1. Forward Computation

A Forward Computation is used to calculate a point's coordinates from distance and direction data from another set of coordinates:

img31 img32
Figure C-1
Forward Computation
Eqn (C-1)

 

Eqn (C-1) is a combination of the latitude and departure and coordinate equations from the Traverse Computations topic.

  • North Lat is (+), South Lat is (–)
  • East Dep is (+), West Dep is (–)

DirAB 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.

2. Inverse Computation

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

img33
Figure C-2
Inverse Computation

 

img34             Eqn (C-2)
 img37   Eqn (C-3) 
img38   Eqn (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, Eqn (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

 

3. Area Computation

The area of a non-crossing closed polygon, Figure C-5,

img91a
Figure C-5
Polygon Areas

 

can be computed using the coordinates of its vertices:

img91bnew         Eqn (C-5)

 

In surveying terms using North and East coordinates:

img91cnew        Eqn (C-6)

 

An easy way to remember either equation is graphically:  

  X & Y E & N

List coordinates in order around exterior.

Repeat first coordinate pair at end.

img92ab img92bb

Cross multiply.

 

 

img92a img92b
Sum cross-products img98b

 

 Add the cross multiplication sums, divide by two, and take the absolute value to obtain the area:

img98a       Eqn (C-7)

 

The absolute value is used because area could be positive or negative depending on traverse configuration, direction around it, and coordinate order (eg, E & N or N & E). Positive or negative, the area magnitude will be correct.