CogoKit

Boundary retracement

Corner position recovered from two bearing-tree ties

Two original bearing trees are found and coordinated. The corner is computed independently from each recorded tie, the discrepancy is examined, and the accepted position is checked against the record distance between the trees.

Apply· about 25 minutes by hand· 6 steps

Given

  • An original corner whose monument is gone, but whose two recorded bearing trees were both found, identified by their scribing and by blazes consistent with the original survey.
  • Original field notes: from the corner, a 14 inch oak bears N 32°15′ E, 41.2 links; a 10 inch pine bears S 48°30′ W, 27.6 links.
  • The centers of the two trees were coordinated by the retracement. Oak: N 5,412.71, E 8,930.32. Pine: N 5,377.44, E 8,902.39. Local plane grid, US survey feet, to 0.01 ft.
  • One link is 0.66 ft (one hundredth of a 66 ft chain). Recorded tie bearings are to the nearest quarter of a degree and tie distances to 0.1 link, which is the precision of the original notes.

Required

  • The record tie distances converted to feet, and the direction from each tree back to the corner.
  • The corner position computed independently from each tree, and the discrepancy between them.
  • The accepted corner position, checked against the record.

Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.

Worked solution

This is an obliterated corner, not a lost one

A corner whose position can be recovered from acceptable evidence is obliterated, not lost, and it is restored from that evidence rather than by proportionate measurement. Accessories such as bearing trees are precisely that kind of evidence, and the distinction between an obliterated and a lost corner is drawn in the chapter of the BLM Manual of Surveying Instructions (2009) on the restoration of lost or obliterated corners.

The practical consequence is large. A proportioned position on this line might land many feet from where the original monument stood; two bearing trees put it within a few tenths. So the first duty in the field is to search for accessories, and only a genuinely exhausted search justifies moving to proportion.

Convert the record ties and reverse the bearings

The notes record the direction and distance from the corner to each tree. To compute the corner from the tree, the same distance is run on the reverse bearing, which is the same angle in the opposite quadrant.

Distances in the original notes are in links, one hundredth of a Gunter's chain, so a link is 0.66 ft. Both ties are short, under 30 ft, which is normal: bearing trees were chosen close to the corner precisely so that the tie would not accumulate chaining error.

oak tie = 41.2 links x 0.66 = 27.192 ft ; record bearing N 32°15′ E, reverse S 32°15′ W, azimuth 212°15′00″ pine tie = 27.6 links x 0.66 = 18.216 ft ; record bearing S 48°30′ W, reverse N 48°30′ E, azimuth 48°30′00″

Compute the corner from each tree

Run each reversed tie from its tree by forward computation: the latitude is the distance times the cosine of the azimuth and the departure is the distance times the sine. Each tie gives a complete, independent position for the corner, and they are kept separate on purpose so that they can be compared.

Averaging first and comparing afterwards throws away the only quality check available in this problem.

cos 212°15′00″ = -0.845721 ; sin 212°15′00″ = -0.533612 cos 48°30′00″ = +0.662620 ; sin 48°30′00″ = +0.748956 from the oak: dN = 27.192 x (-0.845721) = -22.997 ; dE = 27.192 x (-0.533612) = -14.510 N = 5,412.71 - 22.997 = 5,389.713 -> 5,389.71 E = 8,930.32 - 14.510 = 8,915.810 -> 8,915.81 from the pine: dN = 18.216 x (+0.662620) = +12.070 ; dE = 18.216 x (+0.748956) = +13.643 N = 5,377.44 + 12.070 = 5,389.510 -> 5,389.51 E = 8,902.39 + 13.643 = 8,916.033 -> 8,916.03
Corner position from each bearing tree
FromTree NTree EAzimuthTie (ft)Corner NCorner E
Oak5,412.718,930.32212°15′00″27.1925,389.718,915.81
Pine5,377.448,902.3948°30′00″18.2165,389.518,916.03

Examine the discrepancy

The two positions differ by 0.30 ft. Whether that is good depends on what the record can support: the tie bearings were recorded to the nearest quarter of a degree, and a quarter of a degree at 27 ft is 0.12 ft, while a tenth of a link is 0.066 ft. Two ties each carrying that much uncertainty can easily differ by three tenths, so 0.30 ft is consistent with the record and gives no reason to doubt either tree.

