BALANCING A GOVERNMENT MEANDER BALANCING A GOVERNMENT MEANDER LINE USING STATE PLANE COORDINATESLINE USING STATE PLANE COORDINATES

JAMES WENGLER, PLS, CFSJAMES WENGLER, PLS, CFS

THE DIRECTION OF LINES

2-17. The direction of each line of the public land surveys is determined with reference to the true meridian as defined by the axis of the earth’s rotation. Bearings are stated in terms of angular measure referred to the true north or south.

By basic law, and the Manual requirements, the directions of all lines are stated in terms of angular measure referred to the true north (or south) at the point of record.

THE GEODESY OF LARGE-SCALE CADASTRAL SURVEYS

Transfer of Azimuth, Station Error, and Curvature

2-74. When carrying forward the direction of lines through intermediate transit stations by the method of fore-and-back sights and deflection angles, two corrections become important where the purpose is to maintain accuracy. First, each station setup involves uncertainty in the maintenance of the direction of a line, or in the value of the angle that may be turned, called "station error." Second, if the line is other than a meridian, its direction will have an increment of curvature; this is applied in order to convert from the forward azimuth to the back azimuth of that same line at the next station.

As solar transit orientation is designed to give the meridian at each station, thereby avoiding cumulative errors of conventional transit methods, the corrections

for station error and curvature do not enter into the ordinary solar transit directions. However, for the purpose of a comparison of the solar transit direction of the chord of a long line, half the value of the convergency of the meridians of the two end stations is applied.

For example, a parallel of latitude as run by solar transit methods is a true latitudinal curve, i. e.-a small circle of the earth, everywhere due east or west. The transit line or chord between any two distant points of the parallel is a great circle, whose mean azimuth, or bearing at midpoint, is due east and west. At one end of the chord the forward azimuth is always northeasterly (or northwesterly) ; at the opposite end, the back azimuth will be northwesterly (or northeasterly). At the end stations of the chord, the difference between the forward (or back) azimuth and due east or west, will be equal to half the value of the curvature counting from the two end stations. At the end stations of the chord, the difference between the forward azimuth and the back azimuth +180° will be the full value of the convergency of the meridians of the two end stations.

3-119. The surveyor commences at one of the meander corners, follows the bank or shore line, and determines the length and true bearing of each course, from the beginning to the next meander corner. All meander courses refer to the true meridian and are determined with precision. “Transit angles” showing only the amount of the deviation from the preceding course are not acceptable in field notes of meanders.

Broken Boundaries

5-43. Angle Points of Nonriparian Meander Lines.

In some cases it is necessary to restore (or possibly to locate for the first time) the angle points, within a section, of the record meander courses for a stream, lake, or tidewater, which may be required under the special rules which are applicable to nonriparian meander lines.

In these cases the positions of the meander corners on the section boundaries are determined first. The record meander courses and distances are then run and temporary angle points are marked. The residual error is shown by the direction and length of the line from the end of the last course to the objective meander corner. The residual is distributed on the same plan as in balancing a survey for the computation of the areas of the lottings as represented on the plat.

The general rule is that the adjustment to be applied to the {latitude / departure} of any course is to the resolved {latitude / departure} of the closing error as the length of the course’ is to total length of all the courses. Each adjustment is applied in a direction to reduce the closure. If the northings are to be increased, then the southings will be decreased. A line due east would then be given a correction to the north (in effect to the left ); a line due west, also to the north (in effect to the right ).

Each incremental correction is determined and applied in proportion to the length of the line.

The field adjustments for the positions of the several angle points are accomplished simply by moving each temporary point on the bearing of the closing error an amount that is its proportion of that line, counting from the beginning. The particular distance to be measured at any point is to the whole length of the closing error as the distance of that point from the starting corner is to the sum of the lengths of all the courses. Figure 72.

The same principle is followed to plot lottings of dependently resurveyed sections in their true relative positions when the record meander line and the true shoreline differ greatly because of distortion.

