Parabolic reflector assembled from triangular shaped petals
BACKGROUND &amp; SUMMARY OF THE INVENTION 
     Low-cost, high-gain antenna reflector designs exist in the prior art. One such design is disclosed in U.S. Pat. No. 3,832,717 issued to the inventor hereof Aug. 27, 1974. That design offered the advantages of low cost, light weight and convenient assembly in the field while providing relatively high gain performance up to approximately 4 GHz. However, that reflector, comprising a plurality of generally triangular petals assembled in slightly overlapping relationship, attained only a &#34;quasi-paraboloid&#34; shape which reduces its gain characteristics above 4 GHz. In order to achieve useable gain at frequencies in the 12 GHz region, a reflector having a conformation more closely approaching a true paraboloid is necessary. 
     True parabolic antennae typically require substantial truss support structure and are expensive to manufacture. The parabolic reflector of the present invention requires no support truss and, by improving the basic design concept of the above-mentioned U.S. patent, achieves substantially parabolic shape over the entire surface of the reflector without appreciably increasing manufacturing cost, complicating field assembly or increasing shipping weight. 
     One embodiment of the present invention comprises a plurality of greatly overlapping, generally triangular-shaped petals having precisely sized and positioned holes in the overlap region through which a set of fasteners are inserted to locate one petal relative to the next and to bend adjacent petals elastically to provide curvilinear transverse shape therein. For this embodiment the petals may be interlaced or alternately overlayed. Another embodiment of the present invention comprises a layer of generally triangular shaped petals coupled to a second fully overlapping layer of similar shaped petals, the layers being rigidly held together in substantially parabolic conformation by fasteners inserted through commonly and precisely sized and positioned holes through the petals of both layers. For each embodiment, a rigid, segmented exterior rim formed to receive the outer edges of the assembled petals provides the necessary mechanical structure for mounting and positioning the reflector and for maintaining mechanical integrity over a wide range of environmental conditions. 
     Before assembly of either configuration reflector, each petal is essentially a flat sheet of light-weight, flexible, relatively thin, electromagnetically reflective material such as aluminum. The ultimate reflector configuration is determined by the precisely positioned fasteners in each petal or, alternatively in the case of the two-layer configuration wherein the inner layer petals edgewise abutt, by the shape of the abutting petal edges. Thus, by controlling the locating of the holes in the petals and the shape of the petal edges, the shape of the reflector is controlled and adjusted as desired. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a perspective view of a microwave antenna incorporating a prior art quasi-paraboloid dish reflector. 
     FIG. 2a is a top view of an eight-petal quasi-paraboloid reflector constructed according to FIG. 1 taken at the intersection of the petals with a plane perpendicular to the focal axis of that reflector prior to assembly. 
     FIG. 2b is a top view of the eight-petal configuration of FIG. 2a after assembly. 
     FIG. 3a is a top view of an eight-petal paraboloid reflector constructed according to one embodiment of the present invention having interlaced petals taken at the intersection of the petals with a plane perpendicular to the focal axis of that reflector prior to assembly. 
     FIG. 3b is a top view of an eight-petal paraboloid reflector constructed according to one embodiment of the present invention having alternately overlayed petals taken at the intersection of the petals with a plane perpendicular to the focal axis of that reflector prior to assembly. 
     FIG. 3c is a top view of the reflector configuration of FIG. 3 during assembly showing curvilinear transverse petal shape 
     FIG. 4a is a side view of two cantilever beams attached to the same anchor wall. 
     FIG. 4b is a side view of the top beam of FIG. 4a in bent configuration. 
     FIG. 5 is a top view of a petal constructed according to one embodiment of the present invention. 
     FIG. 6a is a top view of a two-layer, 16 petal paraboloid reflector constructed according to another embodiment of the present invention taken at the intersection of the petals with a plane perpendicular to the focal axis of that reflector prior to assembly. 
     FIG. 6b is a top view of the reflector configuration of FIG. 6a after assembly. 
     FIG. 7 is a top view of a petal constructed according to another embodiment of the present invention. 
     FIG. 8 is a three-dimensional view of a parabola as it rotates about the z-axis. 
     FIG. 9 is a perspective view of a rectangular plate subjected to uniform bending moments. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In the quasi-parabolic reflector of U.S. Pat. No. 3,832,717, each petal slightly overlapped adjacent petals for conveniently indexing the petals to one another during assembly. The overlap was so small that, after assembly, a small, transverse, petal-to-petal angle resulted (refer to FIGS. 1, 2a and 2b) because the overlapped portion of each petal would simply inelastically bend to the angle of the adjacent petal. No smooth, curved bending occurred along common transverse axes of the adjacent petals. 
