1. Meridian

A meridian is a north-south reference line. It is used as a basis for a direction, which describes a line’s orientation.

Common meridians include, but are not limited to, Figure C-1:

  • True
  • Magnetic
  • Astronomic
  • Grid
  • Geodetic
  • Assumed

c 01

Figure C-1
Meridians

`

Depending on the system, meridians may converge or may stay parallel, Figure C-2.

c 02a

c 02b

(a) Typical of Grid and Assumed systems, meridians are parallel. (b) In a larger earth-based system, such as True, Astronomic, or Geodetic, meridians converge to a point (e.g., North Pole).

Figure C-2
Parallel and Converging Meridians

 

For small project areas such as a single lot survey, meridians can assumed to be parallel, regardless of system, without introducing significant error.

Most meridian systems (even Assumed) are constant over time. However, magnetic meridians change location and their change is not consistent. Magnetic meridians are important as the first instrument extensively used for property surveys was the compass. Although True directions were reported in the early Public Land Surveys, they were first measured by compass then converted using solar observation.

A surveyor will select an appropriate meridian for the project at hand and orient all survey lines to it. Sometimes directions based on one meridian must be converted to a different meridian in order to maintain consistency across surveys. We’ll look at that, along with magnetic direction conversions, in Meridian Conversions later.

We will use parallel meridians for the remainder of this discussion - convergency issues will be covered in a later Topic.


2. Direction

A direction is angle from a meridian to a line. It is similar to a horizontal angle in a traverse except the backsight is always along the meridian, Figure C-3.

Figure C-3
Direction

 

There are two different ways to express line direction: bearing and azimuth.

a. Bearing

A bearing is an angle from the North or South end of the meridian turned to the East or West.  A bearing has three parts:

      • Prefix - N or S indicating which end of the meridian is turned from.
      • Angle
      • Suffix - E or W indicating turning direction from the meridian to the line.

N 66°40' E - from the North end of the meridian, turn 66°40' to the East.

Example

Bearing AB = N 66°40' E

Bearing AC = S 55°32' E

Bearing AD = S 44°21' W

Figure C-4
Bearings
 

 

A bearing falls in one of four quadrants so the angle does not exceed 90°. The angle is to the right (clockwise) in the NE and SW quadrants, to the left (counter-clockwise) in the SE and NW quadrants. A due North direction can be expressed as either N 00°00' E or N 00°00' W; due East as  N 90°00' E or S 90°00' E; similarly for dues South and West.

A back-bearing is reverse of a bearing, that is, Bearing BA is the back-bearing of Bearing AB. Because the meridians are parallel at both ends of the line, the bearing angle is the same but quadrant is reverse. This is true only when meridians are parallel. Where meridians converge, the forward and back bearing angles will differ by the total convergence. More on this later.

  

 

 

Bearing AB = N 66°40' E


Bearing BA = S 66°40' W

Figure C-5
Back Bearing
 

 

b. Azimuth

An azimuth is an angle to the right (clockwise) from the meridian to the line. In most cases the azimuth is turned from the north meridian end; earlier control surveys used the south end. An azimuth varies from 0° to 360°.

Example

Azimuth AB = 66°40'

Azimuth AC = 124°28'

Azimuth AD = 224°21'

Azimuth AE = 322°26'

Figure C-6
Azimuths
 

 

A back-azimuth is reverse of a azimuth: Azimuth CA is the back-azimuth of Azimuth AC. Because the meridians are parallel at both ends of the line, the back-azimuth and forward azimuth differ by 180°. As with bearings, this is true only when meridians are parallel. Where meridians converge, the forward and back azimuths will differ by (180° ± total convergence). More on this later.

Example

 

 

 

Azimuth AC = 124°28'

Azimuth CA = 124°28' + 180°00' = 304°28'

Figure C-7
Back Azimuth

 

 

c. Converting Between Bearings and Azimuths

Since Bearings and Azimuths are both referenced to a meridian it is simple to convert one to the other.

To convert from bearings to azimuths:

Table C-1
Quadrant From Bearing To Azimuth
NE N β E β
SE S β E 180° - β
SW S β W 180° + β
NW N β W 360° - β

Example

Azimuth AB = 66°40'

Azimuth AC = 180°00' - 55°32' = 124°28'

Azimuth AD = 180°00' + 44°21' = 224°41'

Azimuth AE = 360°00' - 37°34' = 322°26'

Figure C-8
Azimuths from Bearings
 

 

To convert from an azimuth, α, to a bearing:

Table C-2
Quadrant To Bearing

NE

N α E

SE

S (180° - α) E

SW

S (α - 180°) W

NW

N (360° - α) W

Example

Bearing AB = N 64°40' E

Bearing AC = S (180°00'-124°28') E = S 55°32' E

Bearing AD = S (224°21' - 180°00') W = S 44°21' W

Bearing AE = N (360°00' - 322°26') W = N 37°34' W

Figure C-9
Bearings from Azimuths

 

 

Rather then memorize tables, drawing a sketch will help determine correct conversion logic to use.

d. Bearing or Azimuth?

