F. Coordinates
1. General
Point position can be expressed in relative or absolute terms.
a. Relative position
A relative position is the either the (a) length and direction, or, (b) latitude and departure between two points.
Give the adjusted traverse of Figure F1:
Figure F1 Adjusted Loop Traverse 
Lengths and directions around a traverse define the relative locations of successive traverse points.
Point B is 472.72 feet from point A at a bearing of S 68°05'27"W.
Point C is 216.12 feet from point B at a bearing of N 19°46'15"W.
In terms of latitudes and departures:
Point B is 176.39 feet South and 438.57 feet West of point A.
Point C is 203.38 feet North and 73.10 feet West of point B.
Where is point C relative to point A? Because the two points are not directly connected on the traverse, it requires a little more computing.
Lat A to C: [472.72 ft x cos(68°05'27") + 216.12 ft x cos(19°46'15")] = +26.99 ft
Dep A to C: [472.72 ft x sin(68°05'27")  216.12 ft x sin(19°46'15")] = 511.68 ft
Point C is 28.99 ft North and 511.68 ft West of point A. We could compute the lats and deps through point D instead of point B  the distances from point A to point C would be the same.
b. Absolute Position
An absolute position is a distance from a datum. In the case of a traverse point, two horizontal lines serve as the data. One line corresponds with the meridian, the other is perpendicular to it, Figure F2.
Figure F2 Horizontal Datum 
The meridional line is called either the Y or North (N) axis; the other the X or East (E) axis.
A point position is expressed as a coordinate pair are represent perpendicular distances from the two axes.
For example:
In terms of an X and Y system, Figure F3, the coordinates of point P are X=225.64' and Y=320.95'  
Figure F3 X and Y System 

In terms of an North and East system, Figure F4, the coordinates of point P are E=225.64' and N=320.95' 

Figure F4 North and East System 
2. Coordinates Computations
a. Forward Computation
A forward computation uses a starting coordinate pair along with a distance and direction to determine another coordinate pair.
In Figure F5, starting with coordinates at P, compute the coordinates at Q.
Figure F5 Forward Computation 
Using Equations D1 and D2, the latitude and departure of the line are:
LatPQ= LPQx cos(DirPQ) DepPQ = LPQ x sin(DirPQ) 
L: line length 
To compute X and Y coordinates:
Y_{Q} = Y_{P} + Lat_{PQ} X_{Q} = X_{P} + Dep_{PQ} 
Equations F1 and F2 
To compute N and E coordinates:
NQ = NP + LatPQ EQ = EP + DepPQ 
Equations F3 and F4 
For a complete traverse, Figure F6:
Figure F6 Coordinates Around a Loop Traverse 
Starting with known coordinates at T (N_{T}, E_{T}) and applying Equations F3 and F4 around the traverse:
Compute coordinates of Q:  
Compute coordinates of R:  
Compute coordinates of S:  
Compute coordinates of T: 
Computing back into T gives a math check: the end coordinates should be the same as the start coordinates.
In order for the math check to be met, adjusted lats and deps must be used.
Where do the start coordinates come from? They can be assumed or they could be from a formal coordinate system. We'll discuss formal coordinate systems in a later topic.
b. Inverse Computation
An inverse computation is used to determine the distance and direction between two coordinate pairs. The computations involved are basically the same as those for determining a line's new length and direction from its adjusted lats and deps.
For the traverse shown in Figure F7, how would we determine the length and direction of the line from point T to point R?
Figure F7 Length and Direction Between Nonadjacent Points 
Knowing the coordinates of the two points, we can determine the latitude and departure of the line from the coordinate differences, Figure F8.
Figure F8 
Equations F5 and F6 
Note that the differences are the To point values minus the From point values.
Equation F7  
and  
Equation F8  
where  
90° ≤ ß ≤ 90° 
The mathematic signs on the coordinate differences determine the direction quadrant, Figure F9.
Figure F9 Converting ß to a Direction 
If X and Y coordinates are used, remember that Y corresponds to N and X corresponds to E.
3. Examples
a. Traverse 1
(1) Forward Computation
Figure F10 
Adjusted  
Line  Lat (ft)  Dep (ft) 
AB  176.386  438.574 
BC  +203.382  73.105 
CD  +192.340  +198.635 
DE  219.336  +313.044 
The coordinates of point A are 500.000' N, 2000.000' E. Compute the coordinates of the remaining points.
A simple way is to set up a table with North coordinates and latitudes in one column, East coordinates and departures in another.
(2) Inverse Computation
What are the length ad bearing of the line A to C in the diagram below?
Figure F11 Length anf Direction Determination 
From Equations F5 and F6:
Substituting into Equation F7
and Equation F8
Because ΔN is North and ΔE is West: N86°58'47.6"W
Line AC: 512.39' at N86°58'48"W
b. Traverse 2
(1) Forward Computation
The crossing traverse in Figure F12 was previously adjusted with the results shown below.
Figure F12 Crossing Traverse Forward Computation 
The coordinates of point E are 200.000' X, 1000.000' Y.
Compute the coordinates of the remaining points.
Arranging the computations in a table:
(2) Inverse Computation
Determine the length and azimuth of the line from point F to point H.
Remember: Y=>N, X=>E; and it's To minus From.
Because ΔY is North and ΔX is West:
Line FH: 275.25' and azimuth of 270°40'49".
4. Closing
Although including coordinates involves addition computations, they provide ability to indirectly determine other values. We saw examples here where the distance and direction can be determined between points not directly connected on a traverse. This is very useful where lines may be obstructed but we still need their length and direction. We will see in the next section how coordinates facilitate area computation.
Later still in the Coordinate Geometry topic, we'll see how integrating forward and inverse computations with trigonometric principles allow for greater field measurement and computational flexibility.