     Referring now to FIGS. 3a and 3c, as the interlaced overlap is increased and the petals are fastened with two fasteners, such as rivets or nuts and bolts, at each fastening position in the overlap region, the petals are forced to bend in the transverse direction as shown. The number of fasteners used at each position is essentially arbitrary taking into consideration petal material and thickness. The transverse direction (i.e. axis) is perpendicular to the longitudinal axis of the petal. If the hole positions are located accurately, the petals will form a substantially parabolic reflector. Thus the surface of a reflector need not be preformed using expensive dies. No metal spinning operations are needed nor stretch forming. These operations are costly, particularly for reflectors having a diameter greater than ten feet. When fasteners are installed in the alternately overlayed petal configuration of FIG. 3b, transverse bending is obtained in essentially the same manner. 
     The reflector design of FIGS. 3a and 3b results from the combined effects of petal overlap, fastener hole positions, the number of petals and the thickness of the petal material. The transverse bending or curvilinear shape of the petals may be explained by analogy to the bending of beams. 
     Suppose two cantilever beams are attached to the same wall one above the other as shown in FIG. 4a. The bottom beam is shorter than the top beam. Also suppose b is the ratio of the length of the bottom beam to the length of the top beam. (b is analogous to the amount of overlap of one petal over the adjacent petal.) As the top beam is bent such that the end is bent down by an angle φ as shown in FIG. 4b, a gap or space S, will begin to develop between the top and bottom beams (or petals). The maximum amount of this gap is given by the following relation derived from beam theory. ##EQU1## where l is the length of the top beam 
     This space between beams (or petals) can be made as small as possible by reducing φ, l, and/or b so that there is no abrupt change in the slope of the inner surface from one beam (petal) to the next. Reducing φ may be achieved by using a greater number of petals. Reducing l may be achieved by reducing the diameter of the antenna or using more petals (increasing n). Reducing b will reduce the amount of overlap bu may not be desirable because this is what cause the transverse bending. Thus, some trade off among the variables to achieve the best combination is necessary. A complete derivation of equation A is given in Appendix A to this specification. 
     If The local yield point stress of the petal material is exceeded during assembly, the petal will be permanently deformed which is undesirable because any such local yielding would indicate a non-uniform stress and/or bending moment which would cause non-uniformity in the curvature of the reflector surface. The maximum stress throughout the petal after assembly should be kept below the endurance limit stress or yield point. The maximum stress in the petals is defined by the relationship: ##EQU2## where ν is Poisson&#39;s ration, Z o  is focal length, E is the elastic modulus of the material and t is the thickness of the petals. A derivation of this equation is given in Appendix B. 
     After selecting the number of sections and overlap b, φ may be determined from ##EQU3## where r is the radial distance to any point on the line Z = r 2  /4Z o  rotated about the Z axis and Z o  is the focal length of the parabolic surface of revolution. The maximum value of r is the radius of the antenna. 
     l may be determined from ##EQU4## It is apparent from relationships A and B that the trade off is between φ, l, n, and t, where S max  &lt; t to provide smooth transition from petal-to-petal on the reflector surface. 
     Knowing the radius of the dish, the maximum space that will exist between the overlapping edges of adjacent petals may be calculated using equation A. Using equation B, the maximum stress may be determined after selecting material E of thickness t. The thickness t is reduced until the maximum stress is substantially below the elastic limit and/or yield point. By keeping S max  less than t the overlap will provide a close and essentially smooth transition from petal-to-petal. This will enhance the conditions necessary for an accurate parabolic surface. If after making the calculation with equation A, the S max  may be larger than desirable; if so, it may be desirable to reduce S max  by increasing the number of petals, n. 
     The maximum transverse deflection or bending of a petal along any transverse axis at its intersection with the longitudinal axis is given by: ##EQU5## where n = the number of petals, 
     Z o  = the focal length of the reflector, and δ &lt; t to preclude permanent deformation of the petals. The derivation of equation (C) is given in Appendix C. 
     The positions of the holes in the petals are now determining from the following relation. If, for example, the radius of the antenna (9.2 fee) is divided into 23 equally spaced lengths, the distance R to each of those 23 positions on the surface of the parabola from the center can be determined by: ##EQU6## where Z =  r 2  14 Z o  , dZ/dr = r/2Z o  and r = projection of R on the r-axis after petal is bent longitudinally. 
     Thus, for each r an R may be calculated. Referring to FIG. 5, a value of θ is now calculated for each r from the following equation: 
     
