
29                                                                        
30                                                                        
31       FPA OUT    B34 ms     2*B14 - 2*B24                              
                               + B44ms                                    
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         ______________________________________                                    
CHART 1C - Process Step 1.2                                               
           FPM        FPM       FPM                                       
Cycle      A IN       B IN      C OUT                                     
______________________________________                                    
32         B34 ls     -2 ls                                               
33         B33 ms     -2 ms                                               
34         B33 ls     -2 ls                                               
35         B32 ms     -2 ms                                               
36         B32 ls     -2 ls                                               
37         B31 ms     -2 ms                                               
38         B31 ls     -2 ls                                               
39                                                                        
40                                                                        
41         B24 ms     3 ms      -2*B34 ms                                 
42         B24 ls     3 ls      -2*B34 ls                                 
43         B23 ms     3 ms      -2*B33 ms                                 
44         B23 ls     3 ls      -2*B33 ls                                 
45         B22 ms     3 ms      -2*B32 ms                                 
46         B22 ls     3 ls      -2*B32 ls                                 
47         B21 ms     3 ms      -2*B31 ms                                 
48         B21 ls     3 ls      -2*B31 ls                                 
49                                                                        
50                                                                        
51         B14 ms     -3 ms     3*B24 ms                                  
52         B14 ls     -3 ls     3*B24 ls                                  
53         B13 ms     -3 ms     3*B23 ms                                  
54         B13 ls     -3 ls     3*B23 ls                                  
55         B12 ms     -3 ms     3*B22 ms                                  
56         B12 ls     -3 ls     3*B22 ls                                  
57         B11 ms     -3 ms     3*B21 ms                                  
58         B11 ls     -3 ls     3*B21 ls                                  
59                                                                        
60                                                                        
61                              -3*B14 ms                                 
62                              -3*B14 ls                                 
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         ______________________________________                                    
CHART 1D                                                                  
Process Step 1.2                                                          
      FPA       FPA       FPA                                             
Cycle A IN      B IN      C OUT                                           
______________________________________                                    
32    FPA OUT   B34 ls    2*B14  -2*B24 +B44 ls                           
33    FPA OUT   B33 ms    2*B13  -2*B23 +B43 ms                           
34    FPA OUT   B33 ls    2*B13  -2*B23 +B43 ls                           
35    FPA OUT   B32 ms    2*B12  -2*B22 +B42 ms                           
36    FPA OUT   B32 ls    2*B12  -2*B22 +B42 ls                           
37    FPA OUT   B31 ms    2*B11  -2*B21 +B41 ms                           
38    FPA OUT   B31 ls    2*B11  -2*B21 +B41 ls                           
39                                                                        
40                                                                        
41    B44 ms    FPM OUT   C14 ms                                          
42    B44 ls    FPM OUT   C14 ls                                          
43    B43 ms    FPM OUT   C13 ls                                          
44    B43 ls    FPM OUT   C13 ls                                          
45    B42 ms    FPM OUT   C12 ms                                          
46    B42 ls    FPM OUT   C12 ls                                          
47    B41 ms    FPM OUT   C11 ms                                          
48    B41 ls    FPM OUT   C11 ls                                          
49                                                                        
50                                                                        
51    FPA OUT   FPM OUT   -2*B34 -B44 ms                                  
52    FPA OUT   FPM OUT   -2*B34 -B44 ls                                  
53    FPA OUT   FPM OUT   -2*B33 - B43 ms                                 
54    FPA OUT   FPM OUT   -2*B33 -B43 ls                                  
55    FPA OUT   FPM OUT   -2*B32 -B42 ms                                  
56    FPA OUT   FPM OUT   -2*B32 -B42 ls                                  
57    FPA OUT   FPM OUT   -2*B31 -B41 ms                                  
58    FPA OUT   FPM OUT   -2*B31 -B41 ls                                  
59                                                                        
60                                                                        
61    FPA OUT   FPM OUT   3*B24  -2*B34 -B44 ms                           
62    FPA OUT   FPM OUT   3*B24  -2*B34 -B44 ls                           
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         ______________________________________                                    
CHART 1E                                                                  
PROCESS STEP 1.2                                                          
           FPM       FPM         FPM                                      
CYCLE      A IN      B IN        C OUT                                    
______________________________________                                    
63                               -3*B13 ms                                
64                               -3*B13 ls                                
65                               -3*B12 ms                                
66                               -3*B12 ls                                
67                               -3*B11 ms                                
68                               -3*B11 ls                                
69                   -2 ms                                                
70                   -2 ls                                                
71                   -2 ms                                                
72                   -2 ls                                                
73                   -2 ms                                                
74                   -2 ls                                                
75         FPA OUT   -2 ms                                                
76         FPA OUT   -2 ls                                                
77                                                                        
78                                                                        
79         C23 ms    -2 ms                                                
80         C23 ls    -2 ls                                                
81                   2 ms                                                 
82                   2 ls                                                 
83         C11 ms    2 ms                                                 
84         C11 ls    2 ls                                                 
85         C21 ms    2 ms        -2*C22 ms                                
86         C21 ls    2 ls        -2*C22 ls                                
87         C22 ms    3 ms                                                 
88         C22 ls    3 ls                                                 
89                               -2*C23 ms                                
90                               -2*C23 ls                                
91         C12 ms    3 ms                                                 
92         C12 ls    3 ls                                                 
93                               2*C11 ms                                 
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         ______________________________________                                    
CHART 1F                                                                  
PROCESS STEP 1.2                                                          
      FPA        FPA       FPA                                            
Cycle A IN       B IN      C OUT                                          
______________________________________                                    
63    FPA OUT    FPM OUT   3*B23 - 2*B33 - B43 ms                         
64    FPA OUT    FPM OUT   3*B23 - 2*B33 - B43 ls                         
65    FPA OUT    FPM OUT   3*B22 - 2*B32 - B42 ms                         
66    FPA OUT    FPM OUT   3*B22 - 2*B32 - B42 ls                         
67    FPA OUT    FPM OUT   3*B21 - 2*B31 - B41 ms                         
68    FPA OUT    FPM OUT   3*B21 - 2*B31 - B41 ls                         
69                                                                        
70                                                                        
71               FPA OUT   C24 ms                                         
72               FPA OUT   C24 ls                                         
73    FPA OUT              C23 ms                                         
74    FPA OUT              C23 ls                                         
75                         C22 ms                                         
76                         C22 ls                                         
77                         C21 ms                                         
78                         C21 ls                                         
79                                                                        
80                                                                        
81                                                                        
82                                                                        
83               FPA OUT   C23 ms + C24 ms                                
84               FPA OUT   C23 ls + C24 ls                                
85    FPM OUT                                                             
86    FPM OUT                                                             
87                                                                        
88                                                                        
89    C24 ms     FPM OUT                                                  
90    C24 ls     FPM OUT                                                  
91                                                                        
92                                                                        
93    C14 ms     FPM OUT                                                  
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         ______________________________________                                    
CHART 1 G                                                                 
PROCESS STEP 1.2                                                          
           FPM      FPM          FPM                                      
CYCLE      A IN     B IN         C OUT                                    
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94                                   2*C11  ls                            
95         C11    ms    3      ms    2*C21  ms                            
96         C11    ls    3      ls    2*C21  ls                            
97         C13    ms    -2     ms    3*C22  ms                            
98         C13    ls    -2     ls    3*C22  ls                            
99         C21    ms    -3     ms                                         
100        C21    ls    -3     ls                                         
101                                  3*C12  ms                            
102                                  3*C12  ls                            
103                                                                       
104                                                                       
105        C12    ms    -2     ms    3*C11  ms                            
106        C12    ls    -2     ls    3*C11  ls                            
107                                  -2*C13 ms                            
108                                  -2*C13 ls                            
109        C32    ms    -2     ms    -3*C21 ms                            
110        C32    ls    -2     ls    -3*C21 ls                            
111        C31    ms    -2     ms                                         
112        C31    ls    -2     ls                                         
113        C33    ms    -2     ms                                         
114        C33    ls    -2     ls                                         
115                                  -2*C12 ms                            
116                                  -2*C12 ls                            
117                                                                       
118                                                                       
119                                  -2*C32 ms                            
120                                  -2*C32 ls                            
121                                  -2*C31 ms                            
122                                  -2*C31 ls                            
123                                  -2*C33 ms                            
124                                  -2*C33 ls                            
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 1 H                                                                 
PROCESS STEP 1.2                                                          
        FPA       FPA       FPA                                           
CYCLE   A IN      B IN      C OUT                                         
______________________________________                                    
94      C14 ls    FPM OUT                                                 
95      FPA OUT   FPM OUT   -2*C22 + C23 + C24 ms                         
96      FPA OUT   FPM OUT   -2*C22 + C23 + C24 ls                         
97                FPM OUT                                                 
98                FPM OUT                                                 
99      FPA OUT             -2*C23-C24 ms                                 
100     FPA OUT             -2*C23- C24 ls                                
101     C14 ms    FPM OUT                                                 
102     C14 ls    FPM OUT                                                 
103     FPA OUT   C13 ms    2*C11 + C14 ms                                
104     FPA OUT   C13 ls    2*C11 + C14 ls                                
105               FPM OUT   D21 ms                                        
106               FPM OUT   D21 ls                                        
107     FPM OUT                                                           
108     FPM OUT                                                           
109     FPA OUT   FPM OUT   3*C22 - 2*C23 - C24 ms                        
110     FPA OUT   FPM OUT   3*C22 - 2*C23 - C24 ls                        
111                         3*C12- C14 ms                                 
112                         *C12- C14 ls                                  
113               FPA OUT   2*C11 + C13 + C14 ms                          
114               FPA OUT   2*C11 +  C13 + C14 ls                         
115     FPM OUT                                                           
116     FPM OUT                                                           
117     A REG 2   FPA OUT   -3*C11 - 2*C13 ms                             
118     A REG 2   FPA OUT   -3*C11 - 2*C13 ls                             
119     C34 ms    FPM OUT   D22 ms                                        
120     C34 ls    FPM OUT   D22 ls                                        
121     C33 ms    FPM OUT                                                 
122     C33 ls    FPM OUT                                                 
123     C34 ms    FPM OUT                                                 
124     C34 ls    FPM OUT                                                 
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         ______________________________________                                    
CHART 1 I                                                                 
PROCESS STEP 1.2                                                          
           FPM       FPM          FPM                                     
CYCLE      A IN      B IN         C OUT                                   
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125        C32 ms    3      ms                                            
126        C32 ls    3      ls                                            
127        C42 ms    3      ms                                            
128        C42 ls    3      ls                                            
129                                                                       
130                                                                       
131                                                                       
132                                                                       
133                                                                       
134                                                                       
135        C31 ms    -3     ms    3*C32  ms                               
136        C31 ls    -3     ls    3*C32  ls                               
137                               3*C42  ms                               
138                               3*C42  ls                               
139        C41 ms    -3     ms                                            
140        C41 ls    -3     ls                                            
141        C42 ms    2      ms                                            
142        C42 ls    2      ls                                            
143        C41 ms    2      ms                                            
144        C41 ls    2      ls                                            
145                               -3*C31 ms                               
146                               -3*C31 ls                               
147                                                                       
148                                                                       
149        C43 ms    -2     ms    -3*C41 ms                               
150        C43 ls    -2     ls    -3*C41 ls                               
151                               2*C42  ms                               
152                               2*C42  ls                               
153                               2*C41  ms                               
154                               2*C41  ls                               
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         ______________________________________                                    
CHART 1 J                                                                 
PROCESS STEP 1.2                                                          
       FPA        FPA       FPA                                           
CYCLE  A IN       B IN      C OUT                                         
______________________________________                                    
125                         D11 ms                                        
126                         D11 ls                                        
127                         D12 ms                                        
128                         D12 ls                                        
129               FPA OUT   -2*C32 + C34 ms                               
130               FPA OUT   -2*C32 + C34 ls                               
131    FPA OUT              2*C31 + C33 ms                                
132    FPA OUT              2*C31 + C33 ls                                
133                         -2*C33- C34 ms                                
134                         -2*C33- C34 ls                                
135    A REG 2    FPM OUT                                                 
136    A REG 2    FPM OUT                                                 
137    C44 ms     FPM OUT                                                 
138    C44 ls     FPM OUT                                                 
139                                                                       
140                                                                       
141                         D 31 ms                                       
142                         D 31 ls                                       
143                                                                       
144                                                                       
145    FPA OUT    FPM OUT   3*C32 - 2*C33 - C34 ms                        
146    FPA OUT    FPM OUT   3*C32 - 2*C33 - C34 ls                        
147                         3*C42- C44 ms                                 
148                         3*C42- C44 ls                                 
149    A REG 2    FPM OUT                                                 
150    A REG 2    FPM OUT                                                 
151    C44 ms     FPM OUT                                                 
152    C44 ls     FPM OUT                                                 
153    C43 ms     FPM OUT                                                 
154    C43 ls     FPM OUT                                                 
155                         D 32 ms                                       
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         ______________________________________                                    
CHART 1 K                                                                 
PROCESS STEP 1.2                                                          
          FPM         FPM     FPM                                         
CYCLE     A IN        B IN    C OUT                                       
______________________________________                                    
156                                                                       
157                                                                       
158                                                                       
159                           -2*C43 ms                                   
160                           -2*C43 ls                                   
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 1 L                                                                 
PROCESS STEP 1.2                                                          
       FPM       FPM       FPM                                            
CYCLE  A IN      B IN      C OUT                                          
______________________________________                                    
156                        D 32 ls                                        
157                                                                       
158                                                                       
159    FPA OUT   FPM OUT   -3*C41 + 3*C42 - C44 ms                        
160    FPA OUT   FPM OUT   -3*C41 + 3*C42 - C44 ls                        
161                        -2*C42 + C44 ms                                
162                        -2*C42 + C44 ls                                
163    A REG 2   FPA OUT   2*C41 + C43 ms                                 
164    A REG 2   FPA OUT   2*C41 + C43 ls                                 
165                                                                       
166                                                                       
167                                                                       
168                                                                       
169                        D42 ms                                         
170                        D42 ls                                         
171                                                                       
172                                                                       
173                        D41 ms                                         
174                        D41 ls                                         
175                        D24 ms                                         
176                        D24 ls                                         
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     Referring to Charts 1A through 1L, in cycles 1 through 8, eight signals 83 representing the ms and ls words of the elements B14 through B11 of the input x coefficient matrix (4) are input from the matrix RAM 52 to the &#34;FPM A IN&#34; input 67. In the same cycles 1 through 8, signals 84 representing 2 ms and 2 ls are input from the F BUS 64 to the &#34;FPM B IN&#34; input 68. Thus, referring to the results (6) of the post-concatenation performed in step 1.2, the first 8 cycles of Chart 1A represent loading the signals 83 and 84 into the floating point multiplier 69 for the multiplication operation that will result in the first term of each of the C concatenated matrix elements C14, C13, C12 and C11. In the 11th through 18th cycles, the floating point multiplier 69 outputs signals 85 representing the products of the multiplicands that were input into the &#34;FPM A IN&#34; input 67 and the &#34;FPM B IN&#34; input 68 during cycles 1 through 8. These products include, for example, the product &#34;2*B14 ms&#34;. These products are loaded into the &#34;FPA B IN&#34; input 73 as indicated by &#34;FPM OUT&#34; at cycles 11 through 18 under the &#34;FPA B IN&#34; heading. Also, during cycles 11 through 18, signals 83 representing the input x coefficient matrix (4) elements B44 through B41 are input from the matrix RAM 52 to the &#34;FPA A IN&#34; input 72 for both the ms and ls words. This conditions the floating point adder 71 to output (in cycles 21 through 28) signals 86 representing, for example, the sum of the first term (B11*2) and the fourth term (B41) of the C11 concatenated matrix (6). 
     In cycles 21 through 28, the floating point multiplier 69 outputs signals 85 representing the products that correspond to the second terms of each of the C11 through C14 elements of the concatenated matrices (6). These are indicated as being input to the &#34;FPA B IN&#34; input 73. In cycles 21 through 28 the signals 86 are output from the floating point adder 71 and are then input to the floating point adder A input 72 to condition the floating point adder 71 for producing output signals 86 representing the sum of the first, second and fourth terms of each of the C11, C12, C13 and C14 elements of the concatenated matrix (6) in cycles 31 through 38. 
     As noted above, the data in the matrix RAM 52 can be overwritten. To condition the x coefficient matrix RAM 52 for being overwritten with the x coefficient matrix data in place of the input x coefficient matrix data, in cycles 3-10, the ms and ls words for the input x coefficient matrix elements B13, B12, B11 and B14 are written into the A REG 1 register 75, the A REG 2 register 76, and the A REG 3 register 77 and successively into the pipeline RAM 56. In cycles 21-28, this data is written into the matrix RAM 52 as the C44 through C41 elements of the x coefficient matrix (6). The following &#34;MTX PTR&#34; addresses indicate writing to the following locations in the x coefficient matrix storage section 55 of the matrix RAM 52: 
     
