# determine_region: Determine confidence region for position In triangulation: Determine Position of Observer

## Description

This function is similar to `determine_position()`except for the fact that it is assumed that the angles are subject to measurement error. Hence a confidence region (error 'ellipse') is returned instead of an exact position.

## Usage

 ```1 2 3 4``` ```determine_region(A, B, C, observer_angle_AB, observer_angle_AC, angle_error = pi/24, number_of_points = 200, output_plot = TRUE, lines_in_plot = FALSE, coordinates_in_plot = FALSE, decimals_in_plot = 2) ```

## Arguments

 `A` A point defined by a vector containing an x- and an y-coordinate `B` A point defined by a vector containing an x- and an y-coordinate `C` A point defined by a vector containing an x- and an y-coordinate `observer_angle_AB` An angle (numeric) expressed in radians `observer_angle_AC` An angle (numeric) expressed in radians `angle_error` A numeric indicating the measurement error in radians `number_of_points` A numeric indicating the number of error points tested `output_plot` Boolean variable indicating whether a plot should be created `lines_in_plot` Boolean variable indicating whether lines should be drawn in the plot `coordinates_in_plot` Boolean variable indicating whether the coordinates should be printet in the plot `decimals_in_plot` Integer indicating the number of decimals used

## Value

Coordinates indicating the outer border of the confidence region. Note that several different regions may exist.

## Examples

 ```1 2 3 4 5``` ```determine_region(A = c(0, 0), B = c(10, 0), C = c(5, 5 * 3^0.5), observer_angle_AB = pi * 2/3, observer_angle_AC = pi * 1/2) determine_region(A = c(0, 0), B = c(10, 0), C = c(5, 5), observer_angle_AB = pi * 5/6, observer_angle_AC = pi * 1/2, lines_in_plot = FALSE) ```

### Example output

```                 [,1]     [,2]     [,3]    [,4]     [,5]     [,6]     [,7]
x-coordinate 7.155201 7.106347 7.069011 7.03771 7.010309 6.985710 6.963259
y-coordinate 2.884320 2.896283 2.902917 2.90677 2.908835 2.909622 2.909437
[,8]     [,9]    [,10]    [,11]    [,12]    [,13]    [,14]
x-coordinate 6.942528 6.923219 6.905115 6.888051 6.871899 6.856558 6.841944
y-coordinate 2.908479 2.906889 2.904768 2.902193 2.899224 2.895909 2.892286
[,15]    [,16]    [,17]    [,18]    [,19]    [,20]    [,21]
x-coordinate 6.82799 6.814638 6.801841 6.789558 6.777752 6.766394 6.755456
y-coordinate 2.88839 2.884245 2.879875 2.875300 2.870536 2.865598 2.860499
[,22]    [,23]    [,24]    [,25]    [,26]    [,27]    [,28]
x-coordinate 6.744916 6.734752 6.724946 6.715481 6.706344 6.697520 6.688998
y-coordinate 2.855250 2.849861 2.844341 2.838699 2.832941 2.827073 2.821102
[,29]    [,30]    [,31]    [,32]    [,33]    [,34]    [,35]
x-coordinate 6.680768 6.672820 6.665145 6.657736 6.650586 6.643688 6.637036
y-coordinate 2.815033 2.808871 2.802621 2.796285 2.789868 2.783373 2.776804
[,36]    [,37]    [,38]    [,39]    [,40]   [,41]    [,42]
x-coordinate 6.630626 6.624454 6.618514 6.612804 6.607320 6.60206 6.597020
y-coordinate 2.770163 2.763452 2.756674 2.749832 2.742927 2.73596 2.728934
[,43]    [,44]    [,45]    [,46]    [,47]    [,48]    [,49]
x-coordinate 6.592200 6.587596 6.583209 6.579037 6.575078 6.571333 6.567802