         θ = (180° r)/nR 
    
     the R and S positions of each hole position are labeled as R 1 , S 1  and R 2 , S 2  and R 3  as shown in FIG. 5. These positions are calculated from the following relationships: 
     
         R.sub.1 = Rcos[θ(1-2b)]-0.7                                         
             R.sub.2 = Rcos θ -0.7                                  
                       R.sub.3 = Rcos[θ(1+2b)]-0.7                  
S.sub.1 = Rsin [θ(1-2b)]                                            
             S.sub.2 = Rsin θ                                       
                       S.sub.3 = Rsin[θ(1+2b)]                      
 
    
     where b is the ratio of the overlap to the width of the petal at the fastening point. The constant 0.7 arises from the presence of a circular hole at the center of the assembled reflector. Table I summarized the precise hole positions for each of 80 petals in the example reflector to achieve curvilinear transverse shape in each of those petals. 
     
                                           TABLE I                                 
__________________________________________________________________________
r       R   R.sub.1                                                       
                 S.sub.1                                                  
                     R.sub.2                                              
                         S.sub.2                                          
                             R.sub.3                                      
                                 S.sub.3                                  
__________________________________________________________________________
1   0.400                                                                 
        0.400   0.010   0.021   0.031                                     
2   0.800                                                                 
        0.800                                                             
            0.100                                                         
                0.021                                                     
                    0.099                                                 
                        0.042                                             
                            0.098                                         
                                0.063                                     
3   1.200                                                                 
        1.201                                                             
            0.501                                                         
                0.031                                                     
                    0.500                                                 
                        0.063                                             
                            0.498                                         
                                0.094                                     
4   1.600                                                                 
        1.603                                                             
            0.902                                                         
                0.042                                                     
                    0.901                                                 
                        0.084                                             
                            0.898                                         
                                0.126                                     
5   2.000                                                                 
        2.006                                                             
            1.305                                                         
                0.052                                                     
                    1.303                                                 
                        0.105                                             
                            1.299                                         
                                0.157                                     
6   2.400                                                                 
        2.410                                                             
            1.709                                                         
                0.063                                                     
                    1.706                                                 
                        0.126                                             
                            1.702                                         
                                0.108                                     
7   2.800                                                                 
        2.815                                                             
            2.114                                                         
                0.073                                                     
                    2.112                                                 
                        0.147                                             
                            2.107                                         
                                0.220                                     
8   3.200                                                                 
        3.223                                                             
            2.522                                                         
                0.084                                                     
                    2.519                                                 
                        0.167                                             
                            2.513                                         
                                0.251                                     
9   3.600                                                                 
        3.633                                                             
            2.931                                                         
                0.094                                                     
                    2.928                                                 
                        0.188                                             
                            2.922                                         
                                0.282                                     
10  4.000                                                                 
        4.045                                                             
            3.343                                                         
                0.105                                                     
                    3.339                                                 
                        0.209                                             
                            3.333                                         
                                0.314                                     
11  4.400                                                                 
        4.459                                                             
            3.758                                                         
                0.115                                                     
                    3.754                                                 
                        0.230                                             
                            3.746                                         
                                0.345                                     
12  4.800                                                                 
        4.877                                                             
            4.175                                                         
                0.126                                                     
                    4.171                                                 
                        0.251                                             
                            4.162                                         
                                0.377                                     
13  5.200                                                                 
        5.298                                                             
            4.596                                                         
                0.136                                                     
                    4.591                                                 
                        0.272                                             
                            4.582                                         
                                0.403                                     
14  5.600                                                                 
        5.722                                                             
            5.020                                                         
                0.147                                                     
                    5.014                                                 
                        0.293                                             
                            5.005                                         
                                0.489                                     
15  6.000                                                                 
        6.149                                                             
            5.447                                                         
                0.157                                                     
                    5.441                                                 
                        0.314                                             
                            5.421                                         
                                0.471                                     
16  6.400                                                                 
        6.581                                                             
            5.879                                                         
                0.168                                                     
                    5.872                                                 
                        0.335                                             
                            5.861                                         
                                0.502                                     
17  6.800                                                                 
        7.816                                                             
            6.314                                                         
                0.178                                                     
                    6.307                                                 
                        0.356                                             
                            6.296                                         
                                0.534                                     
18  7.200                                                                 
        7.456                                                             
            6.753                                                         
                0.188                                                     
                    6.746                                                 
                        0.377                                             
                            6.734                                         
                                0.565                                     
19  7.600                                                                 
        7.900                                                             
            7.197                                                         
                0.199                                                     
                    7.190                                                 
                        0.398                                             
                            7.177                                         
                                0.596                                     
20  8.000                                                                 
        8.348                                                             
            7.646                                                         
                0.209                                                     
                    7.638                                                 
                        0.419                                             
                            7.625                                         
                                0.628                                     
21  8.400                                                                 
        8.802                                                             
            8.099                                                         
                0.220                                                     
                    8.091                                                 
                        0.440                                             
                            8.077                                         
                                0.659                                     
22  8.800                                                                 
        9.260                                                             
            8.557                                                         
                0.230                                                     
                    8.549                                                 
                        0.461                                             
                            8.534                                         
                                0.691                                     
23  9.200                                                                 
        9.724                                                             
            9.021                                                         
                0.241                                                     
                    9.012                                                 
                        0.482                                             
                            8.997                                         
                                0.722                                     
__________________________________________________________________________
 
