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3. Point-to-point computations

Point-to-point 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 A-8.

Equation A-1

Equation A-2

Figure A-8
Forward computation


Equations A-1 and A-2 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 (–)

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.

b. Inverse Computation

An Inverse Computation determines the distance and direction between two coordinate pairs, Figure A-9.

Figure A-9
Inverse computation



Equation A-3

Equation A-4

Equation A-5

Equation A-6

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 A-10.

Figure A-10
Direction quadrants

Table A-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

When ΔN = 0, Equation A-7 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 A-11.

when ΔE is (+), Az = 90°

when ΔE is (-), Az = 270°

Figure A-11
 When ΔN = 0

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 A-12:

Figure A-12
Forward Example 1

Because the bearing is South and East, the Lat is negative and Dep positive. From equations A-1 and A-2:

(2)  Forward 2

Compute the coordinates of point R given the information in Figure A-13:

Figure A-13
Forward Example 2

Because the direction is an azimuth, Equations A-1 and A-2 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 A-14.

Figure A-14
Inverse Example

Substitute the coordinates into Equations A-3 and A-4 (remember, it's to minus from):

Use Equation A-5 to compute the length:

Compute β, the angle from the meridian:

Because ΔN is negative and ΔE is negative, the direction is in the South-West quadrant, so add 180° to β.

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