y-coordinate 2.721851 2.714710 2.707515 2.700265 2.692962 2.685607 2.678199
[,50]    [,51]    [,52]    [,53]    [,54]    [,55]    [,56]
x-coordinate 6.564484 6.561380 6.558491 6.555818 6.553361 6.551124 6.549106
y-coordinate 2.670741 2.663232 2.655674 2.648065 2.640407 2.632699 2.624942
[,57]    [,58]    [,59]    [,60]    [,61]    [,62]    [,63]
x-coordinate 6.547312 6.545743 6.544402 6.543292 6.542418 6.541782 6.541391
y-coordinate 2.617135 2.609278 2.601371 2.593413 2.585404 2.577344 2.569230
[,64]    [,65]    [,66]    [,67]    [,68]    [,69]    [,70]
x-coordinate 6.541248 6.541358 6.541729 6.542366 6.543276 6.544468 6.545949
y-coordinate 2.561062 2.552840 2.544562 2.536225 2.527829 2.519372 2.510852
[,71]    [,72]    [,73]    [,74]    [,75]    [,76]    [,77]
x-coordinate 6.547730 6.549820 6.552231 6.554975 6.558065 6.561518 6.565351
y-coordinate 2.502266 2.493611 2.484885 2.476084 2.467205 2.458242 2.449193
[,78]    [,79]    [,80]    [,81]    [,82]    [,83]    [,84]
x-coordinate 6.569580 6.574229 6.579319 6.584879 6.590937 6.597528 6.604692
y-coordinate 2.440051 2.430810 2.421464 2.412005 2.402424 2.392710 2.382853
[,85]    [,86]    [,87]    [,88]    [,89]    [,90]    [,91]
x-coordinate 6.612473 6.620926 6.630112 6.640107 6.651000 6.662904 6.675958
y-coordinate 2.372839 2.362650 2.352269 2.341672 2.330831 2.319711 2.308269
[,92]    [,93]    [,94]    [,95]    [,96]    [,97]    [,98]
x-coordinate 6.690340 6.706286 6.724118 6.744297 6.767530 6.794997 6.828988
y-coordinate 2.296447 2.284171 2.271338 2.257798 2.243323 2.227526 2.209658
[,99]   [,100]   [,101]   [,102]   [,103]   [,104]   [,105]
x-coordinate 6.875383 6.995525 7.272933 7.387984 7.434089 7.468610 7.497030
y-coordinate 2.187776 2.141462 2.840459 2.776777 2.745131 2.718951 2.695691
[,106]   [,107]   [,108]   [,109]   [,110]   [,111]   [,112]
x-coordinate 7.521484 7.543070 7.562444 7.580033 7.596136 7.610970 7.624702
y-coordinate 2.674356 2.654435 2.635623 2.617718 2.600582 2.584111 2.568228
[,113]   [,114]   [,115]   [,116]   [,117]   [,118]   [,119]
x-coordinate 7.637457 7.649340 7.660433 7.670805 7.680512 7.689604 7.698122
y-coordinate 2.552872 2.537995 2.523558 2.509527 2.495874 2.482578 2.469617
[,120]   [,121]   [,122]   [,123]   [,124]   [,125]   [,126]
x-coordinate 7.706100 7.713571 7.720562 7.727095 7.733193 7.738874 7.744154
y-coordinate 2.456973 2.444633 2.432582 2.420808 2.409300 2.398050 2.387048
[,127]   [,128]   [,129]   [,130]   [,131]   [,132]   [,133]
x-coordinate 7.749049 7.753572 7.757734 7.761547 7.765021 7.768164 7.770984
y-coordinate 2.376287 2.365759 2.355458 2.345378 2.335514 2.325859 2.316411
[,134]   [,135]   [,136]   [,137]   [,138]   [,139]   [,140]
x-coordinate 7.773488 7.775684 7.777576 7.779170 7.780471 7.781483 7.782210
y-coordinate 2.307164 2.298114 2.289258 2.280592 2.272113 2.263817 2.255703
[,141]   [,142]   [,143]   [,144]   [,145]   [,146]   [,147]
x-coordinate 7.782656 7.782822 7.782713 7.782329 7.781673 7.780747 7.779551
y-coordinate 2.247766 2.240005 2.232418 2.225001 2.217754 2.210674 2.203759
[,148]   [,149]   [,150]   [,151]   [,152]   [,153]   [,154]
x-coordinate 7.778087 7.776356 7.774356 7.772090 7.769555 7.766752 7.763680
y-coordinate 2.197007 2.190417 2.183988 2.177718 2.171606 2.165651 2.159851