    
     In the configuration of FIGS. 6a and 6b, a second layer of essentially identical petals has been added to effectively fully overlap each of the petals in the configuration of FIGS. 3a and 3b. Outer layer of petals 60 is fastened to inner layers of petals 62 by two fasteners (for example, 65) at each fastening position in the overlap region of adjacent petals. The precisely sized and positioned holes serve to index the petals of inner layer 62 to each other when fastened to the petals of outer layer 60, and to cause transverse bending of each petal in the reflector which essentially eliminates petal-to-petal angles when tightly fastened. Equation (C) derived for the greatly overlapped configuration also describes the transverse bending of the petals in layers 60 and 62. 
     Outer layer 60 also provides structural support for the reflector, eliminating the need for supporting truss. The petals of inner layer 62 are constructed to essentially edgewise abutt one another to eliminate petal-to-petal discontinuities at the reflecting surface which enhances the gain characteristics of the antenna. 
     Since the sheet materials respond non-linearly when deflection, δ, is greater than thickness, t, much more force is required to achive such deflection than for δ less than t. In addition, there is substantial risk of exceeding yield point stress and permanently deforming the material. Therefore, the bending of the petal material should be kept within the linear (δ &lt; 2t for this fully overlapping configuration) bending range of the material to facilitate field assembly and to avoid support trusses which are necessary to apply greater force yet increase weight and cost. Since the petals of the reflector are to be bent longitudinally as well as transversely, the deflection for a selected number of petals defines t. If t is too large the bending moment to attain transverse petal deflection would increase beyond the limits of field assembly without special tools, jigs and skilled labor. Therefore, n should be increased to reduce δ which in turn reduces t to facilitate field assembly. Table II gives the maximum transverse deflection of the petals at 10 inch increments of r according to equation (C) for a 10 foot diameter reflector comprising up to 80 petals. 
     
                       TABLE II                                                    
______________________________________                                    
n = 20      n = 30   n = 40   n = 60 n = 80                               
______________________________________                                    
r = 20&#34;                                                                   
       0.050    0.022    0.013  0.006  0.003                              
r = 30&#34;                                                                   
       0.110    0.049    0.028  0.012  0.007                              
r = 40&#34;                                                                   
       0.190    0.084    0.047  0.021  0.012                              
r = 50&#34;                                                                   
       0.285    0.127    0.071  0.032  0.018                              
r = 60&#34;                                                                   
       0.392    0.174    0.098  0.044  0.025                              
______________________________________                                    
 
    
     It should be noted also that δ increases along the longitudinal axis toward the outer rim of the reflector. The point at which δ = 2t can be controlled by appropriate selection of the parameters discussed above. For this embodiment of the present invention, this point is at the rim along the longitudinal axis of the petal. 
     The hole positions are determined by the relations given below: 
     
         R.sub.1 = Rcos[θ(1-b)]-A                                            
                 R.sub.2 = Rcos[θ(1+B)]-A                           
S.sub.1 = Rsin[θ(1-B)]                                              
                 S.sub.2 = Rsin[θ(1+B)]                             
 
    
     Where 
     A = the radius of the hole in the center of the dish 
     B = 0.5, the ratio of petal overlap to petal width 
     R is determined by equation (D) θ = πr/nR 
     Referring to FIG. 7, Table III gives the hole positions for the petals in both the outer and inner layers of petals for a 10 foot diameter reflector having a total of 40 petals and a prabolic focal length, Z o , of 48 inches as determined by the above relations. 
     
