LATITUDES AND DEPARTURES

LATITUDES AND DEPARTURES: Background

• The latitude of a line is its projection on the north-south meridian and is equal to the length of the line times the cosine of its bearing.
• The departure of a line is its projection on the east-west meridian and is equal to the length of the line times the sine of its bearing.
• The latitude is the y component of the line (also known asnorthing), and the departure is the x component of the line (also known as easting).

 

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CLOSURE OF LATITUDES AND DEPARTURES

• The algebraic sum of all latitudes must equal zero or the difference in latitude between the initial and final control points
• The algebraic sum of all departures must equal zero or the difference in departure between the initial and final control points
• If the sums of latitudes and departures do not equal zero, corrections must be made.

 

DEGREE and RADIAN MEASURE

Trigonometric functions require input data to be stored in radian measure, but the field measurements are in degrees. Therefore a conversion is necessary. Remember that there are 2*pi radians in a circle.

• To convert from degrees to radians, multiply by azimuth by pi/180
• To convert from radians to degrees, multiply by radians by 180/pi

 

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CALCULATION OF LATITUDES AND DEPARTURES (Using azimuths)

Station	Azimuth	Length	Latitude	Departure
A
	26° 10'	285.10	+255.88	+125.72
B
	104° 35'	610.45	-153.70	+590.78
C
	195° 30'	720.48	-694.28	-192.54
D
	358° 18'	203.00	+202.91	-6.02
E
	306° 54'	647.02	+388.48	-517.41
A
MISCLOSURE			-0.71	+0.53

For example, look at the calculation of latitude for the angle from station A to station B:
With a calculator:

26° 10' = 26.16667°
26.16667° * pi/180 = 0.4566945 rad
cos(0.4566945 rad) = 0.897515
0.897515 + 285.10 ft = 255.88 ft

Or in one operation using R:

cos((26 + 10/60) * pi/180) * 285.1
[1] 255.8815

likewise for departure

sin((26 + 10/60) * pi/180) * 285.1
[1] 125.7245

or in Excel:

SIN((26 + 10/60) * PI()/180) * 285.1 = 125.7245

As you can see, setting this up in Excel is fairly straightforward. You will have a record of your measurements and any transformations of those measurements, and it will allow you to check your work easily.

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ADJUSTMENT OF LATITUDES AND DEPARTURES

In order to calculate corrections for latitude and departure there is a simple formula called the Compass (or Bowditch) Rule, which is used when angles and distances are measured with the same relative accuracy. There are other methods for different measurement accuracy differentials as well, but this method is simple to implement and works well enough for our purposes.

Station	Azimuth					Length	Latitude	Departure
A							+0.08	-0.06
	26° 10'					285.10	+255.88	+125.72
B							+0.18	-0.13
	104° 35'					610.45	-153.70	+590.78
C							+0.21	-0.15
	195° 30'					720.48	-694.28	-192.54
D							+0.06	-0.05
	358° 18'					203.00	+202.91	-6.02
E							+0.18	-0.14
	306° 54'					647.02	+388.48	-517.41
A
TOTALS						2466.05	-0.71	+0.53

For example, look at line AB.
correction in latitude = -total latitude misclosure / traverse perimeter * length of AB = -(-0.71 / 2466.05 * 285.1) = 0.08
correction in departure = -total departure misclosure / traverse perimeter * length of AB = -(0.53 / 2466.05 * 285.1) = -0.06

Once you have calculated the correction factors, simply add these to the original latitudes and departures to get balanced latitude and departure values..

					Balanced	Balanced
Station	Latitude			Departure	Latitude	Departure
A	+0.08			-0.06
	+255.88			+125.72	+255.96	+125.66
B	+0.18			-0.13
	-153.70			+590.78	-153.52	+590.65
C	+0.21			-0.15
	-694.28			-192.54	-694.07	-192.69
D	+0.06			-0.05
	+202.91			-6.02	+202.97	-6.07
E	+0.18			-0.14
	+388.48			-517.41	+388.66	-517.55
A
TOTALS			-0.71	+0.53	0.00	0.00

For example, again look at line AB
original latitude AB + correction = 255.85 + 0.08 = 255.96
original departure AB + correction = 125.72 + (-0.06) = 125.66
Also make sure that your balanced latitudes and departures sum to zero, respectively.

PROCEDURE FOR TRAVERSE CALCULATIONS

TRAVERSE CALCULATIONS

PROCEDURE FOR TRAVERSE CALCULATIONS

Adjust angles or directions

Determine bearings or azimuths

Calculate and adjust latitudes and departures

Calculate rectangular coordinates

BALANCING ANGLES OF CLOSED TRAVERSES

An example of a calculation involving interior angles is available.