         ______________________________________                                    
Chart 2                                                                   
       MTX PTR                                                            
       Address       Location                                             
______________________________________                                    
       B00 ms        B44 ms                                               
       B00 ls        B44 ls                                               
       B03 ms        B43 ms                                               
       B03 ls        B43 ls                                               
       B02 ms        B42 ms                                               
       B02 ls        B42 ls                                               
       B01 ms        B41 ms                                               
       B01 ls        B41 ls                                               
______________________________________                                    
 
    
     Referring again to the concatenated matrix (6), it is observed that no multiply or add operations are required for the C31 through C34 elements of the concatenated matrix (6). Accordingly, they do not have to be moved in or out of or within the matrix RAM 52. 
     In cycles 41 through 48, the floating point adder 71 outputs the FPA output signals 86 that represent the ms and ls words of the C14 through C11 elements of the concatenated matrix (6). In the successive cycles 42 through 49, such FPA output signals 86 are stored in the A REG 1 register 75. In successive cycles 43 through 50 such signals 86 are successively stored in the A REG 2 register 76. In the next cycles 44 through 51, such signals 86 are stored in the A REG 3 register 77. In cycles 44-51 such signals 86 are stored in the pipeline RAM 56. 
     With this description in mind, it is believed that the processing of the matrix elements C21 through C24 of the concatenated matrix (6) may be understood from further reference to Charts 1A through 1L. Such processing results in the outputting (see Chart 1F, &#34;FPA C OUT&#34;) in cycles 71 through 78 of the FPA output signals 86 representing the ms and ls words for the C 21 through C24 elements of the concatenated matrix (6). In cycles 71-78 these signals 86 are written into the matrix RAM 52 as the ms and ls words of the x coefficient elements B24 B21. This completes the processing of the C concatenated matrix (6). 
     STEP 1.3 CONCATENATE MATRIX (6) WITH THE TRANSPOSE OF THE MAGIC MATRIX (5) 
     Referring again to Process 1, it may be understood that when the concatenated matrix (6) is obtained (step 1.2), it is then concatenated (step 1.3) with the transpose of the magic matrix (5). The result is the x coefficient matrix (8) which has elements D11-D14, etc. The concatenation of matrix (6) with the transpose of the magic matrix (5) is done in a manner similar to that described above with respect to the post-concatenation operation corresponding to step 1.2. Thus, step 1.3 is performed by starting to obtain element D21 in the x coefficient matrix (8). Still referring to Charts 1A through 1L, signals 86 representing the C23 and C24 elements are input in cycles 71 through 74 to the FPA A IN and FPA B IN inputs 72 and 73 respectively as shown in Chart 1F. The sums of these elements are output as signals 86 in cycles 83 and 84 (see &#34;FPA C OUT&#34; on Chart 1F). 
     To continue to process the element D21, in cycles 75 and 76 the signals 86 representing the C22 ms and C22 ls elements of the concatenated matrix (6) are read from the matrix RAM 52 and written into the FPM A IN input 67. In those cycles, the -2 ms and -2 ls signals 84 from the F BUS 64 are input to the FPM B IN input 68. This will obtain the second term of the D21 element in cycles 85 and 86. In cycles 85 and 86, the signals 86 representing the C21 ms and C21 ls terms are read from the matrix RAM 52 into the FPM A IN input 67 and the signals 84 from the F BUS 64 are input to the FPM B IN input 68. 
     The output signals 85 from the floating point multiplier 69 in cycles 85 and 86 representing (-2*C22) ms and ls are input to the &#34;FPA A IN&#34; input 72 as indicated by &#34;FPM OUT&#34; under the &#34;FPA A IN&#34; heading in Chart 1F. Signals 86 representing C23+C24 ms and ls (FPA C OUT at cycles 83 and 84) are input to the FPA B IN input 73 in cycles 83 and 84 so that the floating point adder 71 will output in cycles 95 and 96 a signal 86 representing the second through fourth terms of the element D21. This signal 86 is input to the FPA A IN input 72 in cycles 95 and 96 (see Chart 1H) and the signals 85 representing the term 2*C21 are input to the FPA B IN input 73, so that the signals 86 output in cycles 105 and 106 represent the ms and ls words of the D21 elements of the x coefficient matrix (8) (see Chart 1H, FPA C OUT). Similarly, at cycles 119 and 120, the floating point adder 71 outputs signals 86 representing the ms and ls words of the D22 element of the x coefficient matrix (8). In respect to each of the signals 86 representing the ms and ls words for the D21 and D22 matrix elements, when such signals 86 are output from the FPA C OUT output 74 they are (in the next cycle) applied to the A REG 1 register 75, and in succeeding cycles are applied to the A REG 2 register 76, to the A REG 3 register 77, to the pipeline RAM 56, and are ultimately stored in the matrix RAM 52. 
     Referring again to the x coefficient matrix (8), it may be observed that no multiplication, addition or data transfer need be performed to obtain the D13, D23, D33 and D43 elements. These are retained in the matrix RAM 52 at those addresses. To obtain the values for the D14, D24, D34 and D44 elements, that only require shifting to new storage locations in the matrix RAM 52, respective ms and ls words of the C11, C21, C31 and C41 elements are read from the matrix RAM 52 into the A REG 1 register 75 and into the A REG 2 register 76 before being written into the new address of the matrix RAM 52. 
     The signals 86 representing the D11 and D12 elements are output from the floating point adder 71 in cycles 125 through 128 (Chart 1J) and are input to the pipeline RAM 56 via the registers 75, 76, and 77, and then to the matrix RAM 52. The signals 86 representing the D31 element (cycles 141 and 142) are written into the pipeline RAM 56, and then to the matrix RAM 52, whereas the signals 86 representing the D32 elements (cycles 155 and 156) and the D42 and D41 elements (cycles 169, 170, 171, and 172) are written directly into the matrix RAM 52. 
     PROCESSING Y COEFFICIENT MATRIX 
     When the complete x coefficient matrix (8) has been obtained and stored in the x coefficient matrix location of the matrix RAM 52, the push/pop RAM 48 outputs the next address signal 40 to the WCS 36 to cause similar operations of Process 1 to be performed again to obtain the y coefficient matrix (8). The signals 86 representing the B11, etc. elements of the input y coefficient matrix (4) are input to the y location of the matrix RAM 52. The signals 84 representing the 3, -3, 2 and -2 values of the various terms of the magic matrix (5) are also input from the F BUS 64 and the operation is performed as described above. 
     PROCESSING Z COEFFICIENT MATRIX 
     When the complete y coefficient matrix (8) has been obtained and the signals 86 stored in the y coefficient matrix location of the matrix RAM 52, the push/pop RAM 48 outputs the next address signal 40 to the WCS 36 to cause a similar operation to be performed again to obtain the z coefficient matrix (8). The signals representing the B11, etc. elements of the input z coefficient matrix (4) have been input to the z location of the matrix RAM 52. The signals 84 representing the 3, -3, 2 and -2 values of the various terms of the magic matrix (5) have also been input from the F BUS 64 and the above operations of Process 1 are run as described above. 
     STEP 3.0.2 SUN SCREEN SPACE VECTOR PROCESSING 
     When the complete z coefficient matrix (8) has been obtained and stored in the z location of the matrix RAM 52, further instructions process the x, y, and z coordinates of the sun screen space vector, including row transformations indicated by the matrices (28), and then by calculating and squaring the sun object space vector as indicated at equations (29) and (30). 
     