[,155]   [,156]   [,157]   [,158]   [,159]   [,160]   [,161]
x-coordinate 7.760337 7.756723 7.752835 7.748671 7.744229 7.739506 7.734499
y-coordinate 2.154206 2.148714 2.143376 2.138190 2.133156 2.128274 2.123542
[,162]   [,163]   [,164]   [,165]   [,166]   [,167]   [,168]
x-coordinate 7.729204 7.723618 7.717735 7.711552 7.705063 7.698262 7.691143
y-coordinate 2.118962 2.114533 2.110254 2.106127 2.102152 2.098329 2.094658
[,169]   [,170]   [,171]   [,172]   [,173]   [,174]   [,175]
x-coordinate 7.683699 7.675921 7.667802 7.659332 7.650502 7.641299 7.631712
y-coordinate 2.091141 2.087780 2.084574 2.081526 2.078638 2.075912 2.073351
[,176]   [,177]   [,178]   [,179]   [,180]   [,181]   [,182]
x-coordinate 7.621726 7.611328 7.600499 7.589222 7.577475 7.565236 7.552477
y-coordinate 2.070957 2.068733 2.066684 2.064815 2.063130 2.061635 2.060338
[,183]   [,184]   [,185]   [,186]   [,187]   [,188]   [,189]
x-coordinate 7.539168 7.525276 7.510761 7.495578 7.479674 7.462986 7.445439
y-coordinate 2.059246 2.058369 2.057717 2.057304 2.057145 2.057259 2.057667
[,190]   [,191]   [,192]   [,193]   [,194]   [,195]   [,196]
x-coordinate 7.426943 7.407388 7.386632 7.364499 7.340752 7.315070 7.286991
y-coordinate 2.058397 2.059483 2.060967 2.062907 2.065376 2.068476 2.072356
[,197]   [,198]   [,199]   [,200]
x-coordinate 7.255809 7.220343 7.178304 7.123813
y-coordinate 2.077244 2.083524 2.091947 2.104455
[,1]       [,2]     [,3]      [,4]     [,5]       [,6]
x-coordinate 5.850080  1.1092739 5.816477  1.023822 5.790651  0.9661971
y-coordinate 1.646017 -0.7137807 1.642283 -0.664085 1.637652 -0.6290565
[,7]       [,8]     [,9]      [,10]    [,11]      [,12]
x-coordinate 5.768883  0.9231828 5.749729  0.8896301 5.732447  0.8628860
y-coordinate 1.632554 -0.6019497 1.627156 -0.5800905 1.621544 -0.5620759
[,13]      [,14]    [,15]      [,16]    [,17]      [,18]
x-coordinate 5.716594  0.8413676 5.701882  0.8240397 5.688112  0.8101810
y-coordinate 1.615770 -0.5470533 1.609867 -0.5344544 1.603860 -0.5238753
[,19]      [,20]    [,21]      [,22]    [,23]      [,24]
x-coordinate 5.675136  0.7992643 5.662845  0.7908909 5.651154  0.7847503
y-coordinate 1.597764 -0.5150154 1.591595 -0.5076421 1.585361 -0.5015708
[,25]      [,26]    [,27]      [,28]    [,29]      [,30]
x-coordinate 5.639993  0.7805948 5.629308  0.7782233 5.619055  0.7774699
y-coordinate 1.579071 -0.4966513 1.572732 -0.4927589 1.566349 -0.4897885
[,31]      [,32]    [,33]      [,34]    [,35]      [,36]
x-coordinate 5.609194  0.7781955 5.599694  0.7802824 5.590529  0.7836299
y-coordinate 1.559927 -0.4876505 1.553469 -0.4862671 1.546980 -0.4855705
[,37]      [,38]    [,39]      [,40]    [,41]      [,42]
x-coordinate 5.581674  0.7881508 5.573110  0.7937693 5.564819  0.8004190
y-coordinate 1.540462 -0.4855010 1.533918 -0.4860054 1.527349 -0.4870361
[,43]      [,44]    [,45]      [,46]    [,47]      [,48]
x-coordinate 5.556786  0.8080409 5.548998  0.8165831 5.541442  0.8259988
y-coordinate 1.520759 -0.4885502 1.514148 -0.4905090 1.507518 -0.4928771
[,49]      [,50]    [,51]      [,52]    [,53]      [,54]
x-coordinate 5.534108  0.8362461 5.526986  0.8472875 5.520069  0.8590886
y-coordinate 1.500871 -0.4956223 1.494207 -0.4987149 1.487529 -0.5021274
