| RIGID BODY TRANSFORMATION | ||||||||||||
| x | y | X | Y | Wi | X' | Y' | ||||||
| A | 1000.00 | 1000.00 | 999.70 | 1000.05 | 100 | 999.2507337 | 999.6407167 | Ao= | -1.3144 | |||
| B | 1000.00 | 1350.00 | 999.90 | 1350.33 | 100 | 999.4485864 | 1349.640661 | Bo= | 0.20617 | |||
| C | 1350.00 | 1350.00 | 1350.22 | 1350.00 | 1 | 1349.44853 | 1349.442808 | Alf= | -0.032388932 | -0.000565294 | ||
| D | 1350.00 | 1000.00 | 1349.90 | 1000.11 | 1 | 1349.250678 | 999.442864 | |||||
| a= | 202 | Wi(Bo-yi) | Wi(xi-Ao) | Wi(Bo-yi)^2 | Wi(xi-Ao)^2 | Wi Vx | Wi Vy | |||||
| b= | -237235.3771 | -99943.45467 | 100056.5134 | 99886941.32 | 100113058.7 | 44.92663029 | 40.92832538 | |||||
| c= | 202834.14 | -134943.4491 | 100076.2986 | 182097344.5 | 100152655.5 | 45.14135865 | 68.93391762 | |||||
| d= | 488717500 | -1349.236638 | 1350.76293 | 1820439.506 | 1824560.494 | 0.771469509 | 0.557191893 | |||||
| f= | 1298622500 | -999.236694 | 1350.565078 | 998473.9707 | 1824026.029 | 0.649322225 | 0.66713597 | |||||
| g= | 91.48878068 | Wi (Bo-yi)*Vx | Wi (xi-Ao)*Vy | |||||||||
| h= | 111.0865709 | -44901.22638 | 40951.45536 | |||||||||
| m= | 4085.359084 | -60915.30633 | 68986.51327 | |||||||||
| -1040.894927 | 752.6341537 | |||||||||||
| Ao= | -0.863774335 | -648.8265938 | 901.0105435 | |||||||||
| Bo= | 0.758060671 | |||||||||||
| Alf= | -0.000111675 | |||||||||||
| Matrix A | Matrix At | |||||||||||
| 1 | 0 | -999.84383 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | ||
| 0 | 1 | 1001.0144 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | ||
| 1 | 0 | -1350.1238 | -999.84383 | 1001.0144 | -1350.12383 | 1001.2144 | -1349.79383 | 1351.5344 | -999.90383 | 1351.2144 | ||
| 0 | 1 | 1001.2144 | ||||||||||
| 1 | 0 | -1349.7938 | Matrix N | Matrix L | ||||||||
| 0 | 1 | 1351.5344 | 4 | 0 | -4699.66532 | 44.92663029 | ||||||
| 1 | 0 | -999.90383 | 0 | 4 | 4704.981745 | 40.92832538 | ||||||
| 0 | 1 | 1351.2144 | -4699.66532 | 4704.981745 | 11301169.28 | 45.14135865 | ||||||
| 68.93391762 | ||||||||||||
| Matrix N-1 | 0.771469509 | |||||||||||
| 5.878831476 | -5.635199006 | 0.004790836 | 0.557191893 | |||||||||
| -5.635199006 | 5.891573739 | -0.004796256 | 0.649322225 | |||||||||
| 0.004790836 | -0.004796256 | 4.0776E-06 | 0.66713597 | |||||||||
| Matrix U | Matrix X | |||||||||||
| -2.325616483 | -0.581404121 | |||||||||||
| -2.323 | -0.5807 | |||||||||||
| 0.0451 | 7.8E-17 | |||||||||||
| EXAMPLES OF 4 PARAMETER TRANSFORMATION | |||||||||
| 1. TWO POINTS | |||||||||
| 2. WITH OVER DETERMINATION | |||||||||
| Transformations (2 points, no weight) | (similarity 4 parameters) | ||||||||
