A L ESHLEMAN: Aircraft Division, H D MERIWETHER: Missiles and Space Systems Division.
While large scale digital computers have brought about major changes in many areas of the engineering process, these changes have been felt primarily where analytical, accrual, or clerical functions are performed. Here computer programs have been applied to a broad spectrum of problems that have the properties of being well defined, solvable by equations, bounded by machine capability and suited to batch processing. Such applications are by no means exhausted; however, there does exist both a firm precedent for usage, and a set of definable goals for future development and exploitation by computerized techniques. While applications of this nature have received usage in the design areas, they are limited in scope and do not satisfy the essential needs of the designer. The designer does not have the luxury of working with a defined problem; his function is to create the definition.
Until recently, the computer did not seem suitable for pure design applications. Computing machines have not been powerful enough to handle the masses of data that the designer requires. Operational procedures did not produce answers in time to be of use to the designer. But worst of all, there seemed little hope of economically defining a realistic three-dimensional structure by a computer program.
Improvements in its capability have made the computer more attractive as a design tool. Growth in computer speed and memory size, together with the appearance of on-line graphic consoles, now make the development of a versa tile three-dimensional structural representation a worthwhile investment.
The basic product is defined by designers. Their responsibility is to ensure that all parts fit together, without confusion, into a working whole; that individual parts be made and assembled at a minimum cost; that the product withstand a wide range of environments; that it be easily and rapidly serviced; that it be easily repaired, and that it be tolerant of unintentional or extraordinary usage. The design must be accomplished within an extremely short period of time. Little margin for error can be tolerated. Any changes subsequent to design release are extremely costly, and too many changes can kill a program. In short, while a good design cannot by itself guarantee a profitable operation, a poor design can almost guarantee a failure. No proficiencies elsewhere in an organization can salvage a design conceived in haste, executed by expedience, and released before maturity.
Design is an iterative process whereby the basic requirements of a product are satisfied by the definition of a structure. The successful design is more capable of fulfilling its intended functions for less cost than competing designs. In a very real sense, the design process is a race against time, where each iterative step taken before a design is released will, in some manner, produce a product that is superior to its predecessors.
In general, design is not frozen until the cost of the next iterative cycle exceeds the anticipated benefits, or the length of time required to perform it constitutes a default to the competition.
Applying these conditions to today's design environment, it is a rare instance indeed when the original release drawings reflect an adequate number of iterative design cycles. Aerospace companies are devoting enormous sums of money to develop manufacturing and release systems which will adequately cope with needed design changes. These systems are grossly abused when used to compensate for an incomplete design. To a large extent this amounts to the development of a single pass design capability that is simultaneous with the production operation. When design is attempted during this time period, the number of people involved and the immediate disposition of resources and inventories make the resultant product immeasurably more costly than a design matured on the boards.
The only way to minimize these costly corrections is to release a matured design on schedule. Extensive use of computers in the design functions is the way to bring this about.
The development of a capability to increase the number of iterative design cycles hinges upon the availability of a computer graphics system geared to design requirements.
Two basic steps constitute an iteration in the design cycle, a graphical process followed by an analytical process. The graphical process develops a plausible idea through the use of sketches, layouts, and overlays; and clay, soap, paper, or wood models of the proposed design which satisfy a set of geometric and functional constraints. In order to verify its adequacy, the proposed design is subjected to a series of analytical tests that utilize tools ranging from slide rules, through desk top calculators, to large scale computers.
The transformation of data from graphical form into data that will drive a sophisticated general purpose computer program is awkward and time consuming. The advantages of current computer application programs are, for the most part, denied the designer because of the excessive time required to set up a problem, Even after answers are obtained, the computer data must be retransformed into the frame of reference of the original problem before the next step of the iteration can begin, The development of a modeling capability to perform many of the same functions as the designer's sketches and scaled models, and yet provide an easy and immediate entry into analysis programs, will permit the processing of an adequate number of iterations within the time (and cost) span allocated the designer.