[,55]      [,56]    [,57]      [,58]    [,59]      [,60]
x-coordinate 5.513349  0.8716186 5.506818  0.8848491 5.500472  0.8987543
y-coordinate 1.480837 -0.5058348 1.474132 -0.5098134 1.467416 -0.5140415
[,61]      [,62]    [,63]      [,64]    [,65]      [,66]
x-coordinate 5.494305  0.9133104 5.488312  0.9284957 5.482489  0.9442900
y-coordinate 1.460688 -0.5184987 1.453949 -0.5231661 1.447201 -0.5280258
[,67]      [,68]    [,69]      [,70]    [,71]      [,72]
x-coordinate 5.476832  0.9606748 5.471338  0.9776330 5.466004  0.9951487
y-coordinate 1.440443 -0.5330611 1.433677 -0.5382562 1.426903 -0.5435964
[,73]      [,74]    [,75]     [,76]    [,77]      [,78]
x-coordinate 5.460827  1.0132071 5.455804  1.031795 5.450935  1.0508984
y-coordinate 1.420122 -0.5490677 1.413333 -0.554657 1.406538 -0.5603517
[,79]      [,80]    [,81]      [,82]    [,83]     [,84]
x-coordinate 5.446217  1.0705067 5.441648  1.0906086 5.437229  1.111194
y-coordinate 1.399737 -0.5661401 1.392930 -0.5720111 1.386117 -0.577954
[,85]      [,86]    [,87]      [,88]    [,89]      [,90]
x-coordinate 5.432957  1.1322529 5.428832  1.1537771 5.424854  1.1757582
y-coordinate 1.379299 -0.5839589 1.372476 -0.5900161 1.365649 -0.5961166
[,91]      [,92]    [,93]      [,94]    [,95]      [,96]
x-coordinate 5.421023  1.1981888 5.417338  1.2210619 5.413800  1.2443712
y-coordinate 1.358818 -0.6022519 1.351982 -0.6084139 1.345143 -0.6145946
[,97]      [,98]    [,99]     [,100]   [,101]    [,102]
x-coordinate 5.410409  1.2681109 5.407165  1.2922758 5.404071  1.316861
y-coordinate 1.338300 -0.6207868 1.331453 -0.6269835 1.324604 -0.633178
[,103]    [,104]   [,105]     [,106]   [,107]     [,108]
x-coordinate 5.401126  1.341863 5.398332  1.3672766 5.395690  1.3931001
y-coordinate 1.317751 -0.639364 1.310895 -0.6455355 1.304036 -0.6516868
[,109]     [,110]   [,111]     [,112]   [,113]     [,114]
x-coordinate 5.393203  1.4193304 5.390871  1.4459654 5.388698  1.4730036
y-coordinate 1.297175 -0.6578125 1.290311 -0.6639076 1.283445 -0.6699671
[,115]     [,116]   [,117]     [,118]   [,119]     [,120]
x-coordinate 5.386685  1.5004439 5.384836  1.5282861 5.383153  1.5565301
y-coordinate 1.276576 -0.6759865 1.269705 -0.6819616 1.262831 -0.6878882
[,121]     [,122]   [,123]     [,124]   [,125]     [,126]
x-coordinate 5.381639  1.5851770 5.380298  1.6142281 5.379134  1.6436856
y-coordinate 1.255955 -0.6937626 1.249077 -0.6995812 1.242196 -0.7053407
[,127]    [,128]   [,129]     [,130]   [,131]     [,132]
x-coordinate 5.378151  1.673552 5.377354  1.7038323 5.376747  1.7345297
y-coordinate 1.235313 -0.711038 1.228428 -0.7166703 1.221540 -0.7222349
[,133]     [,134]   [,135]    [,136]   [,137]     [,138]
x-coordinate 5.376336  1.7656501 5.376127  1.797200 5.376125  1.8291867
y-coordinate 1.214649 -0.7277295 1.207756 -0.733152 1.200861 -0.7385004
[,139]     [,140]   [,141]     [,142]   [,143]     [,144]
x-coordinate 5.376339  1.8616193 5.376776  1.8945076 5.377443  1.9278632
y-coordinate 1.193962 -0.7437731 1.187060 -0.7489686 1.180155 -0.7540859
[,145]     [,146]   [,147]     [,148]   [,149]     [,150]
x-coordinate 5.378350  1.9616991 5.379507  1.9960303 5.380925  2.0308737
y-coordinate 1.173246 -0.7591241 1.166334 -0.7640826 1.159417 -0.7689611