         ______________________________________                                    
CHART 3 A                                                                 
SUN OBJECT SPACE VECTOR (29)                                              
            FPM       FPM II     FPM                                      
CYCLE       A IN      B IN       C OUT                                    
______________________________________                                    
1           I13 ms    x ms                                                
2           I13 ls    x ls                                                
3           I12 ms    x ms                                                
4           I12 ls    x ls                                                
5           I11 ms    x ms                                                
6           I11 ls    x ms                                                
7           I23 ms    y ms                                                
8           I23 ls    y ls                                                
9           I22 ms    y ms                                                
10          I22 ls    y ls                                                
11                               x * I13 ms                               
12                               x * I13 ls                               
13                               x * I12 ms                               
14                               x * I12 ls                               
15                               x * I11 ms                               
16                               x * I11 ls                               
17          I21 ms    y ms       y * I23 ms                               
18          I21 ls    y ls       y * I23 ls                               
19                               y * I22 ms                               
20                               y * I22 ls                               
21                                                                        
22                                                                        
23                                                                        
24                                                                        
25          I33 ms    z ms                                                
26          I33 ls    z ls                                                
27          I31 ms    z ms       y * I21 ms                               
28          I31 ls    z ls       y * I21 ls                               
29          I32 ms    z ms                                                
30          I32 ls    z ls                                                
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 3 B                                                                 
SUN OBJECT SPACE VECTOR (29)                                              
         FPA          FPA        FPA                                      
CYCLE    A IN         B IN       C OUT                                    
______________________________________                                    
2                                                                         
3                                                                         
4                                                                         
5                                                                         
6                                                                         
7                                                                         
8                                                                         
9                                                                         
10                                                                        
11       0 ms         FPM OUT                                             
12       0 ls         FPM OUT                                             
13       0 ms         FPM OUT                                             
14       0 ls         FPM OUT                                             
15       0 ms         FPM OUT                                             
16       0 ls         FPM OUT                                             
17                                                                        
18                                                                        
19                                                                        
20                                                                        
21       FPA OUT      y * I23 ms x I13 + 0 ms                             
22       FPA OUT      y * I23 ls x I13 + 0 ls                             
23       y *I22 ms    FPA OUT    x I12 + 0 ms                             
24       y *I22 ls    FPA OUT    x I12 + 0 ls                             
25                               x I11 + 0 ms                             
26                               x I11 + 0 ls                             
27       y*I21 ms + x*I11                                                 
                      FPM OUT                                             
28       y*I21 ls + x*I11                                                 
                      FPM OUT                                             
29                                                                        
30                                                                        
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 3 C                                                                 
SUN OBJECT SPACE VECTOR (29)                                              
          FPM       FPM          FPM                                      
CYCLE     A IN      B IN         C OUT                                    
______________________________________                                    
31                                                                        
32                                                                        
33                                                                        
34                                                                        
35                               z * I33 ms                               
36                               z * I33 ls                               
37                               z * I31 ms                               
38                               z * I31 ls                               
39                               z * I32 ms                               
40                               z * I32 ls                               
41                                                                        
42                                                                        
43                                                                        
44                                                                        
45        FPA OUT   FPA OUT                                               
46        FPA OUT   FPA OUT                                               
47        FPA OUT   FPA OUT                                               
48        FPA OUT   FPA OUT                                               
49        FPA OUT   FPA OUT                                               
50        FPA OUT   FPA OUT                                               
51                                                                        
52                                                                        
53                                                                        
54                                                                        
55                               SUN Z.sup.2 ms                           
56                               SUN Z.sup.2 ls                           
57                               SUN Y.sup.2 ms                           
58                               SUN Y.sup.2 ls                           
59                               SUN X.sup.2 ms                           
60                               SUN X.sup.2 ls                           
61                                                                        
62                                                                        
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 3 D                                                                 
SUN OBJECT SPACE VECTOR (29)                                              
         FPA        FPA        FPA                                        
CYCLE    A IN       B IN       C OUT                                      
______________________________________                                    
31                  FPA OUT    y*I23 + x*I13 ms                           
32                  FPA OUT    y*I23 + x*I13 ls                           
33                             y*I22 + x*I12 ms                           
34                             y*I22 + x*I12 ls                           
35       FPM OUT                                                          
36       FPM OUT                                                          
37       FPA OUT    FPM OUT    y*I21 + x*I11 ms                           
38       FPA OUT    FPM OUT    y*I21 + x*I11 ls                           
39       A REG 2    FPM OUT                                               
40       A REG 2    FPM OUT                                               
41                                                                        
42                                                                        
43                                                                        
44                                                                        
45                             Z ms                                       
46                             Z ls                                       
47                             Y ms                                       
48                             Y ls                                       
49                             X ms                                       
50                             X ls                                       
51                                                                        
52                                                                        
53                                                                        
54                                                                        
55                  FPM OUT                                               
56                  FPM OUT                                               
57       FPM OUT                                                          
58       FPM OUT                                                          
59                  FPM OUT                                               
60                  FPM OUT                                               
61                                                                        
62                                                                        
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 3 E                                                                 
SUN OBJECT SPACE VECTOR (29)                                              
         FPA         FPA II    FPA                                        
CYCLE    A IN        B IN      C OUT                                      
______________________________________                                    
63                                                                        
64                                                                        
65                                                                        
66                                                                        
67       FPA OUT               SUN Z.sup.2 + y.sup.2 ms                   
68       FPA OUT               SUN Z.sup.2 + y.sup.2 ls                   
70                                                                        
71                                                                        
72                                                                        
73                                                                        
74                                                                        
75                                                                        
76                                                                        
77                             (SUN DIS. ms).sup.2                        
78                             (SUN DIS. ls).sup.2                        
______________________________________                                    
 
    
     Referring to Charts 3A through 3E, the &#34;I&#34; matrix elements are those in the inverse matrix (28), &#34;x&#34; represents the sun screen space x coordinate, &#34;y&#34; represents the sun screen space y coordinate, &#34;z&#34; represents the sun screen space z coordinate, and &#34;SUN x&#34;, &#34;SUN y&#34;, and &#34;SUN z&#34; represent the respective x, y, and z sun object space coordinates. At cycle 1 a signal 83 representing the I 13 ms word is input to the &#34;FPM A IN&#34; input 67 and a signal 84 representing the sun screen space x ms word is input to the &#34;FPM B IN&#34; input 68. The resulting output signal 85 from the floating point multiplier 69 is indicated at cycle 11 as x*I13 ms, which indicates the sun screen space x times the inverse matrix element I13 corresponding to the first term of the sun object space z equation (29). This output signal 85 is input to the floating point adder B input 73. In the same cycle 11 zero is input to the floating point adder A input 72. The other inputs corresponding to the other multiplication operations that are required to compute the sun object space z equation (29) may be understood from further reference to Charts 3A through 3E. 
     Signals 85 representing the product of the inverse matrix element I23 and the sun screen space y are output from the floating point multiplier output 70 at cycles 17 and 18, are applied to the A REG 1 register 75 and in successive cycles are applied to the pipeline RAM 56 and to A REG 3 register 77. In cycles 21 and 22 these signals 86 are input to the FPA B IN input 73 simultaneously with the inputting to the FPA A IN input 72 of the output signal 86 representing (x*I13+0) from the adder 71. Signals 86 representing the sum (I13*x+I23*y) are output from the floating point adder in cycles 31 and 32 and are stored in the A REG 1 register 75 and the A REG 2 register 76. These signals 86 are then input in cycles 35 and 36 to the FPA B IN input 73. Also, in cycles 35 and 36, the ms and ls words for the third term (I33*z) of the sun object space z are output from the floating point multiplier 69 and are input to the A IN input 72 of the floating point adder 71. Ten cycles after cycle 35 and 36, namely cycle 45 and 46, the signal 86 representing the sun object space z coordinate is output from the FPA C OUT output 74. This is also entered into the A REG 1 register 75 and in successive cycles through cycle 53 is entered in the pipeline RAM 56 for both the ms and ls words. 
     Having the signals 86 representing the sun object space z, y, and x coordinates (cycles 45 through 50) (shown by capital letters z, y and x in Chart 3D), then as indicated at equation (30) the sun object space distance squared is obtained. Referring to Charts 3A and 3E, cycles 45 through 50, signals 30 representing the ms and ls words for the sun object space coordinates are shown input into each of the floating point multiplier inputs 67 and 68 so that the sun object space distance squared equation (30) is processed. Ten cycles later, starting at cycles 55, signals 85 representing the sun object space distance squared for the ms and ls words for the x, y, and z coordinates are output from the floating point multiplier 69. The signals 85 representing the squared sun object space x, y, and z coordinates are output in cycles 55 through 60 (see &#34;SUN Z 2  ms&#34;, for example) and are appropriately input to the floating point adder 71 as shown on Charts 3A through 3E, such that in cycles 67 and 68 the partial sums, and in cycles 77 and 78, the entire sums (see (SUN DIS.) 2 , for example) are output as the signals 86 representing equation (30). These signals 86 are written into the pipeline RAM 56 via the registers 75, 76, and 77. 
     PROCESSING MULTIPLIERS 
     Having performed the operations described above, Process 2 Steps 2.0.1.1 through 2.0.1.3 of Process 2 are performed to obtain the multipliers for the coefficients in equation (11) and for the partial derivatives in equations (22) and (23). Referring to FIGS. 4A, 4B, and 5 in conjunction with the Charts 4A through 4I, the multipliers in equations (10) are first computed. 
     
         ______________________________________                                    
CHART 4A                                                                  
OBTAIN MULTIPLIER (10) AND PARTIAL                                        
DERIVATIVES (22) AND (23)                                                 
           FPM        FPM     FPM                                         
CYCLE      A IN       B IN    C OUT                                       
______________________________________                                    
1          D11 ms     t.sup.3 ms                                          
2          D11 ls     t.sup.3 ls                                          
3          D12 ms     t.sup.2 ms                                          
4          D12 ls     t.sup.2 ls                                          
5          D13 ms     t ms                                                
6          D13 ls     t ls                                                
8                                                                         
9          D14 ms     3 ms                                                
10         D14 ls     3 ls                                                
11         FPM OUT    3 ms    D11*t.sup.3 ms                              
12         FPM OUT    3 ls    D11*t.sup.3 ls                              
13         FPM OUT    3 ms    D12*t.sup.2 ms                              
14         FPM OUT    3 ls    D12*t.sup.2 ls                              
15         FPM OUT    3 ms    D13* t ms                                   
16         FPM OUT    3 ls    D13* t ls                                   
17                                                                        
18                                                                        
19                    t.sup.3 ms                                          
                              3* D14 ms                                   
20                    t.sup.3 ls                                          
                              3* D14 ls                                   
21         D21 ms             3*D11*t.sup.3 ms                            
22         D21 ls             3*D11*t.sup.3 ls                            
23         D22 ms     t.sup.2 ms                                          
                              3*D12*t.sup.2 ms                            
24         D22 ls     t.sup.2 ls                                          
                              3*D12*t.sup.2 ls                            
25                            3*D13*t ms                                  
26                            3*D13*t ls                                  
27         D23 ms     2 ms                                                
28         D23 ls     2 ls                                                
29         D24 ms                                                         
30         D24 ls                                                         
31         FPM OUT            D21*t.sup.3 ms                              
______________________________________                                    
 
    
     
         __________________________________________________________________________
CHART 4B                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
    FPA         FPA       FPA                                             
Cycle                                                                     
    A IN        B IN      C OUT                                           
__________________________________________________________________________
2                                                                         
3                                                                         
4                                                                         
5                                                                         
6                                                                         
7                                                                         
8                                                                         
9                                                                         
10                                                                        
11              FPM OUT                                                   
12              FPM OUT                                                   
13  FPM OUT                                                               
14  FPM OUT                                                               
15              FPM OUT                                                   
16              FPM OUT                                                   
17  D14 ms                                                                
18  D14 ls                                                                
19              FPM OUT                                                   
20              FPM OUT                                                   
21  FPM OUT                                                               
22  FPM OUT                                                               
23              FPM OUT   D11*t.sup.3 +D12*t.sup.2 ms                     
24              FPM OUT   D11*t.sup.3 +D12*t.sup.2 ls                     
25  FPM OUT                                                               
26  FPM OUT                                                               
27                        D13*t+D14 ms                                    
28                        D13*t+D14 ls                                    
29  D11*t.sup.3 +D12*t.sup.2 ms                                           
                D13*t+D14 ms                                              
30  D11*t.sup.3 +D12*t.sup.2 ls                                           
                D12*t+D14 ls                                              
31              FPM OUT   3*D14+3*D11*t.sup.3 ms                          
__________________________________________________________________________
 
    
     
         ______________________________________                                    
CHART 4C                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
         FPM        FPM        FPM                                        
Cycle    A IN       B IN       C OUT                                      
______________________________________                                    
32       FPM OUT               D21*t.sup.3 ls                             
33       FPM OUT               D22*t.sup.2 ms                             
34       FPM OUT               D22*t.sup.2 ls                             
35                                                                        
36                                                                        
37                  FPM OUT    2*D23 ms                                   
38                  FPM OUT    2*D23 ls                                   
39       t ms                  2*D24 ms                                   
40       t ls                  2*D24 ls                                   
41                             2*D21*t.sup.3 ms                           
42                             2*D21*t.sup.3 ls                           
43       D23 ms     t ms       2*D22*t.sup.2 ms                           
44       D23 ls     t ls       2*D22*t.sup.2 ls                           
45                                                                        
46                                                                        
47       D31 ms     t.sup.3 ms                                            
48       D31 ls     t.sup.3 ls                                            
49                             2* D23*t ms                                
50                             2* D23*t ls                                
51                                                                        
52                                                                        
53                             D23 * t ms                                 
54                             D23 * t ls                                 
55                                                                        
56                                                                        
57                             D31 * t.sup.3 ms                           
58                             D31 * t.sup.3 ls                           
59                                                                        
60                                                                        
61                                                                        
62                                                                        
______________________________________                                    
 
