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2. Inverse Computation

An Inverse Computation determines the distance and direction between two coordinate pairs.

img33
Figure C-2
Inverse Computation

 

img34            Equation C-2
 img37   Equation C-3
img38   Equation C-4

 

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. 

img36
Figure C-3
Quadrants

 

Table C-1

 

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 counter-clockwise 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 C-4 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 C-4.

ΔE (+) → Az=90° ΔE (-) → Az=270°
img39 img40
Figure C-4
When ΔN = 0

 

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