Traversing
Chapter 9
Closed Traverse Showing Interior Angles
Open Traverse Showing Deflection Angles
Computation of Interior and Exterior Angle at
STA 1 + 43.26
Computation of Azimuths
Computation of Bearings
Traverse Computations
Chapter 10
Step 1: Check Allowable Angle Misclosure
cK n
where:
c is the allowable misclosure in seconds
K is a constant that depends on the level of accuracy
specified for the survey
n is the number of angles
Federal Geodetic Control Subcommittee (FGCS) recommends:
First order
1.7”
Second-order class I
3”
Second-order class II
4.5”
Third-order class I
10”
Third-order class II
20”
Step 2: Balance interior angles
Sum of the interior angles must equal (n-2)180
degrees!
Procedures for Balancing Angles
Method 1: Multiple Measurements: Use weighted average
based on standard errors of angular measurements.
Method 2: Single Measurements: Apply an average correction
to each angle where observing conditions were
approximately the same at all stations. The correction is
computed for each angle by dividing the total angular
misclosure by the number of angles.
Method 3: Single Measurements: Make larger corrections to
angles where poor observing conditions were present. This
method is seldom used.
Step 3: Computation of Preliminary Azimuths or Bearings
Step 3: Computation of Preliminary Azimuths or Bearings
(continued)
Beginning at the reference meridian compute the azimuth or
bearing for each leg of the traverse.
Always check the last leg of the traverse to make sure that you
compute the same azimuth or bearing as the reference meridian.
Step 4: Compute Departures and Latitudes
Step 4: Compute Departures and Latitudes (continued)
departure = L sin (bearing)
latitude = L cos (bearing)
Step 5: Compute Linear Misclosure
Because of errors in the measured angles and distances
there will be a linear misclosure of the traverse. Another
way of illustrating this is that once you go around the
traverse from point A back to point A’ you will notice that
the summation of the departures and latitudes do not
equal to zero. Hence a linear misclosure is introduced.
Step 5: Compute Linear Misclosure (continued)
Linear misclosure = [(departure misclosure)2 + (latitude misclosure)2]1/2
Step 6: Compute Relative Precision
relative precision = linear misclosure / traverse length
expressed as a number 1 / ?
read as 1’ foot error per ? feet measured
Example:
linear misclosure = 0.08 ft.
traverse length = 2466.00 ft.
relative precision = 0.08/2466.00 = 1/30,000
Surveyor would expect 1-foot error for every 30,000 feet surveyed
1984 FGCS Horizontal Control Survey Accuracy Standards
Order and Class
Relative Accuracy Required
First Order
1 part in 100,000
Second Order
Class I
1 part in 50,000
Class II
1 part in 20,000
Third Order
Class I
1 part in 10,000
Class II
1 part in 5,000
From Table 19-3, page 557 in your textbook
Step 7: Adjust Departures and Latitudes
Methods:
Compass (Bowditch) Rule:
correction in departure for AB = - [(total departure misclosure)/(traverse perimeter)](length of AB)
correction in latitude for AB = - [(total latitude misclosure)/(traverse perimeter)](length of AB)
Adjusts the departures and latitudes of the sides of the traverse in proportion to their lengths.
Least Squares Adjustment:
Rigorous but best method based on probabilistic approach which models the occurrence of random
errors. Angle and distance measurements are adjusted simultaneously. Most computer programs
employ a least square adjustment methodology.
For more information on Least Squares Method see Chapter 15 in your textbook.
Step 8: Compute Rectangular Coordinates
XB = XA + departure AB
YB = YA + latitude AB
Step 8: Compute Rectangular Coordinates (continued)
Compute the coordinates of all points on the traverse taking into
account the signs of the latitudes and departures for each side of the
traverse. Be sure to compute the coordinates of the beginning point
(in this case A) so you can check to see if you have made any errors in
your computations.
Signs of Departures and Latitudes
North
Departure (-)
Departure (+)
Latitude (+)
Latitude (+)
West
East
Departure (-)
Departure (+)
Latitude (-)
Latitude (-)
South
Step 9: Compute Final Adjusted Traverse Lengths and Bearings
To compute adjusted bearing AB:
tan (bearing) AB = (corrected departure AB)/(corrected latitude AB)
To compute adjusted length AB:
length AB = [(corrected departure AB)2 + (corrected latitude AB)2]1/2
Coordinate Geometry (COGO)
Formulas:
tan azimuth (or bearing) AB = departure AB/latitude AB
length AB = departure AB/(sin azimuth (or bearing) AB)
length AB = latitude AB/(cos azimuth (or bearing) AB)
length AB = length AB = [(departure AB)2 + (latitude AB)2]1/2
departure AB = XB – XA = delta X
latitude AB = YB – YA = delta Y
A
Bearing angle
delta Y (latitude)
B
delta X (departure)
Coordinate Geometry (COGO) (continued)
More formulas:
tan azimuth (or bearing) AB = (XB – XA)/(YB – YA) = (delta x)/(delta y)
length AB = (XB – XA)/(sin azimuth (or bearing) AB)
length AB = (YB – YA)/(cos azimuth (or bearing) AB)
length AB = [(XB – XA )2 + (YB – YA )2]1/2
Length AB = [(delta X)2 + (delta Y)2]1/2
These types of calculations are referred to as “inversing” between two
points. i.e. if you know the coordinates of two points you can easily
find the bearing between the two points and the distance between the
two points by using the above formulas.
Area Computations
Chapter 12
Step 10: Compute Area using Coordinate Method
areaE’EDD’E’ =
XE  XD
(YE  YD)
2
YE = E’ – C’
XE  XD
(YE  YD)
2
YD = D’ – C’
Each of the areas of
trapezoids and triangles
can be expressed by
multiplying coordinates
in a similar manner.
Step 10: Compute Area using Coordinate Method (continued)
Double area = +XAYB+XBYc+XcYD+XDYE+XEYA
-XBYA –XcYB –XDYC –XEYD -XAYE
An easy way to remember how to compute
the area using the coordinate method:
Be sure to begin and end at the same
coordinate. The products are computed
along the diagonals with dashed arrows
considered plus and solid ones minus. This
method computes the DOUBLE AREA so you
need to divide the result by 2 to get the
area.