    
     
         __________________________________________________________________________
CHART 4D                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
Cycle                                                                     
    A FPA IN    B FPA IN                                                  
                       C FPA OUT                                          
__________________________________________________________________________
32              FPM OUT                                                   
                       3*D14+*D11*t.sup.3 ls                              
33  FPM OUT                                                               
34  FPM OUT                                                               
35  3*D11*t.sup.3 ms   3*D12*t.sup.2                                      
    +3*D14 ms   FPA OUT                                                   
                       +3*D13*t ms                                        
36  3*D11*t.sup.3 ls   3*D12*t.sup.2                                      
    +3*D14 ls   FPA OUT                                                   
                       +3*D13*t ls                                        
37-38                                                                     
39              FPM OUT                                                   
                       s3 ms                                              
40              FPM OUT                                                   
                       s3 ls                                              
41  FPM OUT                                                               
42  FPM OUT                                                               
43              FPM OUT                                                   
                       D21*t.sup.3 +D22*t.sup.2 ms                        
44              FPM OUT                                                   
                       D22*t.sup.3 +D22*t.sup.2 ls                        
45                     dx/ds s2 ms                                        
46                     dx/ds s2 ls                                        
47-48                                                                     
49  FPM OUT                                                               
50  FPM OUT                                                               
51              D21*t.sup.3                                               
                +D22*t.sup.2 ms                                           
                       2*D21*t.sup.3 +2*D24 ms                            
52              D21*t.sup.3                                               
                +D22*t.sup.2 ls                                           
                       2*D21*t.sup.3 +2*D24 ls                            
53  FPM OUT                                                               
54  FPM OUT                                                               
55              D34 ms                                                    
56              D34 ls                                                    
57  FPM OUT                                                               
58  FPM OUT                                                               
59  2*D21*t.sup.3 +2*D24 ms                                               
                FPA OUT                                                   
                       2*D22*t.sup.2 +2*D23*t ms                          
60  2*D21*t.sup.3 +2*D24 ls                                               
                FPA OUT                                                   
                       2*D22*t.sup.2 +2*D23*t ls                          
61              D24 ms                                                    
62              D24 ls                                                    
__________________________________________________________________________
 
    
     
         ______________________________________                                    
CHART 4E                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
          FPM        FPM         FPM                                      
Cycle     A IN       B OUT       C OUT                                    
______________________________________                                    
63        D32 ms     t.sup.2                                              
64        D32 ls     t.sup.2                                              
65                                                                        
66                                                                        
67        D42 ms                                                          
68        D42 ls                                                          
69        D41 ms     t.sup.3 ms                                           
70        D41 ls     t.sup.3 ls                                           
71        D43 ms     t ms                                                 
72        D43 ls     t ls                                                 
73        D33 ms                 D32*t.sup.2 ms                           
74        D33 ls                 D32*t.sup.2 ms                           
75        2 ms                                                            
76        2 ls                                                            
77                   t.sup.2 ms  D42*t.sup.2 ms                           
78                   t.sup.2 ls  D42*t.sup.2 ls                           
79                               D41*t.sup.3 ms                           
80                               D41*t.sup.3 ls                           
81                               D43*t ms                                 
82                               D43*t ls                                 
83        3 ms                   D33*t ms                                 
84        3 ls                   D33*t ls                                 
85        D22 ms     FPM OUT     2*t ms                                   
86        D22 ls     FPM OUT     2*t ls                                   
87        D12 ms                                                          
88        D12 ls                                                          
89        D32 ms                                                          
90        D32 ls                                                          
91        D42 ms                                                          
92        D42 ls                                                          
93        D21 ms     FPM OUT     3*t.sup.2 ms                             
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 4F                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
Cycle FPA        FPA        FPA                                           
A     IN         B IN       C OUT                                         
______________________________________                                    
63    FPA OUT               D21*t.sup.3 +D22*t.sup.2                      
                            +D23*t ms                                     
64    FPA OUT               D21*t.sup.3 +D22*t.sup.2                      
                            +D23*t ls                                     
65-66                                                                     
67               FPA OUT    D31*t.sup.3 +D34 ms                           
68               FPA OUT    D31*t.sup.3 +D34 ls                           
69                          dx/ds s1 ms                                   
70                          dx/ds s1 ls                                   
71                                                                        
72                                                                        
73    FPM OUT               s2 ms                                         
74    FPM OUT               s2 ls                                         
75-76                                                                     
77               FPM OUT                                                  
78               FPM OUT                                                  
79    FPM OUT                                                             
80    FPM OUT                                                             
81    D44 ms     FPM OUT                                                  
82    D44 ls     FPM OUT                                                  
83    FPA OUT    FPM OUT    D31*t.sup.3 +D32*t.sup.2 +D34 ms              
84    FPA OUT    FPM OUT    D31*t.sup.3 +D32*t.sup.2 +D34 ls              
85-86                                                                     
87-88                                                                     
89                          D41*t.sup.3 +D42*t.sup.2 ms                   
90                          D41*t.sup.3 +D42*t.sup.2 ls                   
91    D41*t.sup.3                                                         
      +D42*t.sup.2 ms                                                     
                 FPA OUT    D43*t+D44 ms                                  
92    D41*t.sup.3                                                         
      +D42*t.sup.2 ls                                                     
                 FPA OUT    D43*t+D44 ls                                  
93                          s1&amp;dx/ds s0 ms                                
______________________________________                                    
 
    
     
         ______________________________________                                    
CHART 4G                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
         FPM       FPM        FPM                                         
Cycle    A IN      B IN       C OUT                                       
______________________________________                                    
94       D21 ls    FPM OUT    3*t.sup.2 ls                                
95                            2*D22*t ms                                  
96                            2*D22*t ls                                  
97                            2*D12*t ms                                  
98                            2*D12*t ls                                  
99                            2*D32*t ms                                  
100                           2*D32*t ls                                  
101                           2*D42*t ms                                  
102                           2*D42*t ls                                  
103      D11 ms               3*D21*t.sup.2 ms                            
104      D11 ls               3*D21*t.sup.2 ls                            
105      D31 ms                                                           
106      D31 ls                                                           
107                                                                       
108                                                                       
109      D41 ms                                                           
110      D41 ls                                                           
111                                                                       
112                                                                       
113                           3*D11*t2 ms                                 
114                           3*D11*t.sup.2 ls                            
115                           3*D31*t.sup.2 ms                            
116                           3*D31*t.sup.2 ls                            
117                                                                       
118                                                                       
119                           3*D41*t.sup.2 ms                            
120                           3*D41*t.sup.2 ls                            
121                                                                       
122                                                                       
123                                                                       
124                                                                       
______________________________________                                    
 
    
     
         __________________________________________________________________________
CHART 4H                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
    FPA           FPA    FPA                                              
Cycle                                                                     
    A IN          B IN   C OUT                                            
__________________________________________________________________________
94                       s1 &amp; dx/ds s0 ls                                 
95  D23 ms        FPM OUT                                                 
96  D23 ls        FPM OUT                                                 
97  D13 ms        FPM OUT                                                 
98  D13 ls        FPM OUT                                                 
99  D33 ms        FPM OUT                                                 
100 D33 ls        FPM OUT                                                 
101 D43 ms        FPM OUT                                                 
                         s0 ms                                            
102 D43 ls        FPM OUT                                                 
                         s0 ls                                            
103               FPM OUT                                                 
104               FPM OUT                                                 
105 FPA OUT              2*D22*t+D23 ms                                   
106 FPA OUT              2*D22*t+D23 ls                                   
107               FPA OUT                                                 
                         2*D12*t+D13 ms                                   
108               FPA OUT                                                 
                         2*D12*t+D13 ls                                   
109                      2*D32*t+D33 ms                                   
110                      2*D32*t+D33 ls                                   
111                      2*D42*t+D43 ms                                   
112                      2*D42*t+D43 ls                                   
113 FPM OUT                                                               
114 FPM OUT                                                               
115               FPM OUT                                                 
                         dx/ds s2 ms                                      
116               FPM OUT                                                 
                         dx/ds s2 ls                                      
117 2*D32*t+D33 ms                                                        
118 2*D32*t+D33 ls                                                        
119 2*D42*t+D43 ms                                                        
                  FPM OUT                                                 
120 2*D42*t+D43 ls                                                        
                  FPM OUT                                                 
121                                                                       
122                                                                       
123                      dx/dt s3 ms                                      
124                      dx/dt s3 ls                                      
__________________________________________________________________________
 
    
     
         ______________________________________                                    
CHART 4I                                                                  
OBTAIN MULTIPLIERS (10) AND PARTIAL                                       
DERIVATIVES (22) AND (23)                                                 
           FPA        FPA       FPA                                       
Cycle      A IN       B IN      C OUT                                     
______________________________________                                    
125                                                                       
126                                                                       
127                             dx/dt s1 ms                               
128                             dx/dt s1 ls                               
129                             dx/dt s0 ms                               
130                             dx/dt s0 ls                               
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     The PROM 78 stores values of &#34;s&#34; and &#34;t&#34;, and the squares and cubes of &#34;s&#34; and &#34;t&#34;, for the respective increments 0.1 and 0.2 shown in FIG. 1. The &#34;s&#34; and &#34;t&#34; terms listed in Charts 4A through 4I are signals 87 output from the PROM 78. The register y 60 is set for the &#34;s&#34; or &#34;t&#34; value that is to be processed to cause the PROM 78 to output the desired value. Referring to cycle 1 on Chart 4A, to Process 2, Step 2.0.1, and to the s3 multiplier of equation (10), it is seen that a signal 83 representing the D11 ms word of the x coefficient matrix is input to the floating point multiplier A input 67 from the matrix RAM 83. Also, the PROM 78 is caused to output the signal 87 representing the ms word of t 3 . The signal 87 is input to the B input 68. The remaining cycles 2 through  6 input the respective signals 83 and 87 representing the remaining terms of the s3 multiplier of equation (10) that must be multiplied. These products for the first, second and third terms of the s3 multiplier equation (10) are output from the floating point multiplier 69 as the signals 85 and are in turn added to the D14 term in cycle 17 such 2 that the sum of the third and fourth terms of the s3 multiplier equation (10) are output from the floating point adder 71 at cycles 27 and 28. The sum of the first and second terms of the multiplier s3 have been output as the signals 86 from the floating point adder 71 at cycles 23 and 24 and are input to the A REG 1 register 75 and the A REG 2 register 76. Signals 88 representing such first and second terms of the s3 equation (10) are input in cycles 29 and 30 from the A register 76 into the floating point adder A input 72. In the same cycles 29 and 30, signals 30 from the pipeline RAM 56 representing the third and fourth terms of the s2 equation (10) are input to the floating point adder B input 73. As a result, in cycles 39 and 40, signals 86 representing the ms and ls words for the complete multiplier s3 are output from the floating point adder output C 74 and are input into the A REG 1 register 75. In cycles  39 through cycle 43, such signals 86 representing the multiplier s3 ms and ls words are written into the pipeline RAM 56 via registers 75 through 77. 
     In a similar manner, the s2, s1, and s0 multipliers of equations (10) are calculated and output as the signals 86 in respective cycles 73-74, 93-94, and 101-102. These signals 86 are stored in the pipeline RAM 56 in successive cycles. 
     Further, and referring now to the multipliers identified in equations (20) and (21), the multiplier (dx/ds s2) is processed starting at cycles 11 and 12, where the signals 85 representing the ms and ls words for the product of D11 times t 3  are input to the floating point multiplier A input 67. At the same cycles, the signals 84 representing the 3 ms and ls words are input from the F BUS 64 to the floating point multiplier B input 68. At cycles 21 and 22 the signals 85 representing the entire first term (e.g. 3*D11*t 3 ) of the (dx/ds s2) multiplier are output from the floating point multiplier 69. 
     Referring to Process 2, Steps 2.0.1.1 through 2.0.1.3, and to Charts 4A through 4I, in a similar manner the other terms of the multipliers shown in equations (20) and (21) are output and stored in the pipeline RAM 56 for use in calculating the values of the x, y, and z coordinates and the point normals relating thereto. For example, in cycles 45 and 46 the floating point adder C output 74 outputs the signal 86 representing the multiplier (dx/ds s2) ms and ls words which are stored in the processor input register. 
     When Process Step 2.0.1.3 has been completed, signals 86 representing all of the multipliers shown at equation (20) and (21) and all of the multipliers s3, s2, s1 and s0, used in equation (11) have been obtained and the apparatus 21 is ready to perform Step 2.0.1.4 to obtain the values of the x, y, and z coordinates and the point normals relating thereto. When the last operation of Step 2.0.1.3 has been completed, the next instruction that is read is to &#34;pop&#34;, which causes the next address signal 40 to be read from the stack RAM 48. This address signal 40 is applied to the WCS 36 which outputs an instruction signal 39 to the control register 35, which calls a subroutine for calculating such values of the coordinates x, y, and z and the related point normals. 
     PROCESS 2, STEP 2.0.1.4 OBTAIN VALUES FOR X POINT COORDINATES AND PARTIAL DERIVATIVES 
     Referring to Charts 5A through 5F in conjunction with FIGS. 4A, 4B, and 5, Step 2.0.1.4 of Process 2 is described for the x coordinate. 
     