[,151]     [,152]   [,153]     [,154]   [,155]     [,156]
x-coordinate 5.382616  2.0662487 5.384593  2.1021772 5.386871  2.1386841
y-coordinate 1.152495 -0.7737596 1.145569 -0.7784787 1.138637 -0.7831191
[,157]     [,158]   [,159]     [,160]   [,161]     [,162]
x-coordinate 5.389466  2.1757979 5.392397  2.2135511 5.395684  2.2519811
y-coordinate 1.131699 -0.7876819 1.124754 -0.7921691 1.117801 -0.7965828
[,163]    [,164]   [,165]     [,166]   [,167]     [,168]
x-coordinate 5.399352  2.291131 5.403425  2.3310498 5.407935  2.3717962
y-coordinate 1.110840 -0.800926 1.103869 -0.8052025 1.096888 -0.8094167
[,169]     [,170]   [,171]     [,172]   [,173]     [,174]
x-coordinate 5.412918  2.4134377 5.418413  2.4560544 5.424470  2.4997417
y-coordinate 1.089894 -0.8135744 1.082886 -0.8176826 1.075862 -0.8217499
[,175]     [,176]   [,177]     [,178]   [,179]     [,180]
x-coordinate 5.431146  2.5446148 5.438510  2.5908140 5.446647  2.6385137
y-coordinate 1.068820 -0.8257872 1.061756 -0.8298079 1.054666 -0.8338292
[,181]     [,182]   [,183]    [,184]   [,185]     [,186]
x-coordinate 5.455664  2.6879338 5.465698  2.739359 5.476927  2.7931655
y-coordinate 1.047546 -0.8378732 1.040389 -0.841969 1.033186 -0.8461557
[,187]     [,188]   [,189]     [,190]   [,191]     [,192]
x-coordinate 5.489598  2.8498745 5.504062  2.9102349 5.520853  2.9753983
y-coordinate 1.025927 -0.8504885 1.018593 -0.8550477 1.011158 -0.8599591
[,193]     [,194]    [,195]     [,196]    [,197]     [,198]
x-coordinate 5.540868  3.0473042 5.5658419  3.1297344 5.6002237  3.2324486
y-coordinate 1.003578 -0.8654374 0.9957635 -0.8719084 0.9874862 -0.8804902
[,199]     [,200]   [,201]     [,202]   [,203]     [,204]
x-coordinate 5.6903559  3.4580469 5.930444  1.3604010 6.008608  1.6669053
y-coordinate 0.9763107 -0.9032552 1.644399 -0.8468364 1.628025 -0.9859063
[,205]    [,206]   [,207]    [,208]   [,209]    [,210]   [,211]
x-coordinate 6.039955  1.807252 6.063468  1.919301 6.082865  2.016282 6.099592
y-coordinate 1.617038 -1.041525 1.606979 -1.082388 1.597421 -1.115244 1.588196
[,212]   [,213]    [,214]   [,215]    [,216]   [,217]    [,218]
x-coordinate  2.103355 6.114392  2.183185 6.127705  2.257366 6.139821  2.326945
y-coordinate -1.142769 1.579219 -1.166367 1.570436 -1.186893 1.561815 -1.204917
[,219]    [,220]   [,221]    [,222]   [,223]    [,224]   [,225]
x-coordinate 6.150940  2.392652 6.161208  2.455025 6.170735  2.514473 6.179608
y-coordinate 1.553332 -1.220843 1.544969 -1.234973 1.536712 -1.247541 1.528552
[,226]   [,227]    [,228]   [,229]    [,230]   [,231]    [,232]
x-coordinate  2.571316 6.187893  2.625811 6.195647  2.678168 6.202915  2.728562
y-coordinate -1.258731 1.520480 -1.268694 1.512490 -1.277555 1.504575 -1.285419
[,233]    [,234]   [,235]    [,236]   [,237]    [,238]   [,239]
x-coordinate 6.209734  2.777139 6.216137  2.824026 6.222151  2.869332 6.227799
y-coordinate 1.496732 -1.292376 1.488955 -1.298503 1.481241 -1.303867 1.473588
[,240]   [,241]    [,242]   [,243]    [,244]   [,245]    [,246]
x-coordinate  2.913150 6.233101  2.955565 6.238076  2.996649 6.242738  3.036469
y-coordinate -1.308527 1.465993 -1.312537 1.458452 -1.315943 1.450965 -1.318786
[,247]    [,248]   [,249]    [,250]   [,251]    [,252]   [,253]
