H. Closed Link Traverse
1. General
In this chapter we will cover computational differences between closed loop and link traverses. Unlike a loop traverse, a link traverse does not close back on itself. In order for a link traverse to be closed, the positions of its endpoints must be known. They must be known either relative to each other, Figure H-1(a) and (b):
(a) |
(b) |
Figure H-1 |
or,
by coordinates in the same coordinate system, Figure H-2(a) and (b):
(a) |
(b) |
Figure H-2 |
2. Angles; Misclosure & Adjustment
a. Concept
Because a link traverse doesn't close back on itself there are is no interior angles sum against which to check.
Typically, angles on a link traverse are either consistently turned in the same direction, Figure H-3, or deflection angles are measured, Figure H-4.
Figure H-3 |
Figure H-4 |
In order to check angle closure, the traverse must start and end with known directions, Figure H-5.
Figure H-5 |
To determine and distribute the angular misclosure is a two step process
1. Using the start direction and measured angles, compute the raw direction of each line. The difference between the computed and known end directions is the angular misclosure.
2. Using whatever desired correction method, correct each raw direction for the misclosure.
The known directions can be explicit (bearing or azimuth) or be computed from coordinates.
If one or other direction at the traverse ends is missing then angle misclosure and adjustment cannot be done. This step would be skipped and the process would continue with latitude and departure computations.
b. Examples
(1) Link Traverse 1
The crossing traverse in Figure H-6 begins and ends on known bearings and uses angles measured to the left at each point.
Figure H-6 |
Starting with Brng_{QB}, the remaining bearings are computed using the measured angles. The angular misclosure is the difference between computed and known bearings of line QB.
Compute raw bearings
QR |
116°48'53" - 60°25'19" |
RS |
180°00'00" - (56°23'34" + 48°18'36") |
ST |
75°17'50" - 32°11'51" |
TY |
91°05'25" - 73°05'59" |
Compare computed and known bearings of line TY, Figure H-7.
Figure H-7 |
Angular misclosure = 47°59'42" - 47°59'26" = +0°00'16"
Since there are four angles, each would be corrected by 0°00'04". We could either:
Apply the correction to each angle and then then recompute the directions, or,
Apply the correction to each direction.
We'll demonstrate the latter method.
From Figure H-7, it can be seen that each direction must be rotated counter-clockwise. The first direction would be rotated 0°00'04", the second 0°00'08", and so on, Figure H-8.
Figure H-8 |
Because bearings are used, we need to examine which quadrant they fall in to determine if the bearing angle increases or decreases. Based on Figure H-7, bearing angles in the SE and NW quadrants will be increased, those in the NE and SW decreased.
Line |
Raw bearing |
Correction |
Adjusted Bearing |
QR |
S 56°23'34"E |
+0°00'04" |
S 56°23'38"E |
RS |
S 75°17'50"W |
-0°00'08" |
S 75°17'42"W |
ST |
N 43°05'59"E |
-0°00'12" |
N 43°05'47"E |
TY |
S 43°59'26"E |
+0°00'16" |
S 43°59'42"E |
If you compute the new angles between the adjusted bearings, you would find that each angle is increased by 0°00'04".
(2) Link Traverse 2
The link traverse in Figure H-9 starts and ends on known azimuths and uses deflection angles measured at each point.
Figure H-9 |
As with the bearing example, the angular misclosure is the difference between the computed and known direction of the last line.
Compute raw azimuths
JK |
154°52'30" - 108°00'18" |
KL |
46°52'12" + 92°13'46" |
LM |
139°05'58" - 65°55'46" |
MW |
73°10'12" - 29°21'42" |
Compare the computed and known direction of line MW, Figure H-10.
Figure H-10 |
Angular misclosure = 102°32'06" - 102°31'54" = +0°00'12"
Since there are four deflection angles, each would be corrected by 0°00'03". We could either:
Apply the correction to each deflection angle and then then recompute the directions, or,
Apply the correction to each direction.
We will again apply the corrections directly to the azimuths. The raw direction must be rotated clockwise into the known direction. Unlike bearings, we don't need to worry about quadrants for azimuth corrections - each azimuth is corrected in the same direction, Figure H-11.
Figure H-11 |
Because the computed closing direction must be rotated clockwise into the known closing direction, each raw azimuth will have a clockwise correction applied.
Line |
Raw azimuth |
Correction |
Adjusted azimuth |
JK |
46°52'12" |
+0°00'03" |
46°52'15" |
KL |
139°05'58" |
+0°00'06" |
139°06'04" |
LM |
73°10'12" |
+0°00'09" |
73°10'21" |
MW |
102°31'54" |
+0°00'12" |
102°32'06" |
3. Traverse Closure; Adjustment
a. Similarities; Differences
Latitudes and departures are computed same as those for a loop traverse:
Equations D-1 and D-2 |
Where the two differ is in how their closure is determined and adjustments made.
On a loop traverse, the closure condition is:
Equations D-3 and D-4 |
But because a link traverse does not close back on itself, that condition does not apply. Instead, we need to know the location, relative or absolute, of the traverse's end points.
If we know the relative location, Figure H--12,
Figure H-12 |
If we have coordinates of the endpoints, Figure H-13,
Figure H-13 |
the closure condition is
Equations H-1 and H-2 |
The latitude and departure errors would be a result of how well the closure condition was met. Linear closure and precision would be determined just as for a loop traverse.
b. Example
Given the link traverse in Figure H-14 with adjusted directions and known end point coordinates:
Figure H-14 |
(1) Compute latitudes and departures
Line | Direction | Length | Lat | Dep |
QR | S 56°23'38"E | 398.75' | -220.700' | +332.104' |
RS | S 75°17'42"W | 422.89' | -107.347' | -409.038 |
ST | N 43°05'47"E | 604.49' | +441.402' | +413.004' |
sums: | 1426.13' | +113.355' | +336.070 |
(2) Compute closure and precision
From the coordinates
The closure and precision are
(3) Adjusting by the Compass Rule
Line | Direction | Length | Lat | Dep | Adj Lat | Adj Dep |
QR | S 56°23'38"E | 398.75' | -220.700' | +332.104' | -220.715' | +332.124' |
RS | S 75°17'42"W | 422.89' | -107.347' | -409.038 | -107.363' | -409.017' |
ST | N 43°05'47"E | 604.49' | +441.402' | +413.004' | +441.379' | +413.034' |
sums: | 1426.13' | +113.355' | +336.070 | +113.301' | +336.141' | |
check | check |
Adjusted lengths and directions would be computed the same as for a loop traverse, as would coordinates.
4. Summary
A link traverse requires a few variations on the computations we covered for a loop traverse:
- The angular misclosure is dependent on available directions at both ends of the traverse - if either is not available, then misclosure can neither be determined nor compensated.
- The relative or absolute position of the endpoints must be known - latitude and departure closure conditions are dependent on the end point relationships.
- A link traverse does not have an area. Although the Area by Coordinates or DMDs equations can be applied and yield a numeric result, that result will not make sense as an area.
All other computations; lats and deps, traverse adjustment, adjusted lengths and directions, and coordinate determination; are performed in the same manners as for a loop traverse.