Which one should a surveyor use to express directions? It shouldn't matter as both express the same thing abeit using a different format.

The bearing's biggest advantage is that it's immediately recognizable as a dirrection: N 24°18'30"E, S 56°05'24"W. An azimuth, on the other hand looks like an angle (which it is) with no indication what it represents unless it's specifically called out as a direction: 242°45'36" vs 242°45'36" Az.

Bearings are a more traditional way of expressing directions, espacially in property surveys. Check any metes and bounds description or subdivision plats and chances are 10 times out of 9 the bearings will be used for directions.

Azimuths have a computational edge. Bearing angles are limited to a maximun of 90° and can be clockwise or counterclockwise measured from either end of the meridian. Azimuths always start from North and are clockwise. As we'll see shortly, that makes them easier to compute going around a traverse. And we'll see later how azimuths are a little more efficient for other traverse computations.

So which direction format to use is basically up to the surveyor.


3. Angles and Directions

a. Angles to Directions

Starting with a direction for one traverse line, directions of the others can be computed from the horizontal angles linking them. The process of addition or subtraction is dependent on the type of horizontal angle (interior, deflection, etc), turn direction (clockwise or counterclockwise), and direction type (bearing or azimuth).

(1) Examples - Bearing

Example 1

The bearing of line GQ is S 42°35' E. The angle right at Q from G to S is 112°40'. What is the bearing of the line QS?

Sketch:

;

Figure C-10

 

Add meridian at Q and label angles:

;

Figure C-11

 

At Q, the bearing to G is N 42°35' W.

Subtracting 42°35' from 112°40' gives the angle, β, from North to the East for line QS.

Bearing QS = N 70°05" E

Example 2

The bearing of line LT is N 35°25' W. The angle left at T from L to D is 41°12'. What is the bearing of the line TD?

Sketch:

 Figure C-12

 

Add meridian at T and label angles

 Figure C-13

 

At T the bearing TL is S 35°25' E.

The bearing angle, σ, is 35°25' + 41°12' = 76°37'

Bearing TD = S 76°37' E.

(2) Examples - Azimuth

Example 1

The azimuth of line WX is 258°13'. At X the deflection angle from W to L is 102°45' L. What is the azimuth of line XL?

Sketch:

Figure C-14

 

A deflection angle is measured from the extension of a line. The azimuth of the  extension is the same as that of the line. To compute the next azimuth, the deflection angle is added directly to the previous azimuth.

Because this is a left deflection angle, you would add a negative angle.    

Add meridian at X:

Figure C-15

 

Azimuth XL = 258°13' + (-102°45') = 155°28'

Example 2

The azimuth of line BP is 109°56'. The angle right at P from B to J is 144°06'. What is the azimuth of line PJ?

Sketch:

Figure C-16

 

At P, add the meridian and extend the line BP

Figure C-17

 

To get the azimuth of line PB: 109°56' + 180°00'= 289°56'.

Since 144°06' is to the right (+), add it to the azimuth of PB to compute azimuth of PJ: 144°06' + 289°56' = 434°02'

Figure C-18

 

Why is the azimuth greater than 360°? Because we've gone past North.

To normalize the azimuth, subtract 360°00': 434°02' - 360°00' = 74°02'

 

b. Directions to Angles

Given directions of two adjacent lines, it is a simple matter to determine the angle between the lines.

(1) Example - Bearings

The bearing of line HT is N 35°16' W , the bearing of line TB is N 72°54' E. What is the angle right at T from B to H?

Sketch:

Figure C-19

Label the back-direction at T and angle to be computed, δ.

Figure C-20

 

Based on the sketch, the desired angle is what’s left over after both bearing angles are subtracted from 180°00'.

δ = 180°00' - (72°54' + 35°16') = 71°50"

(2) Example - Azimuths

The azimuth of line MY is 106°12', the azimuth of line YF is 234°06'. What is the angle right at Y from F to M?

Sketch:

Figure C-21

 

Label the back-azimuth at Y and angle to be computed, ρ.