         ______________________________________                                    
CHART 5A                                                                  
STEP 2.0.1.4, OBTAIN VALUES FOR x                                         
POINT COORDINATES AND PARTIAL DERIVATIVES                                 
          FPM         FPM      FPM                                        
Cycle     A IN        B IN     C OUT                                      
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1                     s.sup.3 ms                                          
2                     s.sup.3 ls                                          
3         s3 ms       s.sup.3 ms                                          
4         s3 ls       s.sup.3 ls                                          
5         dx/dt s3 ms s.sup.3 ms                                          
6         dx/dt s3 ls s.sup.3 ls                                          
7                     s.sup.2 ms                                          
8                     s.sup.2 ls                                          
9         dx/ds s2 ms s.sup.2 ms                                          
10        dx/ds s2 ls s.sup.2 ls                                          
11        s2 ms       s.sup.2 ms                                          
12        s2 ls       s.sup.2 ls                                          
13                    s.sup.2 ms                                          
                               s3* s.sup.3 ms                             
14                    s.sup.2 ls                                          
                               s3* s.sup.3 ls                             
15                    s.sup.2 ms                                          
                               dx/dt s3*s.sup.3 ms                        
16                    s.sup.2 ls                                          
                               dx/dt s3*s.sup.3 ls                        
17        dx/dt s2 ms s.sup.2 ms                                          
18        dx/dt s2 ls s.sup.2 ls                                          
19                             dx/ds s2*s.sup.2 ms                        
20                             dx/ds s2*s.sup.2 ls                        
21        A           s ms     s2*s.sup.2 ms                              
22                    s ls     s2*s.sup.2 ls                              
23        s1 ms       s ms                                                
24        s1 ls       s ls                                                
25        dx/ds s1 ms s ms                                                
26        dx/ds s1 ls s ls                                                
27        dx/dt s1 ms s ms     dx/dt s2*s.sup.2 ms                        
28        dx/dt s1 ls s ls     dx/dt s2*s.sup.2 ls                        
29                    s.sup.3 ms                                          
30                    s.sup.3 ls                                          
31        s3 ms       s.sup.3 ms                                          
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         __________________________________________________________________________
CHART 5B                                                                  
STEP 2.0.1.4, OBTAIN VALUES FOR x                                         
POINT COORDINATES AND PARTIAL DERIVATIVES                                 
     FPA          FPA    FPA                                              
Cycles                                                                    
     A IN         B IN   C IN                                             
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2                                                                         
3                                                                         
4                                                                         
5                                                                         
6                                                                         
7                                                                         
8                                                                         
9                                                                         
10                                                                        
11                                                                        
12                                                                        
13   s0 ms        FPM OUT                                                 
14   s0 ls        FPM OUT                                                 
15   dx/dt s0 ms  FPM OUT                                                 
16   dx/dt s0 ls  FPM OUT                                                 
17                                                                        
18                                                                        
19   dx/ds s0 ms  FPM OUT                                                 
20   dx/ds s0 ls  FPM OUT                                                 
21                FPM OUT                                                 
22                FPM OUT                                                 
23   FPA OUT             s3*s.sup.3 +s0 ms                                
24   FPA OUT             s3*s.sup.3 +s0 ls                                
25                       dx/dt(s3*s.sup.3 +s0)ms                          
26                       dx/dt(s3*s.sup.3 +s0)ls                          
27   dx/dt(s3*s.sup.3 +s0)ms                                              
                  FPM OUT                                                 
28   dx/dt(s3*s.sup.3 +s0)ls                                              
                  FPM OUT                                                 
29                       dx/ds(s2*s.sup.2 +s0)ms                          
30                       dx/ds(s2*s.sup.2 +s0)ls                          
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         ______________________________________                                    
CHART 5C                                                                  
STEP 2.0.1.4, OBTAIN VALUES FOR x                                         
POINT COORDINATES AND PARTIAL DERIVATIVES                                 
         FPM           FPM     FPM                                        
Cycle    A IN          B IN    C OUT                                      
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32       s3 ls                                                            
33       dx/dt s3 ms           s1 * s ms                                  
34       dx/dt s3 ls           s1 * s ms                                  
35                     s.sup.2 ms                                         
                               dx/ds s1*s ms                              
36                     s.sup.2 ls                                         
                               dx/ds s1*s ls                              
37       dx/ds s2 ms           dx/dt s1*s ms                              
38       dx/ds s2 ls           dx/dt s1*s ls                              
39       s2 ms                                                            
40       s2 ls                                                            
41                             s3*s.sup.3 ms                              
42                             s3*s.sup.3 ls                              
43                             dx/dt s3*s.sup.3 ms                        
44                             dx/dt s3*s.sup.3 ls                        
45       dx/dt s2 ms                                                      
46       dx/dt s2 ls                                                      
47                             dx/ds s2*s.sup.2 ms                        
         IF MORE &#34;s&#34;:                                                     
48       GO TO A               dx/ds s2*s.sup.2 ls                        
         IF LAST &#34;s&#34;,                                                     
49       THEN DO 50-66 s ms    s2 * s.sup.2 ms                            
50                     s ls    s2 * s.sup.2 ls                            
51       s1 ms                                                            
52       s1 ls                                                            
53       dx/ds s1 ms                                                      
54       dx/ds s1 ls                                                      
55       dx/dt s1 ms           dx/dt s2*s.sup.2 ms                        
56       dx/dt s1 ls           dx/dt s2*s.sup.2 ls                        
57                                                                        
58                                                                        
59                                                                        
60                                                                        
61                             s1 * s ms                                  
62                             s1 * s ls                                  
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                       CHART 5D                                                    
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STEP 2.0.1.4, OBTAIN VALUES FOR x                                         
POINT COORDINATES AND PARTIAL DERIVATIVES                                 
      FPA        FPA       FPA                                            
Cycle A IN       B IN      C OUT                                          
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32                                                                        
33    FPA OUT    FPM OUT   s3*s.sup.3 +s2*s.sup.2 + s0 ms                 
34    FPA OUT    FPM OUT   s3*s.sup.3 +s2*s.sup.2 + s0 ls                 
35    A REG 2    FPM OUT                                                  
36    A REG 2    FPM OUT                                                  
37    FPA OUT    FPM OUT   dx/dt(s3*s.sup.3 +s2*s.sup.2 +s0)ms            
38    FPA OUT    FPM OUT   dx/dt(s3*s.sup.3 +s2*s.sup.2 +s0)ls            
39                                                                        
40                                                                        
41    s0 ms      FPM OUT                                                  
42    s0 ls      FPM OUT                                                  
43    dx/dt s0 ms                                                         
                 FPM OUT   x ms                                           
44    dx/dt s0 ls                                                         
                 FPM OUT   x ls                                           
45                         dx/ds ms                                       
46                         dx/ds ls                                       
47    dx/ds s0 ms                                                         
                 FPM OUT   dx/dt ms                                       
48    dx/ds s0 ls                                                         
                 FPM OUT   dx/dt ls                                       
49               FPM OUT                                                  
50               FPM OUT                                                  
51    FPA OUT              s3*s.sup.3 + s0 ms                             
52    FPA OUT              s3*s.sup.3 + s0 ls                             
53                         dx/dt(s3*s.sup.3 +s0)ms                        
54                         dx/dt(s3*s.sup.3 +s0)ls                        
55    dx/dt(s3*s.sup.3 +s0)ms FPM OUT                                     
56    dx/dt(s3*s.sup.3 +s0)ls FPM OUT                                     
57                 dx/ds(s2*s.sup.2 +s0)ms                                
58                 dx/ds(s2*s.sup.2 +s0)ls                                
59                                                                        
60                                                                        
61    FPA OUT    FPM OUT   s3*s.sup.3 +s2*s.sup.2 +s0 ms                  
62    FPA OUT    FPM OUT   s3*s.sup.3 +s2*s.sup.2 +s0 ls                  
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                       CHART 5E                                                    
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STEP 2.0.1.4, OBTAIN VALUES FOR x                                         
POINTS COORDINATES AND PARTIAL DERIVATIVES                                
           FPM       FPM       FPM                                        
Cycle      A IN      B IN      C OUT                                      
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63                     dx/ds s1*s ms                                      
64                     dx/ds s1*s ls                                      
65                     dx/dt s1*s ms                                      
66                     dx/dt s1*s ls                                      
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                       CHART 5F                                                    
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STEP 2.0.1.4, OBTAIN VALUE FOR x                                          
POINTS COORDINATES AND PARTIAL DERIVATIVES                                
      FPA       FPA       FPA     TRANS RAM                               
Cycle A IN      B IN      C OUT   PTR DATA                                
______________________________________                                    
63    A REG 2   FPM OUT                                                   
64    A REG 2   FPM OUT                                                   
65    FPA OUT   FPM OUT   dx/dt(s3*s.sup.3 +s2*s.sup.2 +s0)ms             
66    FPA OUT   FPM OUT   dx/dt(s3*s.sup.3 +s2*s.sup.2 s0)ls              
67                                                                        
68                                                                        
69                                                                        
70                                                                        
71                x ms                                                    
72                x ls                                                    
73                dx/ds ms                                                
74                dx/ds ls                                                
75                dx/ds ms                                                
76                dx/ds ls                                                
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 FPA C OUT cycles 71-76 = last vertex 9 in row                            
 