ADJUSTING ANGLES

Adjustments applied to angles are independent of the size of the angle

Methods of adjustment:

Make larger corrections where mistakes were most likely
Apply an average correction to each angle
Or a combination

Never make an adjustment that is smaller than the measured accuracy

DETERMINING BEARINGS OR AZIMUTHS

Requires the direction of at least one line within the traverse to be known or assumed

For many purposes, an assumed direction is sufficient

A magnetic bearing of one of the lines may be measured and used as the reference for determining the other directions

For boundary surveys, true directions are needed

LATITUDES AND DEPARTURES

The latitude of a line is its projection on the north-south meridian and is equal to the length of the line times the cosine of its bearing

The departure of a line is its projection on the east-west meridian and is equal to the length of the line times the sine of its bearing

The latitude is the y component of the line and the departure is the x component of the line

LATITUDES AND DEPARTURES

CLOSURE OF LATITUDES AND DEPARTURES

The algebraic sum of all latitudes must equal zero or the difference in latitude between the initial and final control points

The algebraic sum of all departures must equal zero or the difference in departure between the initial and final control points

CALCULATION OF LATITUDES AND DEPARTURES

Using bearings

Station
Bearing
Length
Latitude
Departure
A
N 26° 10'E
285.10
+255.88
+125.72
B
S 75° 25'E
610.45
-153.70
+590.78
C
S 15° 30'W
720.48
-694.28
-192.54
D
N 1° 42'W
203.00
+202.91
-6.02
E
N 53° 06'W
647.02
+388.48
-517.41
A
MISCLOSURE
-0.71
+0.53

CALCULATION OF LATITUDES AND DEPARTURES

Using azimuths

Station
Azimuth
Length
Latitude
Departure
A
26° 10'
285.10
+255.88
+125.72
B
104° 35'
610.45
-153.70
+590.78
C
195° 30'
720.48
-694.28
-192.54
D
358° 18'
203.00
+202.91
-6.02
E
306° 54'
647.02
+388.48
-517.41
A
MISCLOSURE
-0.71
+0.53

ADJUSTMENT OF LATITUDES AND DEPARTURES

Compass (Bowditch) Rule 

ADJUSTMENT OF LATITUDES AND DEPARTURES

Station
Azimuth
Length
Latitude
Departure
A
+0.08
-0.06
26° 10'
285.10
+255.88
+125.72
B
+0.18
-0.13
104° 35'
610.45
-153.70
+590.78
C
+0.21
-0.15
195° 30'
720.48
-694.28
-192.54
D
+0.06
-0.05
358° 18'
203.00
+202.91
-6.02
E
+0.18
-0.14
306° 54'
647.02
+388.48
-517.41
A
TOTALS
2466.05
-0.71
+0.53

ADJUSTMENT OF LATITUDES AND DEPARTURES

Balanced
Balanced
Station
Latitude
Departure
Latitude
Departure
A
+0.08
-0.06
+255.88
+125.72
+255.96
+125.66
B
+0.18
-0.13
-153.70
+590.78
-153.52
+590.65
C
+0.21
-0.15
-694.28
-192.54
-694.07
-192.69
D
+0.06
-0.05
+202.91
-6.02
+202.97
-6.07
E
+0.18
-0.14
+388.48
-517.41
+388.66
-517.55
A
TOTALS
-0.71
+0.53
0.00
0.00

RECTANGULAR COORDINATES

Rectangular X and Y coordinates of any point give its position with respect to a reference coordinate system

Useful for determining length and direction of lines, calculating areas, and locating points

You need one starting point on a traverse (which may be arbitrarily defined) to calculate the coordinates of all other points

A large initial coordinate is often chosen to avoid negative values, making calculations easier.

CALCULATING X AND Y COORDINATES

Given the X and Y coordinates of any starting point A, the X and Y coordinates of the next point B are determined by:

COORDINATES

Balanced
Balanced
Station
Latitude
Departure
Y-coord
X-coord
A
10000.00
10000.00
+255.96
+125.66
B
10255.96
10125.66
-153.52
+590.65
C
10102.44
10716.31
-694.07
-192.69
D
9408.37
10523.62
+202.97
-6.07
E
9611.34
10517.55
+388.66
-517.55
A
10000.00
10000.00
TOTALS
0.00
0.00

LINEAR MISCLOSURE

The hypotenuse of a right triangle whose sides are the misclosure in latitude and the misclosure in departure.

TRAVERSE PRECISION

The precision of a traverse is expressed as the ratio of linear misclosure divided by the traverse perimeter length.

expressed in reciprocal form

Example

0.89 / 2466.05 = 0.00036090
1 / 0.00036090 = 2770.8

Precision = 1/2771