x-coordinate 6.247101  3.075083 6.251177  3.112543 6.254977  3.148899 6.258510
y-coordinate 1.443529 -1.321106 1.436142 -1.322935 1.428803 -1.324306 1.421511
[,254]   [,255]    [,256]   [,257]   [,258]   [,259]    [,260]
x-coordinate  3.184192 6.261785  3.218464 6.264809  3.25175 6.267590  3.284084
y-coordinate -1.325247 1.414263 -1.325783 1.407060 -1.32594 1.399899 -1.325739
[,261]    [,262]   [,263]    [,264]   [,265]    [,266]   [,267]
x-coordinate 6.270133  3.315497 6.272443  3.346017 6.274526  3.375670 6.276386
y-coordinate 1.392780 -1.325200 1.385701 -1.324343 1.378662 -1.323185 1.371662
[,268]   [,269]    [,270]   [,271]    [,272]   [,273]    [,274]
x-coordinate  3.404481 6.278026  3.432473 6.279451  3.459667 6.280664  3.486083
y-coordinate -1.321743 1.364701 -1.320032 1.357776 -1.318066 1.350888 -1.315858
[,275]    [,276]   [,277]    [,278]   [,279]    [,280]   [,281]
x-coordinate 6.281667  3.511738 6.282462  3.536651 6.283052  3.560836 6.283439
y-coordinate 1.344036 -1.313421 1.337219 -1.310767 1.330437 -1.307906 1.323689
[,282]   [,283]    [,284]   [,285]    [,286]   [,287]    [,288]
x-coordinate  3.584310 6.283624  3.607085 6.283608  3.629176 6.283393  3.650594
y-coordinate -1.304848 1.316974 -1.301605 1.310292 -1.298184 1.303642 -1.294594
[,289]    [,290]   [,291]    [,292]   [,293]    [,294]   [,295]
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y-coordinate -1.278700 1.270862 -1.274378 1.264396 -1.269929 1.257959 -1.265359
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y-coordinate 1.207471 -1.225021 1.201281 -1.219568 1.195116 -1.214036 1.188977
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y-coordinate -1.208427 1.182863 -1.202743 1.176774 -1.196989 1.170709 -1.191164
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y-coordinate 1.164670 -1.185273 1.158655 -1.179316 1.152663 -1.173296 1.146697
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y-coordinate -1.167213 1.140753 -1.161069 1.134834 -1.154865 1.128939 -1.148603
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y-coordinate 1.123067 -1.142282 1.117219 -1.135903 1.111395 -1.129467 1.105594
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y-coordinate 1.082634 -1.096405 1.076955 -1.089610 1.071302 -1.082748 1.065675
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y-coordinate -1.075819 1.060076 -1.068816 1.054504 -1.061737 1.048961 -1.054575
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y-coordinate 1.043449 -1.047324 1.037969 -1.039974 1.032523 -1.032516 1.027115
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x-coordinate  3.955809 6.003594  3.942532 5.987813  3.927368 5.970973  3.910034
y-coordinate -1.024935 1.021748 -1.017215 1.016426 -1.009335 1.011156 -1.001265
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x-coordinate 5.952894  3.8901503 5.933332  3.867198 5.9119359  3.8404269
y-coordinate 1.005946 -0.9929655 1.000807 -0.984383 0.9957558 -0.9754371
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x-coordinate 5.8881700  3.8086856 5.8611388  3.7700342 5.8291075  3.7206906
y-coordinate 0.9908192 -0.9660024 0.9860416 -0.9558645 0.9815107 -0.9445981
[,399]     [,400]
x-coordinate 5.7876274  3.6508949
y-coordinate 0.9774555 -0.9310857
```

triangulation documentation built on May 2, 2019, 5:34 a.m.