                       TABLE III                                                   
______________________________________                                    
       r        R       R.sub.1                                           
                               S.sub.1                                    
                                     R.sub.2                              
                                           S.sub.2                        
______________________________________                                    
1       7.500    7.508   3.127                                            
                              0.294  3.081                                
                                          0.882                           
2      10.000   10.018   5.635                                            
                              0.393  5.574                                
                                          1.175                           
3      12.500   12.535   8.151                                            
                              0.491  8.074                                
                                          1.469                           
4      15.000   15.061  10.674                                            
                              0.589 10.582                                
                                          1.763                           
5      17.500   17.596  13.208                                            
                              0.687 13.101                                
                                          2.057                           
6      20.000   20.144  15.753                                            
                              0.785 15.631                                
                                          2.351                           
       r        R       R.sub.1                                           
                               S.sub.1                                    
                                     R.sub.2                              
                                           S.sub.2                        
7      22.500   22.704  18.312                                            
                              0.883 18.175                                
                                          2.645                           
8      25.000   25.280  20.886                                            
                              0.982 20.733                                
                                          2.939                           
9      27.500   27.872  23.476                                            
                              1.080 23.309                                
                                          3.232                           
10     30.000   30.481  26.084                                            
                              1.178 25.902                                
                                          3.526                           
11     32.500   33.111  28.711                                            
                              1.276 28.514                                
                                          3.820                           
12     35.000   35.761  31.359                                            
                              1.374 31.148                                
                                          4.114                           
13     37.500   38.433  34.030                                            
                              1.472 33.804                                
                                          4.408                           
14     40.000   41.129  36.724                                            
                              1.570 36.484                                
                                          4.702                           
15     42.500   43.850  39.443                                            
                              1.669 39.190                                
                                          4.996                           
16     45.000   46.597  42.189                                            
                              1.767 41.921                                
                                          5.290                           
17      47.5000 49.373  44.962                                            
                              1.865 44.681                                
                                          5.584                           
18     50.000   52.176  47.765                                            
                              1.963 47.469                                
                                          5.878                           
19     52.500   55.010  50.597                                            
                              2.061 50.288                                
                                          6.172                           
20     55.000   57.876  53.460                                            
                              2.159 53.138                                
                                          6.466                           
21     57.500   60.773  56.356                                            
                              2.258 56.021                                
                                          6.760                           
22     60.000   63.704  59.286                                            
                              2.356 58.938                                
                                          7.054                           
______________________________________                                    
 
    
     The material used for this embodiment is 5052 H32 aluminum sheet, having thickness, t = 0.050 inches. Holes S 1  and S 2  are located within ± 0.002 inches and S 3  dimension is located within ± 0.002 inches; holes R 1  and R 2  and dimension R 3  are located within ± 0.002 inches for positions 1-8, within ± 0.005 inches for positions 9-12 and within ± 0.010 for positions 13-22. R 1 , S 1  and R 2 , S 2   holes are 0.187/0.189 inch. 
     For the fully overlapping configuration, where the abutting petal edges are essentially rectilinear, the fastener hole positions and sizes must be precisely located to achieve paraboloidal shape. However, if the petal edges are slightly curved outwardly from the longitudinal axis of the petals, precise hole positions would be required only near the center and the outer rim of the assembled reflector (i.e. near the narrowest and widest portions of each petal, respectively) to achieve the same shape. Precise abutting of the curved petal edges rather than precise intermediate hole locations and sizes establishes the paraboloidal shape after assembly. Then, since the intermediate fasteners now merely maintain the transverse bending of the petal in the overlap region, those fasteners could be of smaller diameter or the holes therefor could be larger to facilitate assembly. After installation of fasteners at the narrowest and widest portions of the petals, the assembler simply bends the petal appropriately so that the petals of the inside layer abutt and the intermediate fasteners are inserted. 
     The shape of the slightly curved petal edges are defined by: 
     
         R.sub.3 = Rcos [θ (1+2B)] -A 
    
     
         s.sub.3 = rsin ]θ(1+2B)] 
    
     where A, B, R and θ are defined as above. Again referring to FIG. 7, Table IV gives values for the R 3  and S 3  dimensions corresponding to the R 1 , S 1  and R 2 , S 2  hole positions given in Table III for a petal having curved edges for uses in the reflector configuration of FIGS. 6a and 6b. 
     