Many varieties of three-dimensional modeling techniques exist, or are being developed, The task of selecting one of them for use at a given installation is a difficult one and should be based on a carefully thought out investigation of how the technique will be used and who it must serve.
Prominent among the modeling techniques is a parametric bi-cubic representation of three-dimensional surfaces [1] [2]. A special form of this technique [3] has been selected for use by the Structures Design sections of the McDonnell Douglas Corporation. The features that make it attractive as a design tool are described in this paper. (A brief description of the theory may be found in the Appendix.)
All surfaces are represented by sets of surface patches defined by parametric cubics, These patches have the desirable properties of defining essentially unlimited varieties of geometric shapes in a convenient and practical manner.
Every patch is represented by an array of coefficients which are stored in an identical matrix form. This consistency of notation permits simplification in mathematics and in the coding of computer programs.
The parametric cubics are the lowest order polynomial having the property of being able to produce curves that twist through space. The use of third order polynomials permits economy in computing costs while retaining the ability to form patch boundary curves which can be nonplaner.
This is a feature not available with the conic curves used in traditional lofting techniques, since by definition they are plane curves, There are 12 coefficients (4 for each of the X, Y, and Z coordinates) available to define a parametric cubic curve. A particular curve is uniquely defined by satisfying a set of 12 requirements in terms of end points, slopes, and curvature. If fewer than 12 items are specified, the remaining degrees-of-freedom may be used, for example, to minimize the distances from a set of data points to a curve, or to maximize the minimum radius of curvature. In the most general case, seven items of data specify the state of a curve at a given point: three point coordinates, two directions of the tangent vector, the direction of the normal vector, and the radius of curvature. Thus, a single parametric cubic curve is deficient by two coefficients (14-12=2) of completely satisfying all seven conditions at two fully defined end points.
However, two curves can be joined with the restraint that the curves have common points, tangents, normals, and curvature (seven constraints). The combined curve can now satisfy the seven end restraints at each of the two ends with three fewer requirements than available coefficients (21-24=-3). The remaining degrees-of-freedom can be used to optimize some property of the curve, or they can be used to specify the three coordinates at the joint between the curves. This curve will create a surface patch of the standard form.
The X, Y, and Z coordinates of a surface patch are each single valued cubic polynomial functions of two variables, u and w, which are defined for the region between zero and one. The coefficients of these polynomials can be expressed in terms of end points, partial derivatives and cross-derivatives with respect to the u and w parameters. See the appendix for a more detailed explanation. All communication to and from the graphics system is in terms of the points, slopes, and curvatures of physical space coordinates. The computer program performs the conversion to parameter space coordinates without the user needing to know their values.
All mathematical operations are performed on patch coefficients rather than on the explicit X, Y, and Z coordinates. Included are such things as: translation and rotation of coordinate axies, change of scale along one or more axes, bending a curved line into a new shape, warping a surface into a new shape, duplicating a patch, blending from one boundary curve to another, and forming surface normals, etc.
Each patch array contains all the information necessary to completely and continuously define a surface coordinate, tangent, or normal. The method of extracting any of these values consists of a pre-and post-multiplication of a 4 by 4 matrix by a row vector of u and a column vector of w. Since the display requirements vary, this technique makes the pattern used to draw the patch a simple option of the user (see Figure 1).
Another option is the ability to generate either normal or unit normal surface vectors. When needed, the coefficients of an auxilliary patch of unit normals are computed only once and stored in a matrix of the standard form, The unit normals are computed directly from the parametric cubic form rather than by the time consuming and costly extraction of square roots associated with vector arithmetic. This saving is especially valuable when displaying on-line graphics, or for animated motion in perspective views of an object where surfaces are made invisible if the unit normals point away from the observer.