    
     Referring also to Equation (1), it will be recalled that the D11, etc. matrix elements have been obtained and combined with the &#34;t&#34; values to obtain the s0 through s3 multipliers in Equations (10). Here, such multipliers and the multipliers of Equations 20 and 21 are combined as in Equations (13), (14), (22), and (23). This outputs the signals 86 representing coordinate values for an entire vertex row 12 representing the partial derivatives with respect to &#34;s&#34; and &#34;t&#34; for each x coordinate value. 
     The &#34;s&#34;, &#34;t&#34; PROM 78 still stores s 3  and s 2  for a given value of &#34;s&#34;. The y register 60 is set for the &#34;s&#34; value of the first V1 vertex 9 in the 0 vertex row 12 so that on the first cycle the PROM 78 outputs the signal 87 representing the ms word for s 3  and on the second cycle the signal 87 representing the ls word for s 3 . The s 3  ms and s 3  ls signals 87 are input to the floating point multiplier B input 68 where they remain until that B input 68 is loaded again with a different signal. With those signals 87 at the floating point multiplier B input 68, signals 30 representing the values of s3 ms and s3 ls are input from the pipeline RAM 56 (Chart 4D, cycles 39 and 40) to the floating point multiplier A input 67 in the next two clock cycles (3 and 4). Thus, the floating point multiplier 69 is now conditioned to output the signal 85 representing the first term &#34;s 3  (s3)&#34; of equation (11) for the x coordinate of the V1 vertex 9 in the 0 vertex row 12. 
     To start the processing of equation (23), in clock cycles 5 and 6 signals 30 representing the constants (dx/dt s3 ms) and (dx/dt s3 ls) (Chart 4H, cycles 123 and 124) are input from the pipeline RAM 56 to the floating point multiplier A input 67. In the next two cycles, the value of &#34;s&#34; for the first or V1 vertex 9 in the 0 vertex row 12 is input into the y register 60 and a bit is set to cause the PROM 61 to output the signals 87 representing the value of s 2  ms and s 2  ls. These signals 87 are sequentially input to the floating point multiplier B input 68. During the next four cycles (9-12), signals 30 representing the constants (dx/ds s2) ms, dx/ds s2 ls, s2 ms and s2 ls are read from the pipeline RAM 56 and are input to the floating point multiplier A input 67. The floating point multiplier 69 is now conditioned to output signals representing the first term &#34;(dx/ds s2)s 2  &#34; of equation (22) and the second term &#34;s 2  (s2)&#34; of equation (11). 
     In cycles 13 and 14, signals 85 representing the product of the ms and ls words of &#34;s3&#34; and &#34;s 3  &#34; are output from the floating point multiplier C output 70. Referring to equation (11), this is the first term that required a multiply operation. The signals 85 representing the (s3) s 3  ms and ls words are input to the floating point adder input B register 73 in the same clock cycle. At the same time, signals 87 representing the value of s0 ms and ls words are input to the A input 72 of the floating point adder 71. The floating point adder 71 is now conditioned to add the first and fourth terms of equation (11). 
     In a similar manner, at cycles 15 and 16 the next signals 85 output from the floating point multiplier 69, are input to the floating point adder B input 73. These are the ms and ls words for (dx/dt s3) times the ms and ls words for (s 3 ), respectively. The signals 85 output in cycles 15 and 16 thus are the result of processing the ms and ls words of the first term &#34;(dx/dt s3)*s 3  &#34; of equation (23). 
     In cycles 17 and 18, signals 86 representing the ms and ls words of the second term of equation (23), &#34;(dx/dt s2)&#34;, are input to the floating point multiplier A input 67. Since the signals 87 representing the ms and ls words of s 2  are still input to the floating point multiplier B input 68, cycles 17 and 18 condition the floating point multiplier 69 for processing the second term of equation (22). 
     In cycles 19 and 20, the floating point multiplier 69 outputs the signals 85 representing the ms and ls words for the product &#34;(dx/ds s2)*s 2  &#34;, which are output from the floating point multiplier C output 70 and are input to the floating point adder B input 73. These signals 85 from the floating point multiplier C output 70 present the computation of the first term of equation (22). In cycles 19 and 20, the pipeline RAM 56 also supplies to the floating point adder A input 72 signals 30 representing the ms and ls words for (dx/ds s0). 
     In cycles 21 and 22, the floating point multiplier B input 68 receives signals 84 representing the values of the &#34;s&#34; ms word and the &#34;s&#34; ls word from the PROM 78. In those same two cycles, signals 85 representing the s2*s 2  ms and ls words are output from the floating point multiplier C output 70 and are input to the floating point adder B input 73. This reflects the processing of the second term of equation (11) for the x coordinate. 
     In cycles 23 and 24, signals 30 representing the s1 ms and ls words are input to the floating point multiplier A input 72 from the pipeline RAM 56. In the same two cycles, the floating point adder 71 outputs the signals 86 representing the ms and ls words of the sum of &#34;(s3)s 3  +s0&#34;. This reflects the addition of the first and last terms of equation (11). These signals 86 are input to the floating point adder A input 72. 
     In the next cycles 25 and 26, signals 30 representing the ms and ls words for the term &#34;(dx/ds s1)&#34; are input to the floating point multiplier A input 67 to initiate processing of the second term of equation (22). In the same two cycles, the floating point adder C output 74 outputs signals 86 representing the ms and ls words of the sum of the words that were input at cycles 13 and 14, thus representing the sum of the first and fourth terms of equation (23). Those signals 86 output from the floating point adder C output 74 are stored in the A REG 1 register 75 and then in the A REG 2 register 76 in cycles 26 and 27. In cycles 27 and 28 , the floating point multiplier C output 70 outputs signals 85 representing the result of the multiplication of the second term of equation (23), i.e. (dx/dt s2)*s 2 . Those signals 85 are input to the floating point adder B input 73 in cycles 27 and 28 and the signals 86 representing the sum of the first and fourth terms of equation (23) that were stored in the A REG 2 register 76 are input into the floating point adder A input 72 to enable the addition of the first and fourth terms of equation (23) with the second term (see cycles 37 and 38). In the next two cycles, signals 30 representing the ms and ls words for (dx/dt s1) are input to the floating point multiplier A input 67 representing the third term of equation (23). Since signals representing the ms and ls words for &#34;s&#34; are still applied to the floating point multiplier FPM B IN input 68, cycles 25-28 condition the floating point multiplier 69 to generate signals 85 (see cycles 35-38) representing &#34;dx/ds s1*s&#34; of equation (22) and &#34;dx/dt s1*s&#34; of equation (23). 
     At cycle 28, all necessary inputs have been provided to the floating point multiplier 69 and to the floating point adder 71 with respect to the initial value of &#34;s&#34; for the first or V1 vertex 9 in the 0 vertex row 12. Thus, at this juncture, the signals 87 input to the floating point multiplier 69 starting at cycle 29 represent the next value of &#34;s&#34;. This corresponds to Process 2, Step 2.0.1.4 that indicates that the substeps 2.0.1.4.1 through 2.0.1.4.3 are performed for each &#34;s&#34; step size. Using the next &#34;s&#34; step value, equations (11), (22) and (23) are then processed to obtain the x coordinate value and the partial derivatives for the second vertex V3 in the 0 vertex row 12. Thus, the signals 87 input in cycles 29 through 50 for the second &#34;s&#34; value in the 0 vertex row 12 are for the same terms of the equations (11), (22) and (23) as were input in cycles 1 through 28. 
     To provide signals 87 representing the next value of &#34;s&#34;, &#34;s 2  &#34; and &#34;s 3  &#34;, the value of &#34;s&#34; in the processor 61 is incremented by &#34;s&#34; step size (i.e. 0.1) and the new &#34;s&#34; value is applied to the y register 60 to address the PROM 78 so that it will output the next values of s 3 , s 2 , and s as the signals 87. 
     Cycles 29 through 48 also continue the processing of the first or V1 vertex 9 which is, as described above based on the first &#34;s&#34; value. In particular, the floating point adder C output 74 outputs the signals 86 representing the ms and ls words for the term &#34;(dx/ds s2)*s 2  +(dx/ds s0)&#34;, which reflects the sum of the first and third terms of equation (22). In cycles 33 and 34, the floating point adder C output 74 outputs the signals 86 represeting the ms and ls words for the sums of the first, second and fourth terms of equation (11). This output of such signals 86 representing the first, second and fourth terms is at the same time input to the floating point adder A input 72. In these same cycles 33 and 34, the floating point multiplier 69 has output the signals 85 representing the product that is the third term of equation (11). In cycles 33 and 34, these signals 85 are input to the floating point adder B input 73 to facilitate the final addition step of equation (11), which is output in cycles 43 and 44 as the signals 86 representing the ms and ls words of the x coordinate for the V1 vertex 9 in the 0 vertex row 12. 
     Considering cycles 35 and 36, the floating point multiplier C output 70 outputs the signals 85 representing the ms and ls words for the product that is the second term of equation (22). In the same two cycles 35 and 36 these signals 85 are input to the floating point adder B input 73 to facilitate obtaining the sum of all of the terms of equation (22). To obtain the final sum of equation (22), on the same two cycles 35 and 36, the A REG 2 register 76 outputs the signals 88 representing the sum of the first and third terms of equation (22), which are applied to the floating point adder A input 72. 
     In cycles 37 and 38, the floating point multiplier C output register 70 outputs the signals 85 representing both the ms and ls words for the product that is the third term of equation (23). These signals 85 are input to the floating point adder B input 73. In the same two cycles 37 and 38, the signals 86 representing the ms and ls words of the first, second and fourth terms of equation (23) are output from the floating point adder C output 74 and are input to the floating point adder A input 72 to facilitate completion of the sums of the terms of equation (23). 
     In cycles 43-48, the signals 86 representing the ms and ls words for the x coordinate (equation (11)) of the V1 vertex 9 and the partial derivatives of that coordinate (equations (22) and (23)) are generated by the floating point adder 71. The signals 86 are stored in the font RAM 22. 
     At this juncture (cycle 48), the first value of &#34;s&#34; has been processed and the second value of &#34;s&#34; (for the V2 vertex 9 in the 0 vertex row 12) is still being processed. If at the end of cycle 48 there are additional &#34;s&#34; values to be processed in order to process all of the x coordinates and associated x partial derivatives for the V1 to V21 vertices 9 along the 0 vertex row 12, then the next address input to the WCS 36 will be that which causes cycles 21 through 48 to occur again to finish processing the current &#34;s&#34; and then to start processing the next &#34;s&#34; value for the 0 vertex row 12 (see &#34;A&#34; in Chart 5A). However, during the processing of the successive &#34;s&#34; values across the 0 vertex row 12 for the vertices V1, V3, V5, etc. through Vn-1, the processor 61 has been decrementing from an initial value representing the total number &#34;n&#34; of &#34;s&#34; values in the 0 vertex row 12. When the n-1 &#34;s&#34; value has been processed, the processor 61 outputs a control signal 89 to the branch logic circuit 42 to cause a new next address signal 89 to be input to the WCS 36. The new next address signal 43 initiates a &#34;last vertex routine&#34; for processing the last &#34;s&#34; value in the 0 vertex row 12. 
     This last &#34;s&#34; value routine is described in Charts 5C through F, starting at cycle 49. The routine is the same as the above routines for the prior &#34;s&#34; values, but does not include any further inputs for an &#34;s&#34; value following the last &#34;s&#34; value. Accordingly, the last vertex routine may be understood by reference to Charts 5C through 5F from cycles 49 through 76. The last &#34;s&#34; routine results in outputs at cycles 71 through 76 of the ms and ls words for the x coordinate and the partial derivatives with respect to &#34;s&#34; and &#34;t&#34; for the last value of &#34;s&#34; based on processing equations (11), (22) and (23). 
     COMPUTATION OF Y POINT COORDINATES AND PARTIAL DERIVATIVES 
     The above description of operations to obtain the signals 86 representing the x point coordinates and the partial derivatives is based on a subroutine that conforms to Process Step 2.0.1.4 and that was called by an instruction from the WCS 36. When the last computation is completed for the last &#34;s&#34; value of the 0 vertex row 12, which is for the last or V21 vertex 9 in the example shown in FIGS. 2 and 3, the next instruction that is read by the push/pop RAM 48 is a statement to pop, which causes the next address signal 40 to be read from the push/pop RAM 48. The address read is a next address signal 40 for an instruction for performing Process Steps 2.0.1.1 through 2.0.1.3 for the y coordinate to calculate signals 86 representing the multipliers in equation (13) and in the y coordinate equations corresponding to equations (22) and (23). These signals are written into the pipeline RAM 58. When all of the y coefficient multipliers have been written into the pipeline RAM 58, the push/pop RAM 48 pops to initiate the above-described operation of the WCS 36 to output the next address signal 40 to the control register 35. The instruction signal 39 output from the control register 35 calls the subroutine for Process 2 Steps 2.0.1.4, so that the signals 30 representing the y coefficient multipliers are read from the pipeline RAM 58. Also, the signals 87 representing the &#34;s&#34;, &#34;s 2  &#34; and &#34;s 3  &#34; values are read from the &#34;s&#34;, &#34;t&#34; PROM 78 as described above. The signals 86 are stored in the FONT RAM 22 and represent the values of the y coordinates for the V1 through V21 vertices 9 in the 0 vertex row 12. Also, the signals 86 representing the corresponding partial derivatives with respect to &#34;s&#34; and &#34;t&#34; for each such vertex V1 through V21 in the 0 vertex row 12 are stored in the font RAM 22. 
     COMPUTATION OF Z POINT COORDINATES AND PARTIAL DERIVATIVES 
     The same operation then occurs to end the above y coordinate computation and initiate the z coordinate computation as was described above for the x to y coordinate transition. Step 2.0.1.4 has been completed when the z coordinate computation is completed. The push/pop RAM 48 then reads a statement to pop and the next address signal 40 is sent to the WCS 36 to cause the control register 35 to call the subroutine for performing step 2.0.2, which is described below in respect to Charts 6A through 6E. 
     OBTAIN POINT NORMAL VECTORS 
     Referring to Charts 6A through 6E and the point normal vector equation (15), it is recalled that the signals 86 representing the partial derivatives of x, y, and z with respect to &#34;s&#34; and &#34;t&#34; for each of the V1 through V21 vertices 9 in the 0 vertex row 12 have been stored in the font RAM 22. 
     Based on the descriptions above of Charts 5A through 5F, the following Charts 6A through 6E may be understood as describing how the signals 86 representing the ms and ls words of the point normal V1 vertex 9 of the 0 vertex row 12 are obtained. As an example, as shown below signals 86 are indicated as &#34;NORM×ms&#34; and &#34;NORM×ls&#34; at cycles 27 and 28 representing point normal x in Equation (15) for the V1 vertex 9. 
     As indicated at cycle 43, until the point normal z for the Vn-1 vertex 9 in the given vertex row 12 has been processed, the subroutine loops and performs operations noted at cycles 17 through 43 in respect to each of the vertices 9 of the 0 vertex row 12 up to, but not including, the Vn vertex 9. In the example shown in FIGS. 2 and 3, the x, y, and z point normal coordinates for the V1, V3-V19 vertices 9 are processed. When the V21 vertex is next, a subroutine is called via the push/pop RAM 48 and the WCS 36 to perform the &#34;last one&#34; cycles 44-67 to output the signals 86 representing the x, y, and z point normals for the V21 vertex 9. 
     
