Boundary retracement
Retracing a metes-and-bounds description that will not close
A four-course deed fails to close by more than nine feet. Locate the error from the direction of the misclosure, decide what to hold, correct the record, and adjust the retraced parcel by the compass rule.
Given
- A deed description of a four-sided parcel, beginning at an iron pipe and running: (1) N 88°15′00″ E, 412.65 ft to an iron pipe; (2) S 02°06′00″ E, 318.44 ft to an iron pipe; (3) S 87°52′00″ W, 411.08 ft to an iron pipe; (4) N 02°23′00″ W, 312.17 ft to the point of beginning.
- All four corner pipes were recovered in the field and each is undisturbed and consistent with the age of the deed.
- The point of beginning is assigned the working coordinates N 1,000.00, E 1,000.00. Distances to 0.01 ft, bearings to 1″, areas to 0.0001 acre.
- Bearings are treated as being on a single consistent basis; no rotation between the deed basis and the retracement basis is at issue.
Required
- Latitudes and departures of the record courses, and the misclosure and precision.
- A diagnosis of where the error lies, and a decision on what to hold.
- Adjusted coordinates, final courses and the area of the parcel.
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
Latitudes and departures of the record
Compute each course's latitude and departure from its bearing and distance and sum them. A closed figure returns to its starting point, so both sums must be zero, and whatever they actually come to is the misclosure in latitude and in departure.
The sums are badly out in latitude and almost perfect in departure. That asymmetry is itself the first piece of evidence: an error that lives almost entirely in the north-south component did not come from an east-west course.
latitude = distance x cos(azimuth)
departure = distance x sin(azimuth)
misclosure = sqrt(9.026^2 + 0.350^2) = 9.03 ft
precision = 1,454.34 / 9.03 = 1 : 161| Course | Bearing | Distance (ft) | Latitude (ft) | Departure (ft) |
|---|---|---|---|---|
| 1 | N 88°15′00″ E | 412.65 | +12.602 | +412.458 |
| 2 | S 02°06′00″ E | 318.44 | -318.226 | +11.669 |
| 3 | S 87°52′00″ W | 411.08 | -15.302 | -410.795 |
| 4 | N 02°23′00″ W | 312.17 | +311.900 | -12.982 |
| Sums | - | 1,454.34 | -9.026 | +0.350 |
Read the direction of the misclosure
A misclosure is a vector, and its direction is far more informative than its length. The vector needed to return from the computed end point to the point of beginning is +9.026 in latitude and -0.350 in departure, which is a bearing of about N 02°13′ W.
Course 4 runs N 02°23′00″ W. The misclosure is within about ten minutes of being exactly parallel to it. A misclosure aligned with a single course almost always means the distance of that course is wrong; a misclosure perpendicular to a course points instead at that course's bearing. Ten minutes of divergence over 9 ft is well inside what the rounding of the other three courses can account for.
closing vector = (+9.026 lat, -0.350 dep)
its bearing = N 02°13′ W
course 4 bearing = N 02°23′ W
difference = 0°10′ -> the error is a distance in course 4Solve the course that closes the figure
Take the first three courses as written, compute where they leave the traverse, and inverse from that position back to the point of beginning. That is the course the deed needed in order to close, and comparing it with the course the deed recites tells you exactly what went wrong.
The required bearing is N 02°22′43″ W against a record of N 02°23′00″ W, a difference of seventeen seconds, which is nothing at all for a deed written to the nearest minute. The required distance is 321.20 ft against a record of 312.17 ft, a difference of 9.03 ft.
position after courses 1-3: latitude -320.9269, departure +13.3313 from the POB
inverse back to the POB: 321.20 ft, N 02°22′43″ W
record course 4: 312.17 ft, N 02°23′00″ W
bearing differs by 0°00′17″ ; distance differs by 9.03 ftWhat to hold
The four corner pipes were all recovered, undisturbed, and consistent with the deed's age. Found monuments that a description calls for are the strongest evidence of where the parcel lies, and the courses and distances in the deed are a description of the lines between them. So the retracement holds the pipes, and the record dimensions are read as evidence of, rather than as a substitute for, those lines.
That settles what to correct. The bearing of course 4 is confirmed by the geometry to within seventeen seconds and is kept. The distance is not: 312.17 ft cannot reach the pipe at the point of beginning. Reading 312.17 as a transposition of 321.17 explains the whole 9.03 ft in a single stroke of the pen, which is a far more plausible history than a 9 ft chaining blunder on a 312 ft line while the other three courses check to hundredths.