There are eight independent coefficients available to define a planer curve. The desired values of the X-coordinate, the Y-coordinate, the slope, and the radius of curvature can be specified at both initial and terminal ends of a curve, Detailed analysis of these conditions shows that a critical value for radius of curvature can be computed, for which a solution exists when both end radii are larger or smaller than the critical value. A solution may not exist if a radius greater than critical is specified at one end of the curve, and a radius less than critical is specified at the other end. This is probably a rare condition and if it occurs, it can be handled by using a pair of joined curves over the same interval.
Another procedure of defining a curve is to specify the end points, the end slopes, and an intermediate point on the desired curve (end radii not specified). In this case one degree-of-freedom remains which could be used to maximize the minimum radius of curvature or to set the slope of the curve at the intermediate point.
One of the advantages of parametric curves is the nonsensitivity to infinite slopes. No special procedures are required in the vicinity of an infinite slope. The coordinate system can be rotated with no danger of introducing singular points where the slope is undefined. This means that a set of surface patches defining a three-dimensional body can be in the same basic coordinate system without the messy procedure of using an auxiliary local coordinate system to avoid singular points.
The typical patch has four boundary curves. A three-sided, or even a two-sided patch is handled in the identical matrix form as a standard patch. The only difference is that the length of a boundary curve becomes zero. The coding of the computer program, that computes the coefficients in matrix array for the surface normals, recognizes the degenerate curve and performs the appropriate vector arithmetic on the two curves that actually intersect at the point. Once this matrix array is computed, no further special processing is required for degenerate boundary curves. This form of matrix array cannot be distinguished from that of a standard four-sided patch.
Since patches are defined independently, two adjacent patches can share a common boundary curve, but have completely different slopes along the boundary. For example, a cube can be represented by six patches of the standard form (see Figure 2).
The cross-derivatives are important to the parametric cubic patch system. Higher accuracy is attained by their use, When a 90-degree quadrant of a sphere is formed by approximating a true circle along the boundary curves and then using cubic blending functions between the boundaries, a maximum deviation in δR/R of approximately 0.003 exists when the cross-derivatives are equal to zero. If the circles of the boundary curves are approximated by parametric cubics, and if the appropriate values for the cross-derivatives are used, a discrepancy of 0.00025 is obtained in the maximum deviation in δR/R. If a sphere is formed in 45-degree segments a value of less than 0.000005 exists (see Figure 3, Figure 4, and Figure 5 for supplemental information on accuracy).
A second important use for the cross-derivatives permits adjustment of the surface normal slope along a boundary. Detailed analysis shows that this can be accomplished independently for each boundary curve of a patch without altering the shape of any boundary curve nor the slope of the surface normal vector on the other three boundaries. Figure 6 is a reproduction, of a series of movie frames, that shows a time dependent surface deformation of a rotating sphere. The modeling system, used to create this apparently simple sequence of pictures, had to take into account infinite slopes, three-sided patches, twisted space curves, and coordinate transformations while maintaining slope continuity across patch boundaries.
Under certain conditions the continuity of slope across a boundary between two adjacent parametric cubic patches may not be continuous even though they match at the two ends of the boundary curve. By taking advantage of the cross-derivatives, the slopes can be adjusted to guarantee matching at three points along the curve, Under these conditions the maximum deviation from true match should be satisfactory, except for very rare cases where the solution would be to subdivide the patch in order to reduce the discrepancy to an acceptable value.
The ability to represent an entire model by a system of surface patches that share common boundaries is a significant feature. When classical unbounded analytical surfaces are used for a modeling technique, the complete model is found by computing intersections (often not unique) between them, and then in some manner determining which portion of the analytical surface is actually used. These difficulties do not occur in the parametric surface patch system. Techniques for using the connectivity between patches, together with the knowledge of the surface properties within individual patches, provide a complete surface definition for any three-dimensional model. To fully appreciate the value of this feature, consider that any desired analysis program may be written to process an entire model using a single set of data as input.