GLO RECORD MEASUREMENTS

N 33°45’E 19.85 CH

S 55°E 3.80 CH

32

29

GL2

GL4

N 81°E

4.0 CH

N 61°E

5.5 CH

N 67°

E

6.20 C

H

LAT.= 47° 42’ 29.96541” CONV.= -1° 17’ 54”

GLO RECORD MEASUREMENTS

N 33°45’E 19.85 CH

S 55°E 3.80 CH

32

29

GL2

GL4

N 81°E

4.0 CH

N 61°E

5.5 CH

N 67°

E

6.20 C

H

LAT.= 47° 42’ 29.96541” CONV.= -1° 17’ 54”

1310.10’

CHAINS X 66 = FEET

250.80’ 264.00’363.00’

409.20

’

DEPARTURES

N 33°45’E 19.85 CH

S 55°E 3.80 CH

32

29

GL2

GL4

N 81°E

4.0 CH

N 61°E

5.5 CH

N 67°

E

6.20 C

H LAT.= 47° 42’ 29.96541”

CONV.= -1° 17’ 54”

1310.10’

250.80’ 264.00’363.00’

409.20

’

SIN 33°45’ X 1310.10’ = 727.85’

SIN 55° X 250.80’ = 205.44’

SIN 81° X 264.00’ = 260.75’

SIN 61° X 363.00’ = 317.49’

SIN 67° X 409.20’ = 376.67’

TOTAL DEPARTURE = 1888.20’

727.85’ 205.44’ 260.75’ 317.49’ 376.67’

1888.20’

TABLE 11 DATA: CONVERGENCY OF MERIDIANS , SIX MILES LONG AND SIX MILES APART

LATITUDE 47° N = 5’ 34” LATITUDE 48° N = 5’

46” DIFFERENCE = 0’ 12”

12” = X”______ 60’ 42’ 29.96541”

12” = X”_____ 60’ 42.4994235’X=8.5” (INTERPOLATED DIFFERENCE IN CONVERGENCE BETWEEN LAT 47° & 48° NORTH)

29.96541”/60 = .4994235’

5’ 34” (CONVERGENCE AT N 47°) + 0’ 8.5” (INTERPOLATED DIFFERENCE IN CONVERGENCE) = 5’ 42.5” (CONVERGENCE AT N 47°42’29.96541”)5’ 42.5” / 31,680 FEET (6 MILES) = CONV. PER FOOT°

.095138889° / 31,680’ = 0.000003003°

0.000003003° X 727.85’=.002185734° >HMS = 08”

0.000003003° X 205.44’=.000616936° >HMS = 02”

0.000003003° X 260.75’=.000620870° >HMS = 03”

0.000003003° X 317.49’=.000953422° >HMS = 03”

0.000003003° X 376.67’=.001131140° >HMS = 04”

0.000003003° X 1888.20’=.005670265° >HMS = 20”

LATITUDE = 47° 42’ 29.96541” N

727.85’

205.44’

260.75’

317.49’

376.67’

CONVERGENCE = 8”

2”3”

3”

4”

CONVERGENCE OF MERIDIANS

N 33°45’E 1310.10’

S 55°E 250.80’

32

29

GL2

GL4

N 81°E

264.00’

N 61°E

363.00’

N 67°

E

409.20

’

CONV.= -1° 17’ 54”

CONV.= -1° 17’ 46”

CONV.= -1° 17’ 44”

CONV.= -1° 17’ 41”

CONV.= -1° 17’ 38”

CONV.= -1° 17’ 34”

-1° 17’ 54” + 8” = -1° 17’ 46”

-1° 17’ 46” + 2” = -1° 17’ 44”

-1° 17’ 44” + 3” = -1° 17’ 41”

-1° 17’ 41” + 3” = -1° 17’ 38”

-1° 17’ 38” + 4” = -1° 17’ 34”

CONVERGENCE = 8”

2”

3”

3”

4”

FOR ALL NORTHEAST AND SOUTHEAST BEARINGS, THE CORRECTION FOR CURVATURE IS APPLIED CLOCKWISE.

FOR ALL NORTHWEST AND SOUTHWEST BEARINGS, THE CORRECTION FOR CURVATURE IS APPLIED COUNTERCLOCKWISE.