                       TABLE IV                                                    
______________________________________                                    
R.sub.3          S.sub.3                                                  
______________________________________                                    
 3.040           1.173                                                    
 5.520           1.564                                                    
 8.007           1.955                                                    
10.502           2.347                                                    
13.007           2.738                                                    
15.524           3.129                                                    
18.055           3.520                                                    
R.sub.3          S.sub.3                                                  
20.600           3.911                                                    
23.163           4.302                                                    
25.743           4.694                                                    
28.343           5.085                                                    
30.964           5.476                                                    
33.607           5.867                                                    
36.275           6.259                                                    
38.968           6.650                                                    
41.687           7.042                                                    
44.435           7.433                                                    
47.211           7.824                                                    
50.018           8.216                                                    
52.857           8.607                                                    
55.728           8.999                                                    
58.633           9.390                                                    
______________________________________                                    
 
    
     Referring again to FIG. 7, Table V gives the same data as Table III for a 40 foot diameter reflector having a total of 80 petals and a focal length, Z o , of 192 inches. 
     
                                           TABLE V                                 
__________________________________________________________________________
    r   R   R.sub.1                                                       
                S.sub.1                                                   
                   R.sub.2                                                
                       S.sub.2                                            
                           R.sub.3                                        
                               S.sub.3                                    
__________________________________________________________________________
1   30.00                                                                 
        30.03                                                             
            2.52                                                          
                0.59                                                      
                   2.48                                                   
                       1.77                                               
                           2.44                                           
                               2.35                                       
2   37.50                                                                 
        37.56                                                             
            10.05                                                         
                0.74                                                      
                   9.99                                                   
                       2.21                                               
                           9.94                                           
                               2.94                                       
3   45.00                                                                 
        45.10                                                             
            17.59                                                         
                0.88                                                      
                   17.52                                                  
                       2.65                                               
                           17.46                                          
                               3.53                                       
4   52.50                                                                 
        52.66                                                             
            25.15                                                         
                1.03                                                      
                   25.07                                                  
                       3.09                                               
                           25.00                                          
                               4.12                                       
5   60.00                                                                 
        60.24                                                             
            32.73                                                         
                1.18                                                      
                   32.64                                                  
                       3.53                                               
                           32.56                                          
                               4.71                                       
6   67.50                                                                 
        67.85                                                             
            40.33                                                         
                1.33                                                      
                   40.23                                                  
                       3.97                                               
                           40.14                                          
                               5.30                                       
7   75.00                                                                 
        75.47                                                             
            47.96                                                         
                1.47                                                      
                   47.84                                                  
                       4.42                                               
                           47.74                                          
                               5.88                                       
8   82.50                                                                 
        83.13                                                             
            55.61                                                         
                1.62                                                      
                   55.49                                                  
                       4.86                                               
                           55.38                                          
                               6.47                                       
9   90.00                                                                 
        90.82                                                             
            63.30                                                         
                1.77                                                      
                   63.16                                                  
                       5.30                                               
                           63.04                                          
                               7.06                                       
10  97.50                                                                 
        98.54                                                             
            71.02                                                         
                1.91                                                      
                   70.87                                                  
                       5.74                                               
                           70.74                                          
                               7.65                                       
    r   R   R.sub.1                                                       
                S.sub.1                                                   
                   R.sub.2                                                
                       S.sub.2                                            
                           R.sub.3                                        
                               S.sub.3                                    
11  105.00                                                                
        106.29                                                            
            78.77                                                         
                2.06                                                      
                   78.61                                                  
                       6.18                                               
                           78.47                                          
                               8.24                                       
12  112.50                                                                
        114.09                                                            
            86.57                                                         
                2.21                                                      
                   86.40                                                  
                       6.62                                               
                           86.25                                          
                               8.83                                       
13  120.00                                                                
        121.93                                                            
            94.40                                                         
                2.36                                                      
                   94.22                                                  
                       7.06                                               
                           94.06                                          
