## 1. Forward Computation

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

 Figure C-1 Forward Computation Equation 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.

Because point B has two unknowns (NB and EB) it must be connected to point A with two measurments,  distance and direction. Having only one or the other isn't sufficient to determine B's coordinates.

## 2. Inverse Computation

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

 Figure C-2 Inverse Computation

 Equation C-2 Equation C-3 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.

 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° Figure C-4 When ΔN = 0

## 3. Area Computation

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

 Figure C-5 Polygon Areas

can be computed using the coordinates of its vertices:

 Equation C-5

In surveying terms using North and East coordinates:

 Equation 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. Cross multiply. Sum cross-products

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

 Equation 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.

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