CONV. AT TRUE MEANDER CORENR = -1° 17’ 33”

CONVERGENCE OF MERIDIANS

N 33°45’E 1310.10’

S 55°E 250.80’

32

29

GL2

GL4

N 81°E

264.00’

N 61°E

363.00’

N 67°

E

409.20

’

CONV.= -1° 17’ 54”

CONV.= -1° 17’ 46”

CONV.= -1° 17’ 44”

CONV.= -1° 17’ 41”

CONV.= -1° 17’ 38”

CONV.= -1° 17’ 34”

CONVERGENCE = 8”

2”

3”

3”

4”

N33° 45’ 00”E – (-1° 17’ 50.0”) = N35° 02’ 50.0”E

S55° 00’ 00”E – (-1° 17’ 45.0”) = S53° 42’ 15.0”E

AVG. CONV.=-1°17’50”

AVG.= -1° 17’ 45”

AVG.= -1° 17’ 42.5”

AVG.= -1° 17’ 39.5”

AVG.= -1° 17’ 36”

N81° 00’ 00”E – (-1° 17’ 42.5”) = N82° 17’ 42.5”E

N61° 00’ 00”E – (-1° 17’ 39.5”) = N62° 17’ 39.5”E

N67° 00’ 00”E – (-1° 17’ 36.0”) = N68° 17’ 36.0”E

CONVERT GROUND DISTANCES TO

GRID DISTANCES

N 35°02’50” E

250.79’

32

29

GL2

GL4

263.99’362.99’

409.19

’1310.10’ X .99996807 = 1310.06’

1310.06’

S 53° 42’ 15.0” E

N 82° 17’ 42.5” E

N 62° 17’ 39.5” E

N 68° 17’ 36.0” E

250.80’ X .99996807 = 250.79’

264.00’ X .99996807 = 263.99’

363.00’ X .99996807 = 362.99’

409.20’ X .99996807 = 409.19’

Start 300 MC 29&32 263237.98569 1211149.55429IN N 35 02'50.000" E 1310.0600 344 264310.50434 1211901.85804IN S 53 42'15.000" E 250.7900 345 264162.04806 1212103.98759IN N 82 17'42.500" E 263.9900 346 264197.44127 1212365.59424IN N 62 17'39.500" E 362.9900 347 264366.20622 1212686.96650Raw: NE 68 17'36.000" 409.1900 TC N 68 17'36.000" E 409.1900 Calculated Closing Point 264517.54713 1213067.14066  TC S 13 23'59.518" E 56.0058 419 MC FINAL 264463.06603 1213080.11975

Enter the Closing Foresight Pt. = UnknownAngle right at 419 from 347 to fixed foresight = Unknown Precision Ratio = 1: 46Length Traversed = 2597.0200Length To Close = 56.0058 Error in Latitude = 54.4811Error in Departure = 12.9791

FINAL BALANCED LINE

N 35°59’07” E 1291.49’

32

29

GL2

GL4

S 52°55’ 04” E

254.94’

N 83°31’ 18’ E

264.62’

N 63°29’ 52” E

361.14’

N 69°31’11" E

408.01’

CLOSURE = S 13°23’ 59.518" E 56.0058’

33.6604’

39.3534’

1310.06’ = X’__ 2597.02’ 56.0058’

X=28.2520’

47.1815’

1560.85’ = X’__ 2597.02’ 56.0058’

250.79’263.99’

362.99’

409.19

’

1310.06’

28.2520’

X=33.6604’

1824.84’ = X’__ 2597.02’ 56.0058’

X=39.3534’

2187.83’ = X’__ 2597.02’ 56.0058’

X=47.1815’

COMPARISON OF MEANDER LINES WITH AND WITHOUT

CORRECTION FOR MERIDIAN

CORRECT CLOSURE = S 13°23’ 59.518“ E

56.0058’

INCORRECT CLOSURE = S 23°28’ 27.070" E

106.3482’

S 62° 51’ 56” W 10.82’

S 40° 28’ 44” W 5.54’

S 24° 27’ 46” W 3.25’

S 32° 36’ 11” W 2.09’

UNADJUSTED MEANDER LINE CORRECTED FOR MERIDIAN

UNADJUSTED MEANDER LINE NOT CORRECTED FOR MERIDIAN

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