Substituting 321.17 ft leaves a misclosure of 0.043 ft in a perimeter of 1,463.34 ft, a precision of 1 : 34,391. Residual of that size is exactly what bearings rounded to the nearest minute would produce, and it confirms the diagnosis.
adopt course 4 = N 02°23′00″ W, 321.17 ft
new sums: latitude -0.035 , departure -0.025
misclosure = 0.043 ft in 1,463.34 ft = 1 : 34,391Compass-rule adjustment and final coordinates
With a defensible figure in hand, distribute the small remaining misclosure by the compass rule, which apportions the corrections in proportion to course length on the assumption that the angles and the distances are of comparable quality. That assumption is reasonable here: the bearings are given to the minute and the distances to the hundredth, and neither is obviously the weaker.
The closure removed is +0.035 ft in latitude and +0.025 ft in departure. Individual corrections are shown to 0.001 ft, so the column entries do not quite add to those totals; the adjusted latitudes and departures are what must sum to zero, and they do. The corrections are a hundredth of a foot at most, so the adjustment changes nothing of consequence and simply makes the coordinates consistent.
A (POB) N 1,000.00 E 1,000.00
B N 1,012.61 E 1,412.46
C N 694.39 E 1,424.14
D N 679.10 E 1,013.35| Course | Distance (ft) | Lat correction (ft) | Dep correction (ft) | Adjusted latitude | Adjusted departure |
|---|---|---|---|---|---|
| 1 | 412.65 | +0.010 | +0.007 | +12.612 | +412.464 |
| 2 | 318.44 | +0.008 | +0.005 | -318.219 | +11.674 |
| 3 | 411.08 | +0.010 | +0.007 | -15.293 | -410.788 |
| 4 | 321.17 | +0.008 | +0.005 | +320.900 | -13.350 |
| Sums | 1,463.34 | - | - | 0.000 | 0.000 |
Final courses and area
Inverse between the adjusted coordinates to get the courses as retraced. Every one agrees with the deed to within about a hundredth of a foot and five seconds, apart from course 4, whose distance was the subject of the correction.
The area of the closed figure is 131,713.55 square feet, or 3.0237 acres. Report it as 3.02 acres, and record in the notes that the deed's fourth call was read as 321.17 ft, that the reading rests on four found pipes and on the geometry of the other three courses, and that the record recites 312.17 ft. A retracement that silently substitutes a corrected dimension is worth much less than one that shows its reasoning.
A to B: 412.66 ft N 88°14′55″ E (deed 412.65, N 88°15′00″ E)
B to C: 318.43 ft S 02°06′04″ E (deed 318.44, S 02°06′00″ E)
C to D: 411.07 ft S 87°52′05″ W (deed 411.08, S 87°52′00″ W)
D to A: 321.18 ft N 02°22′56″ W (deed 312.17, N 02°23′00″ W)
area = 131,713.55 sq ft = 3.0237 acresAnswer
- Record misclosure = 9.03 ft, precision 1 : 161, on a bearing of N 02°13′ W
- The error is the distance of course 4: the deed's 312.17 ft is a transposition of 321.17 ft
- Corrected figure closes at 0.043 ft in 1,463.34 ft, a precision of 1 : 34,391
- Adjusted coordinates: A 1,000.00 / 1,000.00; B 1,012.61 / 1,412.46; C 694.39 / 1,424.14; D 679.10 / 1,013.35
- Area = 131,713.55 sq ft = 3.0237 acres, reported as 3.02 acres
Check
The closing course computed from the first three courses alone is 321.20 ft on N 02°22′43″ W, which reproduces the deed's course-4 bearing to 17 seconds. The bearing is therefore confirmed independently of the distance, isolating the error to the distance.
Substituting 321.17 ft drops the misclosure from 9.03 ft to 0.043 ft, a factor of 210. No other single-figure change to any course produces a comparable improvement, which is what makes the transposition diagnosis rather than a guess.
The adjusted latitudes sum to 0.000 and the adjusted departures sum to 0.000, so the adjusted figure closes exactly.
The final inversed courses reproduce the deed's first three calls to 0.01 ft and 5 seconds, confirming that holding the pipes did not distort the parts of the description that were sound.
More boundary retracement
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- Lost interior section corner by double proportionate measurementAn interior section corner is lost with existent corners a mile or so in each of the four cardinal directions. Restore it by double proportionate measurement and verify that the record latitude and departure ratios are preserved.
- Center of section by intersection of the quarter-corner linesA section is subdivided from eight recovered monuments. The center of section is fixed by intersecting the north-south and east-west quarter-corner lines, and the four quarter-section areas are computed and checked against the whole.