For example, each computer drawn perspective picture illustrated in this paper was produced by a single call to a single subroutine. The only difference between them was the input data array, This same array of data can be used in a single call to another subroutine to compute such things as surface area, volume, center-of-gravity, etc. The essential advantage comes from the fact that since the data array is complete, self-sufficient, and non-ambiguous, it is amenable to direct analysis, without recourse to additional sources of data or logic, in order to define an entire model.
Besides being able to construct empirical shapes, it is possible to approximate any analytical surface definition, to any desired accuracy, by a set of parametric cubic patches. Figure 7 (left) illustrates construction of a conic curve that satisfies the requirements of slope at two end points and also includes an intermediate point. Figure 7 (right) is a parametric cubic curve computed from the same set of constraints.
Typically the eye cannot distinguish between two such curves. In general, the accuracy of approximation of analytical curves depends on the length of the curve represented (see Figure 3). To further investigate accuracy, a sequence of five connected parametric cubics were computed to represent a curve defined by the equation
Z = (2x - x2)3/4
and a system of patches were constructed to represent a body of revolution (see Figure 8). The resulting approximation was true within one part in a hundred thousand. Once the shape was in standard parametric form, it was a trivial matter to produce the aspect ratios shown. This capability allows the intermixing of analytical and empirical shapes, for once patches have been created any one array is indistinguishable from any other and all operators apply equally. Hence, the changes in aspect ratio involved only calls to a single operator.
The parametric bi-cubic surface modeling system currently consists of approximately 40 independent subroutine modules which are called. as required, from any program written in FORTRAN IV. The routines are written in FORTRAN IV, and to date they compile and execute on the IBM 7094, IBM 360, and UNIVAC 1108. When use of a display unit, such as the SC 4020 or IBM 2250, is desired, a minimum amount of hardware-dependent calls from a single subroutine are used for access.
The parametric bi-cubic representation is receiving usage in both the analytical and design sections of the McDonnell Douglas Corporation. A brief description of some examples follow.
Probably one of the most widely used applications of numerical surface fitting techniques appears in the area of hypersonic aerodynamics [4]. Configurations currently being studied generally cannot be represented analytically, and the methods of this paper are superior to the out-dated scheme of creating a model of the true shape constructed entirely of segments of planes, cylinders, cones, and spheres. Furthermore, in the hypersonic flow regime, there exists adequate semi-empirical pressure and heat transfer prediction methods which require only the local vehicle geometry. Utilizing the accuracy available in the parametric representation, several new methods have been devised involving higher order surface properties such as curvature.
Some consideration is being given to the development of a comprehensive vehicle design program. Incorporated in this program will be a parametric model of a lifting spacecraft which represents a very broad family of vehicles (see Figure 9). By picking a skin thickness it is possible to calculate the usable internal volume. Furthermore, by defining simplified representations of men, engines, tankage, and subsystems, it should be possible to carry out a realistic sizing study, for a given mission, with a single computer program.
A definition of the external shape of the delta space vehicle was one of the first applications undertaken using the technique. Figure 10 shows the vehicle and the major assemblies. All of the pictures were obtained from the same patch network and drawn with the same routine.
Output from a 6-degree-of-freedom trajectory simulation computer program furnished the data that were used to drive the model shown in the movie. Sample frames are shown in Figure 11. The program was run on the UNIVAC 1108 and execution consisted of creating the model, positioning it, and writing an SC 4020 plot tape. The movie was generated at the rate of 1850 vectors/ second.
Figure 12 shows the use of the technique as a preliminary layout tool. Three major assemblies have been modeled; a bulkhead, a skirt, and a truss assembly. The dimensions and positions of each assembly may be changed, with respect to each other, by the exercise of a simple option. Additional parts can be added and removed at will.
This concept permits parametric variations of a standard shape, in this instance a bathtub fitting. Only the critical dimensions that match requirements for stress and fit need be entered. Output consists of shapes as shown in Figure 13 and Figure 14.
