2. Inverse Computation
An Inverse Computation determines the distance and direction between two coordinate pairs.
Figure C2 Inverse Computation 
Equation C2  
Equation C3  
Equation C4 
Coordinate differences, Δ’s, are the to point minus the from point.
The algebraic sign on ß and the resulting direction depend on the quadrant of the line.
Figure C3 Quadrants 
Table C1 


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 
A negative ß is a counterclockwise angle.
These should all look familiar as they're the same equations from the Coordinates chapter of the Traverse Computations topic.
When ΔN = 0, Equation C4 has no solution.Technically division by 0 is undefined, but actually the result of any number divided by 0 is infinity. Remember from the plot of the tangent function that tan(90°) = tan(270°) = 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 C4.
ΔE (+) → Az=90°  ΔE () → Az=270° 
Figure C4 When ΔN = 0 