1. Coordinate System
A coordinate system is a datum for expressing point positions. The system consists of two or three reference axes perpendicular to each other.
In a two dimensional system, one axis is oriented in the direction of a meridian defining North, the remaining axis being East, Figure A1.
Figure A1 
In a three dimensional system, the third axis is oriented vertically, in the direction of elevation, Figure X2.
Figure A2 
This section will deal primarily with two dimensional systems although the math involved can be easily expanded into the third dimension.
2. Positions
a. Absolute: Coordinates
The absolute position of a point is expressed as coordinates which are perpendicular distances from the reference axes. In surveying and most mapping, the reference system uses North and East axes so position is expressed as (North, East) (aka, Northing and Easting), Figure A3. Other applications, notably math, use X and Y axes; Y coinciding with the meridian. Positions are expressed as (X, Y), Figure A4.
Figure A3 
Figure A4 
Caution must be exercised when using coordinates provided by others. Are (2000 ft,1000 ft) NorthEast or XY coordinates? A common error when importing coordinates in software is reversing their order: this flips and rotates the positions. This effect is demonstrated in Figures A5 and A6. The former uses coordinates in (N, E) order while the latter in (X, Y).
Figure A5 
Figure A5 
We will use the NorthEast coordinate form unless otherwise specified.
b. Relative
A relative position is where one point is with respect to another. This can be expressed as coordinate differences, Figure A6, or as distance and direction, Figure A7.
Figure A6 
Figure A7 
3. Pointtopoint computations
Pointtopoint computations involve just two points
a. Forward Computation
A Forward Computation is used to calculate a point's coordinates from another set of coordinates using distance and direction data between them, Figure A8.
Equation A1 

Equation A2 

Figure A8 

Equations A1 and A2 are a combination of the latitude and departure and coordinate equations from the Traverse Computations chapter:
 North Lat is (+), South Lat is (–)
 East Dep is (+), West Dep is (–)
Dir_{AB} can be either a bearing or azimuth:
 If a bearing (0° to 90°) is used you must manually determine the correct algebraic sign for the Lat and Dep based on quadrant.
 Using azimuths (0° to 360°) automatically results in correct signs.
b. Inverse Computation
An Inverse Computation determines the distance and direction between two coordinate pairs, Figure A9.



Equation A3 

Equation A4 

Equation A5 

Equation A6 
Coordinate differences, Δ’s, are the to point minus the from point: going from point C to point D means subtracting C's coordinates from D's.
The algebraic sign on β and the resulting direction depend on the quadrant of the line. A positive angle is clockwise (to the right); a negative angle is counterclockwise (to the left). Both are from the north or south end of the meridian, Figure A10.
Figure A10 
Table A1 

Algebraic sign  Direction  
Quadrant  ΔN  ΔE  β  Bearing  Azimuth 
NE  +  +  +  N β E  β 
SE    +    S β E  180°+β 
SW      +  S β W  180°+β 
NW  +      N β W  360°+β 
When ΔN = 0, Equation A7 has no solution. Technically division by 0 is undefined, but actually the result of any number divided by 0 is infinity. So what does this mean? In surveying terms when ΔN = 0 the entire line length is ΔE resulting in a due East or West line, Figure A11.
when ΔE is (+), Az = 90° 
when ΔE is (), Az = 270° 
Figure A11 
If you check the tangent of 90° and 270° on your calculator you'll get either an error or "undefined" response. Try tan(89.99999°); you'll get a huge number. Tan(90°) = tan(270°) = infinity.
c. Examples
(1) Forward 1
Compute the coordinates of point R given the information in Figure A12:
Figure A12 
Because the bearing is South and East, the Lat is negative and Dep positive. From equations A1 and A2:
(2) Forward 2
Compute the coordinates of point R given the information in Figure A13:
Figure A13 
Because the direction is an azimuth, Equations A1 and A2 will automatically compute the correct signs for the Lat and Dep.
(3) Inverse
What are the lengtha and azimuth from point J to K?
Point  North (ft)  East (ft) 
J  1153.65  704.08 
K  988.85  200.75 
Draw a sketch to visualize the line, Figure A14.
Figure A14 
Substitute the coordinates into Equations A3 and A4 (remember, it's to minus from):
Use Equation A5 to compute the length:
Compute β, the angle from the meridian:
Because ΔN is negative and ΔE is negative, the direction is in the SouthWest quadrant, so add 180° to β.