                       CHART 6A                                                    
______________________________________                                    
STEP 2.0.2.1, POINT NORMAL VECTORS                                        
         FPM        FPM        FPM                                        
Cycle    A IN       B IN       C OUT                                      
______________________________________                                    
1                   dz/ds ms                                              
2                   dz/ds ls                                              
3        dy/dt ms                                                         
4        dy/dt ls                                                         
5                   dz/dt ms                                              
6                   dz/dt ls                                              
7        dy/ds ms                                                         
8        dy/ds ls                                                         
9        dx/ds ms                                                         
10       dx/ds ls                                                         
11                  dx/dt ms                                              
12                  dx/dt ls                                              
13       dz/ds ms              dz/ds*dy/dt ms                             
14       dz/ds ls              dz/ds*dy/dt ls                             
15       dy/ds ms                                                         
16       dy/ds ls                                                         
17       A                     dy/ds*dz/dt ms                             
18                             dy/ds*dz/dt ls                             
19                  dy/dt ms   dx/ds*dz/dt ms                             
20                  dy/dt ls   dx/ds*dz/dt ls                             
21       dx/ds ms                                                         
22       dx/ds ls                                                         
23                             dz/ds*dx/dt ms                             
24                             dz/ds*dx/dt ls                             
25       NEXT       dz/ds ms   dy/ds*dx/dt ms                             
26                  dz/ds ls   dy/ds*dx/dt ls                             
27       dy/dt ms                                                         
28       dy/dt ls                                                         
29                  dz/dt ms                                              
30                  dz/dt ls                                              
31                             dx/ds*dy/dt ms                             
______________________________________                                    
 
    
     
                       CHART 6B                                                    
______________________________________                                    
STEP 2.0.2.1, POINT NORMAL VECTORS                                        
         FPA         FPA         FPA                                      
Cycle    A IN        B IN        C OUT                                    
______________________________________                                    
2                                                                         
3                                                                         
4                                                                         
5                                                                         
6                                                                         
7                                                                         
8                                                                         
9                                                                         
10                                                                        
11                                                                        
12                                                                        
13                   FPM OUT                                              
14                   FPM OUT                                              
15                                                                        
16                                                                        
17       FPM OUT                                                          
18       FPM OUT                                                          
19                   FPM OUT                                              
20                   FPM OUT                                              
21                                                                        
22                                                                        
23       FPM OUT                                                          
24       FPM OUT                                                          
25                   FPM OUT                                              
26                   FPM OUT                                              
27                               NORM x ms                                
28                               NORM x ls                                
29                                                                        
30                                                                        
31       FPM OUT                                                          
______________________________________                                    
 
    
     
                       CHART 6C                                                    
______________________________________                                    
STEP 2.0.2.1, POINT NORMAL VECTORS                                        
       FPM             FPM       FPM                                      
Cycle  A IN            B IN      C OUT                                    
______________________________________                                    
32                               dx/ds*dy/dt ls                           
33     dy/ds ms                                                           
34     dy/ds ls                                                           
35     dx/ds ms                                                           
36     dx/ds ls                                                           
37                     dx/dt ms  dz/ds*dy/dt ms                           
38                     dx/dt ls  dz/ds*dy/dt ls                           
39     dz/ds ms                                                           
40     dz/ds ls                                                           
41     dy/ds ms                                                           
42     dy/ds ls                                                           
43     IF MORE GO TO A           dy/ds*dz/dt ms                           
44     LAST ONE                  dy/ds*dz/dt ls                           
45                     dy/dt ms  dx/ds*dz/dt ms                           
46                     dy/dt ls  dx/ds*dz/dt ls                           
47     dx/ds ms                                                           
48     dx/ds ls                                                           
49                               dz/ds*dx/dt ms                           
50                               dz/ds*dx/dt ls                           
51                               dy/ds*dx/dt ms                           
52                               dy/ds*dx/dt ls                           
53                                                                        
54                                                                        
55                                                                        
56                                                                        
57                               dx/ds*dy/dt ms                           
58                               dx/ds*dy/dt ls                           
59                                                                        
60                                                                        
______________________________________                                    
 
    
     
                       CHART 6D                                                    
______________________________________                                    
STEP 2.0.2.1, POINT NORMAL VECTORS                                        
         FPA         FPA         FPA                                      
Cycle    A IN        B IN        C OUT                                    
______________________________________                                    
32       FPM OUT                                                          
33                               NORM Y ms                                
34                               NORM Y ls                                
35                                                                        
36                                                                        
37                   FPM OUT                                              
38                   FPM OUT                                              
39                                                                        
40                                                                        
41                               NORM Z ms                                
42                               NORM Z ls                                
43       FPM OUT                                                          
44       FPM OUT                                                          
45                   FPM OUT                                              
46                   FPM OUT                                              
47                                                                        
48                                                                        
49       FPM OUT                                                          
50       FPM OUT                                                          
51                   FPM OUT                                              
52                   FPM OUT                                              
53                               NORM X ms                                
54                               NORM X ls                                
55                                                                        
56                                                                        
57       FPM OUT                                                          
58       FPM OUT                                                          
59                               NORM Y ms                                
60                               NORM Y ls                                
61                                                                        
______________________________________                                    
 
    
     
                       CHART 6E                                                    
______________________________________                                    
STEP 2.0.2.1, POINT NORMAL VECTORS                                        
           FPA       FPA       FPA                                        
Cycle      A IN      B IN      C OUT                                      
______________________________________                                    
62                                                                        
63                                                                        
64                                                                        
65                                                                        
66                     NORM Z ms                                          
67                     NORM Z ls                                          
______________________________________                                    
 
    
     All of the signals 86 representing the x, y, and z point normal coordinates for the V1 through V21 vertices 9 are stored in the font RAM 22 to complete step 2.0.2.1 for the 0 vertex row 12, which represents the first &#34;t&#34; step. 
     PROCESS VERTEX ROW 1 ACCORDING TO PROCESS 2 
     Referring again to Process 2, Step 2.1 calls for incrementing the &#34;t&#34; step size from 0 to 0.2. This is done by a pop command at the end of step 2.0.2.1 that causes the push/pop RAM 48 to output the next address signal 40 to the WCS 36 to cause the control register 35 to call the subroutine for performing Step 2.1 and to decrement the processor register 26. When the &#34;t&#34; step size has been incremented to 0.2, Step 2.0.1 is followed for that &#34;t&#34; step size so that the 1 vertex row 14 will be processed. As a result of performing Process 2, for each of the V1 through V22 vertices 9 of the vertex rows 12 and 14, the x, y and z coordinates and the corresponding point normal vectors have been obtained. When the last operation of Step 2.0.-1.4 has been completed, the next instruction that is read is to &#34;pop&#34;, which causes the next address signal 40 to be read from the stack RAM 48. This address signal 40 is applied to the WCS 36 which outputs an instruction signal 39 to the control register 35 for performing Process 3. 
     OBTAIN POINT INTENSITY 
     Reference is made to Process 3 and to Charts 7A through 7F, 
     
                       CHART 7A                                                    
______________________________________                                    
PROCESS 3 - TRANFORM POINT AND                                            
OBTAIN POINT INTENSITY                                                    
         FPM        FPM       FPM                                         
Cycle    A IN       B IN      C OUT                                       
______________________________________                                    
1        PNZ ms     PNZ ms                                                
2        PNZ ls     PNZ ls                                                
3        SUN Z ms   PNZ ms                                                
4        SUN Z ls   PNZ ls                                                
5        PNY ms     PNY ms                                                
6        PNY ls     PNY ls                                                
7        SUN Y ms   PNY ms                                                
8        SUN Y ls   PNY ls                                                
9        a33 ms     Z ms                                                  
10       a33 ls     Z ls                                                  
11       a32 ms     Z ms      PNZ.sup.2 ms                                
12       a32 ls     Z ls      PNZ.sup.2 ls                                
13       a31 ms     Z ms      Z SUN*PNZ ms                                
14       a31 ls     Z ls      Z SUN*PNZ ls                                
15                            PNY.sup.2 ms                                
16                            PNY.sup.2 ls                                
17       PNX ms     PNX ms    Y SUN*PNY ms                                
18       PNX ls     PNX ls    Y SUN*PNY ls                                
19       SUN X ms             Z * a33 ms                                  
20       SUN X ls             Z * a33 ls                                  
21       a23 ms     Y ms      Z * a32 ms                                  
22       a23 ls     Y ls      Z * a32 ls                                  
23       a22 ms     Y ms      Z * a31 ms                                  
24       a22 ls     Y ls      Z * a31 ls                                  
25       a21 ms     Y ms                                                  
______________________________________                                    
 
    
     
                       CHART 7B                                                    
______________________________________                                    
PROCESS 3 - TRANSFORM POINT AND                                           
OBTAIN POINT INTENSITY                                                    
Cy-  FPA            FPA       FPA                                         
cle  A IN           B IN      C OUT                                       
______________________________________                                    
2                                                                         
3                                                                         
4                                                                         
5                                                                         
6                                                                         
7                                                                         
8                                                                         
9                                                                         
10                                                                        
11                  FPM OUT                                               
12                  FPM OUT                                               
13                                                                        
14                                                                        
15   FPM OUT                                                              
16   FPM OUT                                                              
17   Z SUN*PNZ ms   FPM OUT                                               
18   Z SUN*PNZ ls   FPM OUT                                               
19   a43 ms         FPM OUT                                               
20   a43 ls         FPM OUT                                               
21   a42 ms         FPM OUT                                               
22   a42 ls         FPM OUT                                               
23   a41 ms         FPM OUT                                               
24   a41 ls         FPM OUT                                               
25                            PNZ.sup.2 + PNY.sup.2 ms                    
26                            PNZ.sup.2 + PNY.sup.2 ls                    
27   PNZ.sup.2 +PNY.sup.2 ms                                              
                    FPM OUT   Z+Y (SUN*PN)ms                              
28   PNZ.sup.2 +PNY.sup.2 ls                                              
                    FPM OUT   Z+Y (SUN*PN)ls                              
29   Z + Y (SUN*PN)ms                                                     
                    FPM OUT   Z*a33 + a43 ms                              
30   Z + Y (SUN*PN)ls                                                     
                    FPM OUT   Z*a33 + a43 ls                              
31                  FPM OUT   Z*a32 + a42 ms                              
______________________________________                                    
 
    
     
                       CHART 7C                                                    
______________________________________                                    
PROCESS 3 - TRANSFORM POINT AND                                           
OBTAIN POINT INTENSITY                                                    
Cy-  FPM           FPM         FPM                                        
cle  A IN          B IN        C OUT                                      
______________________________________                                    
26   a21 ls        Y ls                                                   
27                             PNX.sup.2 ms                               
28                             PNX.sup.2 ls                               
29                             X SUN*PNX ms                               
30                             X SUN*PNX ls                               
31   a13 ms        X ms        Y * a23 ms                                 
32   a13 ls        X ls        Y * a23 ls                                 
33   a12 ms                    Y * a22 ms                                 
34   a12 ls                    Y * a22 ls                                 
35   a11 ms                    Y * a21 ms                                 
36   a11 ls                    Y * a21 ls                                 
37                 SUN DIST.sup.2 ms                                      
38                 SUN DIST.sup.2 ls                                      
39   PN DIST.sup.2 ms                                                     
40   PN DIST.sup.2 ls                                                     
41                             X * a13 ms                                 
42                             X * a13 ls                                 
43   A REG 3       A REG 3     X * a12 ms                                 
44   A REG 3       A REG 3     X * a12 ls                                 
45                             X * a11 ms                                 
46                             X * a11 ls                                 
49                 FPM OUT     SUN*PN DIST.sup.2  ms                      
50                 FPM OUT     SUN*PN DIST.sup.2 ls                       
53                                                                        
      ##STR1##                 Intensity Dot.sup.2 ms                     
54                                                                        
      ##STR2##                 Intensity Dot.sup.2 ls                     
55                                                                        
      ##STR3##     A REG 2                                                
56                                                                        
      ##STR4##     A REG 2                                                
______________________________________                                    
 
    
     
                       CHART 7D                                                    
______________________________________                                    
PROCESS 3 - TRANSFORM POINT AND                                           
OBTAIN POINT INTENSITY                                                    
      FPA       FPA       FPA                                             
Cycle A IN      B IN      C OUT                                           
______________________________________                                    
32    A REG 2   FPM OUT   Z*a32 + a42 ls                                  
33    A REG 2   FPM OUT   Z*a31 + a41 ms                                  
34    A REG 2   FPM OUT   Z*a31 + a41 ls                                  
35    A REG 2   FPM OUT                                                   
36    A REG 2   FPM OUT                                                   
37                        PN DIST.sup.2 ms                                
38                        PN DIST.sup.2 ls                                
39                        Intensity Dot ms                                
40                        Intensity Dot ls                                
41    FPA OUT   FPM OUT   Y*a23+Z*a33+a43ms                               
42    FPA OUT   FPM OUT   Y*a23+Z*a33+a43ls                               
43    FPA OUT   FPM OUT   Y*a22+Z*a32+a42ms                               
44    FPA OUT   FPM OUT   Y*a22+Z*a32+a42ls                               
45    FPA OUT   FPM OUT   Y*a21+Z*a31+a41ms                               
46    FPA OUT   FPM OUT   Y*a21+Z*a31+a41ls                               
47                                                                        
48                                                                        
49                                                                        
50                                                                        
51                                                                        
52                                                                        
53                                                                        
54                                                                        
55                                                                        
56                                                                        
57                        Z&#39; ms                                           
58                        Z&#39; ls                                           
59                        Y&#39; ms                                           
60                        Y&#39; ls                                           
61                        X&#39; ms                                           
62                        X&#39;  ls                                          
______________________________________                                    
 