Parametric cubics can be used to advantage in analytical/experimental investigation such as the notched specimen study described here. With little effort, test coupons containing a number of various notch shapes can be designed and converted into a discrete element mathematical model for analysis by matrix force or matrix displacement techniques (see Figure 15). The results of this analysis can be displayed using parametric cubics, and can be used to determine the location and orientation of test instrumentation. It is quite possible to use the original computer definition to drive numerically controlled machinery that fabricates the test specimens.
The satisfaction of pilot and passenger vision requirements in the design of aircraft structure is an important application. Analysis for the effect of index of refraction on vision characteristics must be considered. This, of course, requires knowledge of the surface normals to the windshield as well as a definition of the perimeter around the edge of the windshield. When the windshield shape is represented by parametric surfaces it is as simple to analyze a curved windshield as it is for a flat windshield. See Figure 16 for a three-view of a DC-9 aircraft windshield drawn using parametric cubics.
Another application using parametric cubics is in radome design. Figure 17 was drawn using a set of parametric cubic surface patches. Figure 18 is a structures preliminary design analysis model shown for one quadrant of the whole structure. The amount of detail representing the actual structure was selected based on the intended use of the analysis. At any given stage of design a model having greater or less detail may be required. This usually means that a completely new set of coordinates is required to define each of the individual points on, or within, the structure. This process is a very difficult and time-consuming operation. The analysis to check the performance of the antenna requires the computation of incidence angles over critical regions of the surface. If the contours are of non-classical shapes these incidence calculations can also be very difficult and time consuming.
The performance of the antenna depends on the surface shape, the interference effects of the internal structure, and the overall rigidity, but internal structure depends on the surface shape and the requirements for rigidity.
The inter-relationships among the design requirements in the simplified statement of this problem illustrate the desirability of a three-dimensional modeling technique which is easily introduced into the iterative design process.
Any comprehensive design application utilizing the computer should have as a planned target, the ability to pass on to downstream disciplines a readily available mathematical definition of the completed structural element or assembly. This mathematical model should contain all the information required to define the coordinates of any point on the model and any vector passing through that point. This information will provide a ready basis for the acquisition of data such as aerodynamic properties, loads, weights, moments of inertia, CG's, surface and cross-sectional areas, temperature profiles, and lumped parameter models for sophisticated static and dynamic analyses, etc. It is quite possible to contain in the computer a mathematical expression of the lofted shape of the entire vehicle, as well as all the information necessary to define each detailed part. Once implemented, this capability can produce benefits to the organization that will ultimately overshadow the production of a carefully thought out design as we know it today.
Consider the joint development of layouts, arrangements and configurations by responsible specialists from all disciplines in an environment where they can keep abreast of, and react to, proposed changes literally as soon as they occur. Consider further the possibility of controlling the weight, balance, and dynamic properties of a vehicle by the judicious placement of components that are now located by a set of insular constraints.
It is quite likely that traditional concepts of engineering will undergo appreciable modification before realizing the full potential of this capability. Its imminence suggests that serious consideration be given to the extent of the changes that will take place in practically every level of the establishment:
The method described in this paper holds promise of creating a superior product, not by computerizing the design process but by unloading a large amount of burdensome clerical labor from the designer. In fact, the superior design can only come about through an intensification of the design effort made possible by suitable programming systems in the hands of a designer.
[1] Coons, S A Surfaces for Computer Aided Design of Space Figures, MIT Preprint No 299, January 1964.
[2] Gillert, G O, Geometric Computing, Machine Design, 18 March and 1 April 1965.
[3] Coons, S A, class lecture notes, Computer Aided Design, Summer Session Course, MIT, 1-12 August 1966.
[4] Timmer, H G, Stokes, T Z, A Numerical Surface Description Technique and Its Application to the Hypersonic Aerodynamics of Arbitrarily Shaped Bodies, Douglas Report DAC 59085, March 1967.