| x | y | X | Y | ||||||
| 1 | 100.69 | 166.39 | 670305 | 224173 | |||||
| 2 | 34.90 | 56.07 | 670764 | 223898 | |||||
| xb | yb | Xb | Yb | xbXb | ybYb | xbYb | ybXb | xb2+yb2 | |
| -65.79 | -110.32 | 459 | -275 | -30197.61 | 30338.00 | 18092.25 | -50636.88 | 16498.83 | |
| a1 = | 0.00850909 | ||||||||
| b1 = | 4.16569809 | ||||||||
| a0 = | 670997.274 | ||||||||
| b0 = | 223752.14 | ||||||||
| Example 1 | X=a0+a1*x -b1*y | 670305 | 670764 | ||||||
| Y=b0+b1*x+a1*y | 224173 | 223898 | |||||||
| Same Problem with 3 points, no weight. | (similarity 4 parameters) | ||||||||
| x | y | X | Y | 5 | Points | ||||
| 1 | 100.69 | 166.39 | 670305 | 224173 | |||||
| 2 | 34.90 | 56.07 | 670764 | 223898 | |||||
| 3 | 171.36 | 58.87 | 670741 | 224444 | |||||
| 4 | 100.69 | 166.39 | 670305 | 224173 | |||||
| 5 | 100.69 | 166.39 | 670305 | 224173 | |||||
| SUM | 508.33 | 614.11 | 3352420 | 1120861 | |||||
| xs | ys | Xs | Ys | ||||||
| AVR | 101.666 | 122.822 | 670484 | 224172.2 | |||||
| xi | yi | Xi | Yi | xiXi | yiYi | xiYi | yiXi | xi2+yi2 | |
| -0.98 | 43.57 | -179.00 | 0.80 | 174.70 | 34.85 | -0.78 | -7798.67 | 1899.12 | |
| -66.77 | -66.75 | 280.00 | -274.20 | -18694.48 | 18303.40 | 18307.24 | -18690.56 | 8913.53 | |
| 69.69 | -63.95 | 257.00 | 271.80 | 17911.36 | -17382.15 | 18942.83 | -16435.66 | 8947.11 | |
| -0.98 | 43.57 | -179.00 | 0.80 | 174.70 | 34.85 | -0.78 | -7798.67 | 1899.12 | |
| -0.98 | 43.57 | -179.00 | 0.80 | 174.70 | 34.85 | -0.78 | -7798.67 | 1899.12 | |
| SUM | 0 | 0 | 0 | 0 | -259.01 | 1025.81 | 37247.72 | -58522.24 | 23558.01 |
| a1 = | 0.03254935 | ||||||||
| b1 = | 4.06528246 | ||||||||
| a0 = | 670979.997 | ||||||||
| b0 = | 223754.901 | ||||||||
| Accuracy Assessment | |||||||||
| X=a0+a1*x-b1*y | Y=b0+b1*x+a1*y | Vx=Xcom-Xgiv | Vy=Ycom-Ygiv | ||||||
| 670306.85 | 224169.65 | 1.85 | 3.43 | -3.35 | 11.22 | ||||
| 670753.19 | 223898.60 | -10.81 | 116.80 | 0.60 | 0.37 | ||||
| 670746.25 | 224453.44 | 5.25 | 27.58 | 9.44 | 89.19 | ||||
| SUM | -3.70 | 6.70 | |||||||
| SUM^2 | 147.81 | 100.78 | |||||||
| SDxy | 11.15 | ||||||||
| SDp | 15.77 |
| WEIGHTED TRANSFORMATION FROM NAD27 TO NAD83 (6 parameters) | ||||||
| x | y | X (E) | Y (N) | w | ||
| NEWARK 1 | 2125650.302 | 696339.488 | 174184.552 | 212161.942 | 1 | |
| EAST ORANGE 2 | 2127697.804 | 700359.394 | 174810.878 | 213385.955 | 6 | |
| CLINTON | 2128303.873 | 707672.831 | 174999.825 | 215614.614 | 5 | |
| NEWARK 89 1 | 2130907.473 | 694056.613 | 175785.386 | 211463.099 | 2 | |
| NEWARK 89196 | 2134177.740 | 703332.456 | 176787.409 | 214288.325 | 3 | |