A discrepancy of several feet would mean something different, and the likely explanations would be that one tree is not the original, that the notes were transcribed with a bearing quadrant reversed, or that the tree has grown enough that its present center is no longer where its pith was when scribed. Older accessories are tied to the pith for exactly that reason.

discrepancy = sqrt((5,389.713 - 5,389.510)^2 + (8,915.810 - 8,916.033)^2) = 0.30 ft one quarter degree at 27.19 ft = 0.12 ft 0.1 link = 0.07 ft

Accept the mean position

With two ties of comparable quality and no reason to prefer either, the accepted corner is the simple mean of the two computed positions. Weighting would be appropriate if one tie were markedly longer than the other or recorded to a coarser precision, but the two here are close enough in both respects that weighting would move the answer by hundredths.

Set a monument at this position, and record both computed positions along with the mean, so that a later surveyor who finds a third accessory can bring it into the same comparison instead of inheriting a single number with no error budget attached.

N = (5,389.71 + 5,389.51) / 2 = 5,389.61 E = (8,915.81 + 8,916.03) / 2 = 8,915.92 accepted corner: N 5,389.61, E 8,915.92

Check the accepted position against the record

Two checks are available and both should be worked. The first inverses from the accepted corner back to each tree and compares the result with the recorded tie: the oak comes out at 41.24 links on N 31°56′ E against a record of 41.2 links N 32°15′ E, and the pine at 27.57 links on S 48°02′ W against 27.6 links S 48°30′ W. Distances agree to a twentieth of a link and bearings to within half a degree, which is as well as notes recorded to the nearest quarter degree can be expected to do.

The second check does not involve the accepted position at all and is therefore genuinely independent. The two record ties, together with the angle between their directions at the corner, fix the distance between the two trees by the law of cosines. That computed distance can be compared with the distance actually measured between the two found trees. They agree within 0.02 ft, which is strong evidence that both trees are the ones the notes describe.

inverse to the oak: 27.22 ft = 41.24 links, N 31°56′ E (record 41.2 lk, N 32°15′ E) inverse to the pine: 18.20 ft = 27.57 links, S 48°02′ W (record 27.6 lk, S 48°30′ W) angle at the corner between the two record directions = 212°15′ - 48°30′ = 163°45′ record oak to pine = sqrt(27.192^2 + 18.216^2 - 2 x 27.192 x 18.216 x cos 163°45′) cos 163°45′ = -0.960050 = sqrt(739.40 + 331.82 + 951.14) = sqrt(2,022.36) = 44.97 ft measured oak to pine (from the found coordinates) = 44.99 ft difference = 0.02 ft

Answer

  • Tie distances: oak 27.19 ft, pine 18.22 ft; reverse azimuths 212°15′00″ and 48°30′00″
  • Corner from the oak: N 5,389.71, E 8,915.81
  • Corner from the pine: N 5,389.51, E 8,916.03
  • Discrepancy between the two solutions = 0.30 ft
  • Accepted corner position: N 5,389.61, E 8,915.92

Check

Independent geometric check: the record ties of 27.19 ft and 18.22 ft with 163°45′ between them put the two trees 44.97 ft apart by the law of cosines, and the distance inversed between the two found trees is 44.99 ft. The 0.02 ft agreement is independent of the accepted corner position and confirms that both trees match the notes.

Inversing from the accepted corner reproduces the record ties as 41.24 links N 31°56′ E to the oak and 27.57 links S 48°02′ W to the pine, against recorded values of 41.2 links N 32°15′ E and 27.6 links S 48°30′ W.

Error-budget check: a quarter of a degree at 27 ft is 0.12 ft and a tenth of a link is 0.07 ft, so a 0.30 ft spread between two independently computed positions is consistent with the stated precision of the original notes rather than evidence of a wrong tree.

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