The surface modeling technique is based on the special form of parametric cubics described briefly in the following discussion. The X, Y, and Z coordinates of space curves are each expressed as a cubic function of a single parameter, u, defined for the interval 0≤u≤1. Similarly, the coordinates for surface patches are expressed as cubic functions of two parameters, u and w, defined for the region of a unit square over the interval of 0≤u≤1 and 0≤w≤1.
In the above sketch a plane curve in the X-u plane is defined where X(u) is a cubic function of u. This function and its derivative, X'(u), are expressed in terms of four coefficients; A, B, C, and D.
X(u) = Au3 + Bu2 + Cu + D (A1)
X'(u) = 3Au2 + 2Bu + C (A2)
where 0≤u≤1
A set of four equations is obtained from these functions by setting the value of u=0 and u=1.
X(0) = D
X(1) = A + B + C + D (A3)
X'(0) = C
X'(1) = 3A + 2B + C
The solution of this set of equations when substituted in Equation {A1) and Equation (A2) and after collection terms becomes;
X(u) = X(0)(2u3 - 3u2 + 1) + X(1)(-2u3 + 3u2) + X'(0)(u3 - 2u2 + u) + X'(1)(u3 -u2) (A4)
X'(u) = X(0)(6u2 - 6u) + X(1)(-6u2 + 6u) +X'(0)(3u2 - 4u +1) + X'(1)(3u2 - 2u) (A5)
or more compactly as
X(u) = X(0) F1(u) + X(1) F2(u) + X'(0)F3(u) + X'(1)F4(u) (A6)
X'(u) = X(0) F'1(u) + X(1) F'2(u) + X'(0)F'3(u) + X'(1)F'4(u) (A7)
where:
F1(u) = 2u3 - 3u2 + 1
F2(u) = -2u3 + 3u2
F3(u) = u3 - 2u2 + u
F4(u) = u3 -u2 (A8)
F'1(u) = 6u2 - 6u
F'2(u) = -6u2 + 6u
F'3(u) = 3u3 - 4u + 1
F'4(u) = 3u2 -2u
The extension of Equation (A6) and Equation (A7) to include a three-dimensional space curve follows directly.
( X(u) ) ( X(0) ) F1(u) ( X(1) ) F2(u) ( X'(0) ) F3(u) ( X'(1) ) F4(u)
( Y(u) ) = ( Y(0) ) + ( Y(1) ) + ( Y'(0) ) + ( Y'(1) ) (A9)
( Z(u) ) ( Z(0) ) ( Z(1) ) ( Z'(0) ) ( Z'(1) )
( X'(u) ) ( X(0) ) F'1(u) ( X(1) ) F'2(u) ( X'(0) ) F3(u) ( X'(1) ) F'4(u)
( Y'(u) ) = ( Y(0) ) + ( Y(1) ) + ( Y'(0) ) + ( Y'(1) ) (A10)
( Z (u) ) ( Z(0) ) ( Z(1) ) ( Z'(0) ) ( Z'(1) )
or, again, in a more compact manner
V(u) = V(0) F1(u) + V(1) F2(u) + V'(0) F3(u) + V'(1) F4(u) (A11)
V'(u) = V(0) F'1(u) + V(1) F'2(u) + V'(0) F'3(u) + V'(1) F'4(u) (A12)
It is of interest to evaluate the equations (A8) for values of u=0 and u=1.
F1(0) = 1 F1(1) = 0 F'1(0) = 0 F'1(1) = 0
F2(0) = 0 F2(1) = 1 F'2(0) = 0 F'2(1) = 0
F3(0) = 0 F3(1) = 0 F'3(0) = 1 F'3(1) = 0
F4(0) = 0 F4(1) = 0 F'4(0) = 0 F'4(1) = 1
Now by inspection of Equation (A11) and Equation (A12), it is seen that the
equations for both V(u) and V'(u) are satisfied, by definition, at the boundary
points, u=0 and u=1.