                               9.42                                       
14  127.50                                                                
        129.81                                                            
            102.28                                                        
                2.50                                                      
                   102.09                                                 
                       7.51                                               
                           101.92                                         
                               10.00                                      
15  135.00                                                                
        137.73                                                            
            110.21                                                        
                2.65                                                      
                   110.00                                                 
                       7.95                                               
                           109.82                                         
                               10.59                                      
16  142.50                                                                
        145.71                                                            
            118.18                                                        
                2.80                                                      
                   117.96                                                 
                       8.39                                               
                           117.78                                         
                               11.18                                      
17  150.00                                                                
        153.73                                                            
            126.20                                                        
                2.95                                                      
                   125.98                                                 
                       8.83                                               
                           125.78                                         
                               11.77                                      
18  157.50                                                                
        161.81                                                            
            134.28                                                        
                3.09                                                      
                   134.04                                                 
                       9.27                                               
                           133.84                                         
                               12.36                                      
19  165.00                                                                
        169.95                                                            
            142.41                                                        
                3.24                                                      
                   142.17                                                 
                       9.71                                               
                           141.95                                         
                               12.95                                      
20  172.50                                                                
        178.14                                                            
            150.61                                                        
                3.39                                                      
                   150.35                                                 
                       10.16                                              
                           150.12                                         
                               13.54                                      
21  180.00                                                                
        186.39                                                            
            158.86                                                        
                3.53                                                      
                   158.59                                                 
                       10.60                                              
                           158.35                                         
                               14.12                                      
22  187.50                                                                
        194.70                                                            
            167.17                                                        
                3.68                                                      
                   166.89                                                 
                       11.04                                              
                           166.65                                         
                               14.71                                      
23  195.00                                                                
        203.08                                                            
            175.55                                                        
                3.83                                                      
                   175.26                                                 
                       11.48                                              
                           175.01                                         
                               15.30                                      
24  202.50                                                                
        211.53                                                            
            183.99                                                        
                3.98                                                      
                   183.69                                                 
                       11.92                                              
                           183.43                                         
                               15.89                                      
25  210.00                                                                
        220.04                                                            
            192.50                                                        
                4.12                                                      
                   192.19                                                 
                       12.36                                              
                           191.92                                         
                               16.48                                      
26  217.50                                                                
        228.63                                                            
            201.09                                                        
                4.27                                                      
                   200.77                                                 
                       12.81                                              
                           200.49                                         
                               17.07                                      
27  225.00                                                                
        237.28                                                            
            209.74                                                        
                4.42                                                      
                   209.41                                                 
                       13.25                                              
                           209.12                                         
                               17.66                                      
28  232.50                                                                
        246.01                                                            
            218.47                                                        
                4.56                                                      
                   218.13                                                 
                       13.69                                              
                           217.83                                         
                               18.24                                      
29  240.00                                                                
        254.82                                                            
            227.27                                                        
                4.71                                                      
                   226.93                                                 
                       14.13                                              
                           226.62                                         
                               18.83                                      
__________________________________________________________________________
 
    
     APPENDIX A 
     Referring to FIG. 4a, generally, ##EQU7## 
     APPENDIX B 
     Referring to FIG. 9, a rectangular plate is subjected to pure bending by moments that are uniformly distributed along the edges of the plate. In a plate undergoing such pure bending the magnitude of the maximum stresses is given by: ##EQU8## where t = thickness of the plate. 
     In the particular case where M x  = M Y  = M: ##EQU9## r x  = radius of curvature of the plate in x direction r y  = radius of curvature of the plate in y direction 
     E = elastic modulus 
     ν = Poisson&#39;s ratio 
     A standard approximation in the binding of plates is: ##EQU10## where ω = ω(x, y) is an equation for the deflection of the plate. 
     In the case of a parabola: ##EQU11## and therefore: ##EQU12## where Z o  = focal length 
     
         ∴ r.sub.x = r.sub.y = 2Z.sub.o                     (3) 
    
     Combining equations (1) and (2): ##EQU13## 
     Substituting equation (3) into the above equation: ##EQU14## 
     This equation would apply for the configuration having greatly overlapping petals. For the configuration having fully overlapping petals, two petal sheets effectively act as one. Substituting 2t for t in equation (4): ##EQU15## 
     Appendix C 
     Referring to FIG. 8, for paraboloidal deflection in three dimensions, ##EQU16## where n = number of petals 
     r = radius of reflector 
     Z o  = focal length of reflector