Photogrammetric Survey

After a more or less schematic analysis, we now continue with precise measurements in the nadir image PSP_003234_2210 from Mars Reconnaissance Orbiter taken on April 5, 2007, with its HiRISE camera. The image has a resolution of 3 meters per pixel and is 2048 pixel wide and 1532 pixel high, which is the range for pixel coordinates from top left to right and from top to bottom (raw image 20.480 x 15320, 30cm per pixel). On the image southern outer arc, southern inner arc and northern outer arc can be clearly identified as landform marks.

2210

The examination of the arc’s parameters can be done with usual photo editing software. After a download of the image from the internet image catalog page, single pixels are turned to white to indicate the positions of virtual survey stakes in the edges of the slopes. We take the coordinates of the virtual survey stakes and try to determine a circle of best fit to their positions with a Least Square Method. A suitable algorithm for circles of best fit is found in a javascript program of the University of Bayreuth, which uses a linear regression algorithm of  Ian D. Coope, University of Canterbury in New Zealand, published in 1993 in Journal of Optimization Theory and Applications. This program needs a set of points and gives back center and radius for the circle of best fit.

These are the coordinates of the virtual survey stakes in the three curved edges:

Soutern Inner Arc

X Y
959
478
992
458
1041
439
1087
437
1134
451
1142
457
1050
460
1188
447
1196
482
1203
485
1214
494
1221
501
1260
532
1285
570

Northern Outer Arc

X Y
1530
999
1524
1012
1516
1025
1512
1035
1502
1048
1494
1060
1487
1069
1479
1079
1472
1090
1461
1103
1450
1112
1440
1120
1432
1130
1421
1140
1412
1151
1403
1160
1393
1169
1383
1180
1372
1189
1357
1197
1342
1203

Soutern Outer Arc

X Y
767
541
771
528
777
516
786
506
794
498
856
432
868
432
878
412
890
407
898
401
911
393
924
383
935
376
946
368
957
361
969
355
982
350
994
346
1012
342
1028
342
1040
341
1055
340
1067
339
1081
339
1094
339
1117
342
1133
348

Northern Outer Arc

Radius: 461

Xcenter: 1110

Ycenter: 804

Southern Outer Arc

Radius: 339

Xcenter: 1080

Ycenter: 679

Southern Inner Arc

Radius: 234

Xcenter: 1072

Ycenter: 677

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Final Result

These are the data of the circles of best fit for the three arcs. The upper half of middle and right column show that the distance between the centers of both southern circular arcs is very small, only 8 pixels, they are nearly the same. While the distance from these both cen-ters to the center of the northern circular arc is around 130 pixels, respectively 128 and 132. 

The lower half shows that all three radii are almost multiples of a distance of this distance. Dividing the radii by 4, 3 and 2 leads to a quantity of about 115 pixels and multi-plying 115 by 4, 3 and 2 gives the measured radii with a deviation of 1, 6 and 4. So with these results we can paint a new picture. Naming the distance of 115 pixel as ro we get in the north a quarter circle arc with a radius of 4 ro and in the south a third circle arc with a radius of 3 ro and half circle arc with a radius of 2 ro. These angles and radii mean that all three arcs have the same length of 2 π ro, a full circumference of a circle with radius ro. In music a ratio of 4:3:2 is a perfect fourth (4:3), a perfect fifth (3:2) and an octave (4:2) and together they form a harmonic series.

Results

Using ro as a grid constant and the distance between the centers of northern and southern arc as basis, we may draw an underlying grid, which can be supplemented with both mandalas. The southern outer arc is slightly longer than a third arc. The ends are not 30° away from the horizontal grid line through the centers of the southern arcs, but with an angle of 19.5° on both sides. Now we reduce the figure to the real, recognizable slope margins and get that pattern with which I started my marginal notes, but now based on the virtual survey stakes set in the landscape.

Metrics Of Mandala
On Mars

Grid constant (115x3m) ro – 345m

Length overall (8ro) – 2760M

Width and also the length of two outer straight lines   (4ro) – 1952m

Length of northern outer arc and southern inner arc (2pro) – 2168m

Southern outer arc  (2.35pro) – 2549m

3 inner straight lines   (4ro) – 1380m

radii – 690m

2ro – 1035m

 3ro – 1380m

  4ro –  690m

area ~ 6km² (2km x 3km) – 1035m