V(0) = V(0)(1) + V(1)(0) + V'(0)(0) + V'(1)(0)
V(1) = V(0)(0) + V(1)(1) + V'(0)(0) + V'(1)(0)
V'(0) = V(0)(0) + V(1)(0) + V'(0)(1) + V'(1)(0)
V'(1) = V(0)(0) + V(1)(0) + V'(0)(0) + V'(1)(1) (A14)
For this reason Equation (A8), are called blending functions since they will cause a smooth transition between the. boundaries at u= 0 and u=1, for any arbitrary values of the boundary points and the boundary tangents.
The Equation (A11) for a space curve is extended to define a surface patch by applying the blending functions, Equation (A8), to the boundary curves of two opposite sides of a surface patch, as shown in the above sketch. For the curve at w=0 to be blended to the curve at w=1:
V(u,w) = V(u,0)(w)F1(w) + V(u,1)(w)F2(w) + Vw(u,0)(w)F3(w) + Vw(u,1)(w)F4(w) (A15)
For the curve at u=0 to be blended to the curve at u=1:
V(u,w) = V(0,w)F1(u) + V(1,w)(w)F2(u) + Vu(0,w)(w)F3(u) + Vu(1,w)F4(u) (A16)
The four terms of Equation (A15) are expanded in the same manner as Equation (A11) and expressed in matrix form but note that expansion of Equation (A16) also leads to the same result.
V(u,0)F1(w) = [ F1(u) F2(u) F3(u) F4(u) ] | V (0,0) 0 0 0 | ( F1(w) )
| V (1,0) 0 0 0 | ( F2(w) )
| Vu(0,0) 0 0 0 | ( F3(w) ) (A17)
| Vu(1,0) 0 0 0 | ( F4(w) )
V(u,1)F2(w) = [ F1(u) F2(u) F3(u) F4(u) ] | 0 V (0,1) 0 0 | ( F1(w) )
| 0 V (1,1) 0 0 | ( F2(w) )
| 0 Vu(0,1) 0 0 | ( F3(w) ) (A18)
| 0 Vu(1,1) 0 0 | ( F4(w) )
Vw(u,0)F3(w) = [ F1(u) F2(u) F3(u) F4(u) ] | 0 0 Vw(0,0) 0 | ( F1(w) )
| 0 0 Vw(1,0) 0 | ( F2(w) )
| 0 0 Vuw(0,0) 0 | ( F3(w) ) (A19)
| 0 0 Vuw(1,0) 0 | ( F4(w) )
Vw(u,0)F4(w) = [ F1(u) F2(u) F3(u) F4(u) ] | 0 0 0 Vw (0,1) | ( F1(w) )
| 0 0 0 Vw (1,1) | ( F2(w) )
| 0 0 0 Vuw(0,1) | ( F3(w) ) (A20)
| 0 0 0 Vuw(1,1) | ( F4(w) )
The complete surface patch equations in matrix form is the addition of these four terms
Again, using a more compact matrix notation
V(u,w) = F(u) B F(w)γ (A22)
where F(u) and F(w)γ are each a row or column matrix of blending functions and where B is called a boundary matrix since its coefficients are geometric properties of the boundaries of the surface patch. This is the form of the surface equation most convenient to develop the graphics theory and computer programnung.
Computing expense is minimized by the use of another form of the surface patch equation that combines the effect of the blending functions and the boundary matrix into a single matrix.
V(u,w) = U S Wγ (A23)
where
U = [u3 u2 u 1 ]
W = [w3 w2 w 1 ]
S = MBMγ
F(u) = UM
F(w) = WM
The definitions of the U, W and M matrices result frorn treating the blending functions F(u) and F(w) as vectors and extracting the coefficients of u and w see Equation (A8), to form the fourth order M matrix. The surface matrix, S, is then formed by pre-multiplying and post-multiplying the boundary matrix, B, by M and by the M-transpose matrix.