| SUM | 10646737.192 | 3501760.782 | 876568.050 | 1066913.935 | 17 | |
| wx | wy | wX | wY | |||
| NEWARK 1 | 2125650.302 | 696339.488 | 174184.552 | 212161.942 | ||
| EAST ORANGE 2 | 12766186.824 | 4202156.364 | 1048865.268 | 1280315.730 | ||
| CLINTON | 10641519.365 | 3538364.155 | 874999.125 | 1078073.070 | ||
| NEWARK 89 1 | 4261814.946 | 1388113.226 | 351570.772 | 422926.198 | ||
| NEWARK 89196 | 6402533.220 | 2109997.368 | 530362.227 | 642864.975 | ||
| SUM | 36197704.657 | 11934970.601 | 2979981.944 | 3636341.915 | ||
| SUBs | 2129276.745 | 702057.094 | 175293.056 | 213902.466 | ||
| xi | yi | Xi | Yi | wi xiXi (I) | wi yiYi (II) | |
| -3626.443 | -5717.606 | -1108.504 | -1740.524 | 4019924.343 | 9951628.418 | |
| -1578.941 | -1697.700 | -482.178 | -516.511 | 4567977.861 | 5261280.701 | |
| -972.872 | 5615.737 | -293.231 | 1712.148 | 1426378.168 | 48074874.416 | |
| 1630.728 | -8000.481 | 492.330 | -2439.367 | 1605714.631 | 39032212.943 | |
| 4900.995 | 1275.362 | 1494.353 | 385.859 | 21971458.772 | 1476331.089 | |
| SUM | 353.469 | -8524.689 | 102.772 | -2598.393 | 33591453.776 | 103796327.568 |
| wi xiYi (III) | wi yiXi (IV) | wi xi2 (V) | wi yi2 (VI) | wi xiyi (VII) | ||
| 6311908.764 | 6337986.626 | 13151085.419 | 32691020.389 | 20734570.205 | ||
| 4893237.010 | 4911557.261 | 14958319.173 | 17293115.335 | 16083405.693 | ||
| -8328502.220 | -8233527.409 | 4732395.064 | 157682500.356 | -27316952.361 | ||
| -7955889.091 | -7877761.325 | 5318550.690 | 128015398.110 | -26093224.866 | ||
| 5673285.688 | 5717524.102 | 72059269.808 | 4879643.343 | 18751627.561 | ||
| SUM | 594040.151 | 855779.255 | 110219620.153 | 340561677.533 | 2159426.232 | |
| Transformation Parameters | ||||||
| a1 = | 0.305 | a2 = | 0.001 | a0 = | -474026.438 | |
| b1 = | -0.001 | b2 = | 0.305 | b0 = | 1165.690 | |
| CECKING CLOSURES | Vx | Vy | ||||
| X=a0+a1*x+a2*y | Y=b0+b1*x+b2*y | Xcmp-Xgiv | Ycmp-Ygiv | |||
| 174184.553 | 212161.943 | 0.001 | 0.001 | |||
| 174810.877 | 213385.953 | -0.001 | -0.002 | |||
| 174999.826 | 215614.616 | 0.001 | 0.002 | |||
| 175785.388 | 211463.102 | 0.002 | 0.003 | |||
| 176787.408 | 214288.324 | -0.001 | -0.001 | |||
| SUM | 0.002 | 0.003 | ||||
| SUM^2 | 0.000 | 0.000 | ||||
| COMPUTATION OF ADDITIONAL POINTS | ||||||
| STATION | X | Y | E(comp) | N(comp) | ||
| NEWARK 2 | 2125955.479 | 698583.410 | 174278.860 | 212845.676 | ||
| Given -> | 174278.862 | 212845.678 | ||||
| EAST ORANGE 1 | 2126708.815 | 700610.322 | 174509.621 | 213463.007 | ||
| Given -> | 174509.622 | 213463.009 |
| 0.068 |
| 0.007 |