Transcript Document

BALANCING A GOVERNMENT MEANDER
LINE USING STATE PLANE COORDINATES
JAMES 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.
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.
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.
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
GL2
GL4
29
32
LAT.= 47° 42’ 29.96541”
CONV.= -1° 17’ 54”
GLO RECORD
MEASUREMENTS
GL2
CHAINS X 66 = FEET
GL4
29
32
LAT.= 47° 42’ 29.96541”
CONV.= -1° 17’ 54”
DEPARTURES
33°45’
55°
81°
61°
67°
GL2
X 1310.10’ = 727.85’
X 250.80’ = 205.44’
X 264.00’ = 260.75’
X 363.00’ = 317.49’
X 409.20’ = 376.67’
GL4
TOTAL DEPARTURE = 1888.20’
LAT.= 47° 42’ 29.96541”
CONV.= -1° 17’ 54”
SIN
SIN
SIN
SIN
SIN
29
32
727.85’
205.44’
260.75’
1888.20’
317.49’
376.67’
TABLE 11 DATA: CONVERGENCY OF MERIDIANS ,
SIX MILES LONG AND SIX MILES APART
LATITUDE = 47° 42’ 29.96541” N
LATITUDE 47° N = 5’ 34”
LATITUDE 48° N = 5’ 46”
DIFFERENCE = 0’ 12”
12”
60’
=
X”______
42’ 29.96541”
0.000003003°
0.000003003°
0.000003003°
0.000003003°
X
X
X
X
727.85’=.002185734°
205.44’=.000616936°
260.75’=.000620870°
317.49’=.000953422°
>HMS
>HMS
>HMS
>HMS
=
=
=
=
0.000003003° X 376.67’=.001131140° >HMS = 04”
0.000003003° X 1888.20’=.005670265° >HMS = 20”
29.96541”/60 = .4994235’
12”
60’
=
X”_____
42.4994235’
X=8.5” (INTERPOLATED DIFFERENCE IN
CONVERGENCE BETWEEN LAT 47° & 48° NORTH)
+
=
5’ 34”
0’ 8.5”
5’ 42.5”
(CONVERGENCE AT N 47°)
(INTERPOLATED DIFFERENCE IN CONVERGENCE)
(CONVERGENCE AT N 47°42’29.96541”)
5’ 42.5” / 31,680 FEET (6 MILES) = CONV. PER FOOT°
.095138889° / 31,680’ = 0.000003003°
08”
02”
03”
03”
260.75’
727.85’
205.44’
317.49’
376.67’
29
32
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.
GL4
CONV. AT TRUE MEANDER CORENR = -1° 17’ 33”
CONV.= -1° 17’ 54”
-1° 17’ 41” + 3” = -1° 17’ 38”
-1° 17’ 38” + 4” = -1° 17’ 34”
GL2
CONV.= -1° 17’ 34”
CONV.= -1° 17’ 38”
CONV.= -1° 17’ 41”
CONV.= -1° 17’ 44”
-1° 17’ 54” + 8” = -1° 17’ 46”
-1° 17’ 46” + 2” = -1° 17’ 44”
-1° 17’ 44” + 3” = -1° 17’ 41”
CONV.= -1° 17’ 46”
CONVERGENCE OF
MERIDIANS
29
32
CONV.= -1° 17’ 38”
GL4
CONV.= -1° 17’ 34”
AVG.= -1° 17’ 39.5”
CONV.= -1° 17’ 41”
AVG.= -1° 17’ 42.5”
CONV.= -1° 17’ 44”
AVG.= -1° 17’ 45”
CONV.= -1° 17’ 46”
GL2
AVG.= -1° 17’ 36”
CONV.= -1° 17’ 54”
CONVERGENCE OF
MERIDIANS
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
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
264.00’ X .99996807 =
363.00’ X .99996807 =
409.20’ X .99996807 =
29
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263.99’
362.99’
409.19’
N 68° 17’ 36.0” E
N 62° 17’ 39.5” E
N 82° 17’ 42.5” E
1310.10’ X .99996807 = 1310.06’
250.80’ X .99996807 = 250.79’
S 53° 42’ 15.0” E
CONVERT GROUND
DISTANCES TO
GRID DISTANCES
GL2
GL4
Start
300 MC 29&32
IN
N 35 02'50.000" E
344
IN
S 53 42'15.000" E
345
IN
N 82 17'42.500" E
346
IN
N 62 17'39.500" E
347
Raw: NE 68 17'36.000"
TC
N 68 17'36.000" E
Calculated Closing Point
TC
S 13 23'59.518" E
419 MC FINAL
263237.98569
1211149.55429
264310.50434
1211901.85804
264162.04806
1212103.98759
264197.44127
1212365.59424
264366.20622
1212686.96650
264517.54713
1213067.14066
264463.06603
1213080.11975
1310.0600
250.7900
263.9900
362.9900
409.1900
409.1900
56.0058
Enter the Closing Foresight Pt. = Unknown
Angle right at 419 from 347 to fixed foresight = Unknown
Precision Ratio
Length Traversed
Length To Close
=
=
=
1: 46
2597.0200
56.0058
Error in Latitude
Error in Departure
=
=
54.4811
12.9791
FINAL BALANCED LINE
=
X’__
56.0058’
=
X’__
56.0058’
=
X’__
56.0058’
=
X’__
56.0058’
X=28.2520’
CLOSURE = S 13°23’ 59.518" E
56.0058’
X=33.6604’
X=39.3534’
GL2
X=47.1815’
47.1815’
N
29
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63°29’ 52” E
361.14’
N
83°31’ 18’ E
264.62’
33.6604’
N
39.3534’
28.2520’
69°31’11" E
408.01’
GL4
S 52°55’ 04” E
254.94’
1310.06’
2597.02’
1560.85’
2597.02’
1824.84’
2597.02’
2187.83’
2597.02’
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’
UNADJUSTED MEANDER LINE
CORRECTED FOR MERIDIAN
S 32° 36’ 11” W
2.09’
UNADJUSTED MEANDER LINE
NOT CORRECTED FOR MERIDIAN
S 24° 27’ 46” W
3.25’
S 40° 28’ 44” W
5.54’
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