Calculating the traverse that represents a parcel can benefit significantly from something that is not properly taught: The use of azimuths.

What is an azimuth? It is a horizontal angle of a line, with the zero at astronomic north, and measured clockwise. It can range from 0 to 360 degrees. Figure 1 shows that line AB has an astronomic direction of AzAB.

Those familiar with bearings will agree that the continuous handling of bearings in calculations is a pain. If all necessary bearings of a traverse are first converted to decimal azimuths, calculations become much easier.

calculating a traverse using azimuths - Figure 1
Figure 1

Let’s assume that line AB in Figure 1 has a bearing of N 60° 50’23” E. This converts to an azimuth of (60 + 50/60 + 23/3600) = 60.839722°. I like to use six digits for all angles. If I now have another traverse leg to a point C (as shown in Figure 2), and I measured an angle of, say, 70° 00’00” (or 70.000000°), what would be the azimuth of line BC? It is the azimuth of line AB, plus 180° (to get the direction opposite to line AB, or direction BA), minus the measured angle at B: AzBC = AzAB + 180° - 70.000000° = 60.839722° + 180° - 70.000000° = 170.839722°.

calculating a traverse using azimuths - Figure 2
Figure 2

So, that is the azimuth of line BC. Let’s now add the north arrow at point B and draw the azimuth angle for line BC (see Figure 3). This shows that the azimuth of a line shows as much directionality as a bearing does. And, operations with Decimal-Degree (DD) format azimuths are much faster than with Degree-Minutes-Seconds (DMS) format bearings. In addition, the calculations and adjustments of closed traverses are more direct.

calculating a traverse using azimuths - Figure 3
Figure 3

There are two things that need to be heeded. We come out of high school with the idea that a zero direction is along the X (east) axis, while azimuths have the zero direction along the Y (north) axis. This switches the trigonometry, as follows. Look at Figure 4, where I isolated line BC of the traverse. The departure (west-to-east) and latitude (south-to-north) components of line BC are shown as dashed lines and have the following values (using horizontal distance d): Departure = d*sin (AzBC) and Latitude = d*cos (AzBC). This goes contrary to high school trig because the sine is supposed to go up-and-down, not left-and-right. However, if we set the distance d = 500.00’, and calculate the latitude and departure values, the benefits of azimuths become apparent: Departure = 500.00’ x sin(170.839722°) = 79.60’ and Latitude = 500.00 x cos(170.839722°) = -493.62’. The negative value for the latitude tells us that we are going more south than north.

calculating a traverse using azimuths - Figure 4
Figure 4

One of my favorite tools for calculations is Microsoft Excel, if I remember that it does not accept angle input in DMS format, but only in radians. Therefore, in Excel, the following formulae must be entered: Departure = 500.00’ * sin(radians(170.839722°)) and Latitude = 500.00 * cos(radians(170.839722°)). If one wants to convert radians back to angles, one uses the degrees function, like alpha = degrees (angle in radians).

Then, adjusting a closed traverse simply becomes a matter of adding up all those positive and negative departures to get a departure misclosure, adjusting departures, and calculating easting coordinates. This is repeated for latitudes and northing coordinates.

With this, I hope to have illustrated that geodesy is much more than just projections.

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