Figure C-22

 

ρ = 286°12' - 234°06' = 52°06'

 

c. Traverse Direction Computations

(1) Example - Bearings

Given the following traverse and horizontal angles:

Figure C-23
Bearing Traverse

 

Using a bearing of N 36°55' E for line AB, determine the bearings of the remaining lines clockwise around the traverse.

Many survey texts use a tabular approach to compute traverse line directions from angles. For the beginner this can be confusing and lead to erroneous directions. It helps to instead draw sketches in order to visualize the line relationships.

At point B:  

Figure C-24

Label the bearing angle, 36°55', from B to A.

Subtract it from 117°19'.

β = 117°19' - 36°55' = 80°24'Brg BC = S 80°24' E

 

At point C:

 

Figure C-25

Label the bearing angle, 80°24', from C to B.

Subtract it along with 87°42' from 180°00' tooo get bearing angle CD.

γ = 180°00' - (80°24' + 87°42') = 11°54'

Brg CD = S 11°54' W

 

At point D:

 

Figure C-26

Label the bearing angle, 11°54', from D to C.

Subtract it from 93°38' to obtain next bearing angle.

&eta= 93°38' - 11°54' = 81°44'Brg DA = N 81°44' W

 

The directions for all four traverse lines have been computed. Angles at B, C, and D have been used, but that at A has not. For a math check, use the Bearing of DA and the angle at A to compute the bearing we started with.

Figure C-27

Label the bearing angle, 81°44', from A to D.

Subtract it along with 61°21' from 180°00' to get next bearing angle.

α = 180°00' - (61°21' + 81°44') = 36°55

Brg AB = N 36°55' E  check

 

If our computed and initial bearings for AB don’t match it means one of two things:

        • there is a math error in our computations, or,
        • the interior angles weren’t balanced.

For this traverse the angles sum to 360°00' so there is no angular misclosure. If our math check had failed it would have been due to a math error in our computations.

Had the angles not been balanced and if there were no math errors, the math check would be off by the angular misclosure.

Summary:

Line Bearing
AB N 36°55' E
BC S 80°24' E
CD S 11°54' W
DA N 81°44' W

 

(2) Example - Azimuths

Given the following traverse and horizontal angles:

Figure C-28
Azimuth Traverse

 

Using a azimuth of 68°00' for line OP, determine the azimuths of the remaining lines counter-clockwise around the traverse. 

 

At point P:

 

Figure C-29

Line PQ is 92°48' right of Azimuth PO

Az PO is the back azimuth of Az OP

Az PQ = (Az OP + 180°00') + Angle PAz PQ

= (68°00' + 180°00') + 92°48' = 340°48'

 

At point Q:

 

Figure C-30

Line QR is 112°26' to the right from Az QP

Az QP is the back azimuth of Az PQ

Az QR = (Az PQ + 180°00') + Angle QAz QR

= (340°48'+180°00') + 112°26' = 633°14'

Normalize: Az QR = 633°14' - 360°00' = 273°14'

 

At point R:

 

Figure C-31

Line RO is 67°14' right from Az RQ

Az RQ is the back azimuth of Az QR

Az RO = (Az QR + 180°00') + Angle RAz RO

 = (273°14' + 180°00') + 67°14' = 520°28'

Normalize: Az RO = 520°28' - 360°00' = 160°28'

 

The directions for all four traverse lines have been computed. Angles at P, R, and R have been used, but not the angle at O. For a math check, use Azimuth RO and the angle at O to compute the original Az OP.

 

At point O:

 

c 25

Figure C-32

Line OP is 87°32' right of Az OR

Az OR is the back azimuth of Az RO

Az OP = (Az RO + 180°00') + Angle O

Az OP = (160°28' + 180°00') + 87°32' = 428°00'

Normalize: Az OP = 428°00' - 360°00' = 68°00' check!

 Summary:

Line Bearing
OP 68°00'
PQ 340°48'
QR 273°14'
RO 160°28'

 

There's a distinct pattern computing these azimuths:

    New Az = (Previous Az + 180°00') + Angle.

This is true for a loop traverse meeting these conditions:

        • Directions are counter-clockwise around the traverse, and,
        • Angles are interior to the right.

What if the directions are clockwise around the traverse and interior angles counter-clockwise?

    New Az = (Previous Az - 180°00') - Angle

Other patterns exist for clockwise travel with clockwise interior angles, clockwise exterior angles, counterclockwise exterior angles, etc. Rather than memorize the possible patterns, draw a sketch, and begin computing; the pattern will present itself after a few lines.