    
     
                       CHART 7E                                                    
______________________________________                                    
PROCESS 3 - TRANSFORM POINT AND                                           
OBTAIN POINT INTENSITY                                                    
       FPM         FPM       FPM                                          
Cycle  A IN        B IN      C OUT                                        
______________________________________                                    
63                                                                        
                              ##STR5##                                    
64                                                                        
                              ##STR6##                                    
65                 FPM OUT                                                
                              ##STR7##                                    
66                 FPM OUT                                                
                              ##STR8##                                    
73     FPA OUT                                                            
74     FPA OUT                                                            
83                           INTENSITY ms                                 
84                           INTENSITY ls                                 
______________________________________                                    
 
    
     
                       CHART 7F                                                    
______________________________________                                    
PROCESS 3 - TRANSFORM POINT AND                                           
OBTAIN POINT INTENSITY                                                    
       FPA      FPA        FPA                                            
Cycle  A IN     B IN       C OUT                                          
______________________________________                                    
63     2 ms     FPM OUT    A -B                                           
64     2 ls     FPM OUT                                                   
73                                                                        
                            ##STR9##                                      
74                                                                        
                            ##STR10##                                     
______________________________________                                    
 
    
     Referring also to FIGS. 4A, 4B, and 5, it will be recalled that equation (33) for the point intensity included, as the denominator of the fraction, the point normal distance squared times the sun object space distance squared. It is also recalled that equation (31) defined point normal distance squared. Process 3 starts in cycles 1 and 2 by entering the signals 86 representing the ms and ls words for the point normal z for the V1 vertex 9 into both the FPM A IN input 67 and FPM B IN input 68 of the floating point multiplier 69. (see &#34;PNZ ms&#34; and &#34;PNZ ls&#34; in Chart 7A). At cycles 11 and 12, the signals 85 representing the ms and ls words of the point normal z squared (see PNZ 2  ms, for example) are output and are input to the FPA B IN input 73. Thus, the third term of equation (31) has been processed and the signals 85 entered into the floating point adder 71 to initiate adding it to one of the other two terms of equation (31). 
     In cycles 5 and 6, the signals 86 representing the ms and ls words for the point normal y for the V1 vertex 9 are entered into the A and B inputs 67 and 68 of the floating point multiplier 69 to initiate the squaring operation required for the second term of equation (31). Signals 85 representing the ms and ls words of the squared point normal y term are output at cycles 15 and 16 and at the same time are entered into the FPA A IN input 72 to condition the floating point adder 71 to output the signals 86 representing the sum of the second and third terms of equation (31) at cycles 25 and 26. (see PNZ 2  +PNY 2  ms, for example). These signals 86 are entered into the A REG 1 register 75 at cycles 25 and 26 and then to the A REG 2 register 76 in cycles 26 and 27. Thus, the completed second and third terms of the equation (31) are held in the A REG 1 register 75 and the A REG 2 register 76 awaiting squaring of the point normal x, which is the first term of the equation (31). This is done in cycles 17 and 18 where the signals 86 representing the ms and ls words for point normal x of the V1 vertex 9 are entered into the A and B inputs 67 and 68 of the floating point multiplier 69. At cycles 27 and 28, the floating point multiplier 69 outputs the signals 85 representing the ms and ls words of the square of point normal x for the V1 vertex 9. These signals 85 are at the same time input to the FPA B IN input 73. The prior signals 86 representing the second and third terms of equation (31) are input from the A REG 2 register 76 to the FPA A IN input 72. In cycles 37 and 38, the signals 86 representing the ms and ls words for the (equation (31)) point normal distance squared (see PN DIST 2  ms and PN DIST 2  ls) are output from the FPA C OUT output 74 and at the same time are input to the A REG 1 register 75. In successive cycles, these signals are shifted to the A REG 2 register 76. 
     It is also recalled that the point intensity dot that is the numerator of the equation (33) includes the point normals times the sun object space for the x, y and z coefficients. Referring to Chart 7A, it will be recalled that in cycles 1 and 2, the signals 86 representing the ms and ls words of the point normal z for the V1 vertex 9 were entered to the respective FPM A IN and FPM B IN inputs 67 and 68. These signals 86 are still applied to the FPM B IN input 68 at cycles 3 and 4, at which time signals 86 representing the sun object space z ms and ls words are applied to the FPM A IN input 67. (see SUN Z ms, for example). Thus, at the end of cycles 3 and 4, the floating point multiplier 69 is conditioned to multiply the third term of equation (32). The signals 85 representing this term are output at cycles 13 and 14 at the FPM C OUT output 70. These signals 85 output in cycles 13 and 14 are written into the A REG 1 register 75 and in cycles 14 and 17 the signals 85 representing the third term of equation (32) are written into the A REG 2 register 76. 
     By reference to cycles 7, 8 and 19, 20 on Chart 7A, it may be understood that the multiplication of the second and first terms of equation (32) occurs and is output respectively in the form of signals 85 at cycles 17 and 18 representing the second term of equation (32), and at cycles 29 and 30 outputting signals 85 representing the first term of equation (32). These respective signals 85 are input directly to the FPA B IN input 73 with the third term of equation (32) at cycles 17 and 18 to condition the floating point adder 71 to add the third term of equation (32) with the second term of equation (32). At cycles 27 and 28, signals 86 representing the ms and ls words of such second and third terms of equation (32) are output and are input to the A REG 1 register 75. In cycles 28 and 29, such second and third terms are written into the A REG 2 register 76. Referring to Chart 7A, cycles 29 and 30, it may be understood that the signals 85 representing the first term of equation (32) are output from the FPM C OUT output 70 and are input directly to the floating point adder FPA B IN input 73. Also, in cycles 29 and 30, the signals 86 stored in A REG 2 register 76 representing the second and third terms of equation (32) are applied to the FPA A IN input 72 to condition the floating point adder 71 to output signals representing the point intensity dot of equation (32). Such signals 86 are shown output from the floating point adder 71 at cycles 39 and 40 under the &#34;FPA C OUT&#34; heading and are indicated by the respective &#34;intensity dot&#34; ms and ls. These signals 86 are written directly into the A REG 1 register 75 and in the next cycles into the A REG 2 register 76. Then in cycles 41 and 43 they are written into the A REG 3 register 77. 
     To continue processing equation (33), in cycles 39 and 40 (Chart 7C), the signals 85 representing PN DIST 2  ms and ls are read from the A REG 2 register 76 and applied to the FPM A IN input 67 for multiplication with the signals 85 representing sun object space distance squared (see &#34;SUN DIST 2  ms and ls&#34; in Chart 7C, cycles 37 and 38). The product, which is the demoninator of equation (33), is represented by the signals 85 at cycles 49 and 50 (see &#34;SUN 2  * PN DIST 2  &#34; ms and ls). To square point intensity dot, the signals 86 in the A REG 3 register 77 are applied to both the FPM A and B in inputs 67 and 68 (Chart 7C, cycles 43 and 44), and the signals 85 are output at cycles 53 and 54 to represent the numerator of equation (33) (see &#34;Intensity Dot 2  &#34; ms and ls at cycles 53 and 54, Chart 7C). 
     Considering the denominator of equation (33), the signals 85 from the floating point multiplier 69 (cycles 49 and 50, Chart 7C) are applied to the reciprocal PROM 59 in cycles 51 and 52. In cycles 52 and 53, signals 90 representing the approximate reciprocal are output from the PROM 59. To obtain precise results from the division, instead of multiplying the reciprocal of the denominator by the signal 85 representing the intensity dot which is output from the floating point multiplier 69 cycles 53 and 54, the following steps are performed: 
     
         n/d=n/d&#39;*(2-d/d&#39;) 
    
     where 1/d&#39;=approximately 1/(sun object space distance) 2  * (point normal distance) 2  from the reciprocal PROM 59, d=(sum object spaced distance) 2  * (point normal distance) 2 , and n=intensity dot 2 . 
     In cycles 63 and 64, signals 85 are output and represent &#34;d/d&#39;&#34;, in cycles 73 and 74, signals 86 are output to represent &#34;2-d/d&#39;&#34;. 
     To conclude this processing and obtain the point intensity, the output signals 86 from the floating point adder 71 in cycles 73 and 74 are multiplied by the signals 85 from the floating point multiplier in cycles 65 and 66 to obtain the signals 85 and 86 representing the point intensity. 
     In addition to the intensity calculation description above, it may be understood that at cycles 9 through 14 (Chart 7A) at cycles 19 through 24 (Chart 7B), and at cycles 32 through 36 (Chart 7C), signals 24 and 83 are input to the respective floating point multiplier 69 and the floating point adder 71 to transform each vertex 9 using a standard transformation matrix having elements such as a33, a32, and a31 represented by signals 83 from the matrix RAM 52. These signals 83 are input to the FPM A IN input 67 in cycles 9 through 14, for example. In particular, the x, y, and z coordinate signals 86 representing the V1 vertex 9 are read from the font RAM 22 as the signals 24. At the above cycles, the signals 24 and 83 are input as noted above and processed to transform the V1 vertex 9. 
     The transformation results in signals 86 output at cycles 57 and 58 (Chart 7D) for the transformed z (or z&#39;) coordinate, at cycles 59 and 60 for the transformed y (or y&#39;) coordinate and at cycles 61 and 62 for the transformed x (or x&#39;) coordinate. The signals 86 representing the transformed V1 vertex 9 are stored in the FIFO 81. As indicated in Process 3, each of the V2 through V22 vertices 9 is seriatim and signals 86 representing the transformed vertices 9 are output and stored in the FIFO 81 for use in Process 4. 
     Referring now to Process 4, it may be understood that the process of defining the polygons 15 starts with Step 4.1 in which the signals 86 representing the transformed V1 vertex and the corresponding point intensity are read from the FIFO 81 (see &#34;get 1st x, y, z, I&#34;, Step 4.1). These signals 86 are written into a RAM 89 (FIG. 4B). Steps 4.2 and 4.3 are performed and the signals 86 corresponding to the V2 and V3 vertices 9 are written into the RAM 89. In Step 4.4, these signals 86 are output as the vertex signals 10 representing the P1 polygon 15. In particular, when the signals 86 representing the transformed V1, V2 and V3 vertices 9 and the corresponding point intensities have been written into the FIFO, the control circuit 23 outputs an instruction to read such signals 86 representing the V1 vertex 9 from the FIFO 81 into selected storage locations in the RAM 89. The next instruction from the circuit 23 causes the signals 86 representing the V2 vertex 9 to be read and written into the RAM 89. The vertex signals 10 are then output to represent the P1 polygon 15. This output is via the terminal 82 connected to a raster-scan output device 91 having a video screen 92 (FIG. 4B). The raster-scan output device 91 responds to the vertex signals 10 that represent the polygons 15 and displays the object on the video screen 92. 
     The next instruction is to loop, which results in an instruction to read from the FIFO 81 the signals 86 representing the transformed coordinates and the intensity of the V4 vertex 9. These signals 86 are written in the RAM 89 over those of the V1 vertex 9 so that the P2 polygon 15 is represented. In response to an instruction from the circuit 23, the RAM 89 is read and the vertex signals 10 are output to represent the V2, V3 and V4 vertices 9 of the P2 polygon 15 and the corresponding point intensities. This continues as defined in Process 4 until the P20th polygon 15 has been processed. Then, the vertex row 19 is processed as described above and the signals 86 representing the V24, V26 . . . V44 vertices 9 are obtained and written into the font RAM 22 in place of the signals 86 that represented the V1, V3 . . . V21 vertices 9. Processes 3 and 4 are then performed as described above. 
     This continues until the P100th polygon 15 has been processed by outputting the vertex signals 10 representing the three vertices 9 of the P100th polygon 15 and the point normals corresponding thereto. At this time, the register 26 of the processor 61 outputs a &#34;register equals zero&#34; signal 89 which causes the control circuit 23 to stop. 
     While the preferred embodiment has been described in order to illustrate the fundamental relationships of the present invention, it should be understood that numerous variations and modifications may be made to these embodiments without departing from the teachings and concepts of the present invention. Accordingly, it should be clearly understood that the form of the present invention described above and shown in the accompanying drawings is illustrative only and is not intended to limit the scope of the invention to less than that described in the following claims.