Traverse Computations
Purpose
The purpose of this practicum is to use the field data obtained over the past several weeks to perform a set of traverse computations.
Procedure
In your field book, find and record the line distance measurements and adjusted interior angles of your traverse. If you were unable to obtain reasonable data for any line or angle, ask the instructor for a “fill-in” value.
The instructor will provide every group with the coordinates of one of your traverse hubs and also the direction of one of your traverse lines. With this data, complete the following traverse computations using a counter-clockwise computation direction. Use the computation sheets provided to assist in organizing your calculations.
1. Line Directions
You may compute the directions using sketches or the tabulation method. The easiest way to compute line directions in the tabulation method is to start with the known (given) direction and then apply the successive field angles around the traverse until all directions have been computed. Field angles must always be applied to the back azimuth of the previous line (rear line) using a positive sign. In general, the sign convention for angle addition can be stated as follows: When computing directions around the traverse in a counter-clockwise direction, add the field angles. When computing directions around the traverse in a clockwise direction, subtract the field angles. Please note that there is always a math check at the end of the direction computations and this check should be shown. Once the azimuths have been successfully computed, convert them to bearings.
Practice problem2. Latitudes and Departures
With the computed azimuths and measured line distances, compute the latitudes and departures of the traverse. The latitude of a line is its projection on a meridian line, or the north-south component of the line. Similarly, the departure of a line is its projection on a parallel, or the east-west component of the line. The following equations are used to compute latitudes and departures.
Latitude = (Horizontal Distance) cos (Azimuth)
Departure = (Horizontal Distance) sin (Azimuth)
Keep in mind that the cosine and sine functions of the azimuth will automatically give the proper sign of each latitude and departure. This is an advantage of using azimuths as opposed to bearings.
3. Linear error of closure (linear misclosure)
Determine the algebraic sum of the latitudes and departures, respectively. These two errors represent the linear errors in the traverse in the north and east directions. The total linear error in the traverse can be computed using the Pythagorean theorem:
4. Relative error of closure (traverse precision)
Once linear error of closure has been determined, the relative error of closure of the traverse can be easily computed using the following relation:
The relative error of closure is traditionally expressed in the form “1/X,” with X being some number with one or two significant figures. A ratio of 1/X is a descriptive number that provides some feeling for the overall accuracy of the traverse. This ratio can be read as “1 part measurement error to X parts of linear distance measured.”
5. Corrections to latitudes and departures using the compass rule
To force the traverse to close mathematically, corrections to the latitudes and departures must be performed. Several methods are available as described in Elementary Surveying: An Introduction to Geomatics, 10th Ed. The most commonly used method is the compass or Bowditch rule. In using this rule, the correction is computed for the latitude and departure of each line. When the corrected latitudes and departures are summed, they should equal zero if no math error has been made in the computations. The compass rule is given by the following equations:
CorrectionLatitude = -(Horizontal Distance)×(S Latitudes)/(Traverse Perimeter)
CorrectionDeparture = -(Horizontal Distance)×(S Departures)/(Traverse Perimeter)
Take notice of the minus sign in the expression. It is important to remember that the correction is always “minus times the error.” Use these equations to compute corrections.
6. Balanced latitudes and departures
Apply the corrections to balance the latitudes and departures.
7. Coordinates of each traverse hubWith balanced latitudes and departures, coordinates of each traverse hub can be computed. Starting with the coordinates of the known traverse hub, apply the latitude and departure of the adjoining counter-clockwise line in the traverse to determine the coordinates of the next point in the traverse. The specific relations used to compute coordinates are as follows:
XB = XA + (Balanced Departure)AB
YB = YA + (Balanced Latitude)AB
These equations are used repeatedly until all coordinates have been computed for all points in the traverse. It is important to apply the latitude and departure of the last line in the traverse to the last computed coordinate pair, comparing the results to the starting coordinates. This serves as a math check on the coordinate computation process.
8. Area of the traverse by coordinates
Determine the area of the closed traverse by first cross-multiplying the north (Y) coordinate of one point with the east (X) coordinate of the next point. These may be thought of as forward or “plus” cross-multiplications. Remember to perform as many “plus” cross-multiplications as you have lines in the traverse. Sum these products as the sum of “plus” areas. Once the forward operations are performed, cross multiply in the reverse or “minus” direction by multiplying the east coordinate of one point with the north coordinate of the next point. Again, remember to perform as many “minus” cross-multiplications as you have lines in the traverse. Sum these products as the sum of “minus” areas. The algebraic addition of these two sums will equal twice the area in the traverse. To get area, divide this sum by two.
9. Area of the traverse by Simple Geometric Figures
The area of any traverse can be determined by subdividing it into a set of unique triangles. The area of a three-sided traverse can be computed directly with the use of the trigonometric relation
where s = 0.5(a + b + c)
The quantities a, b, and c are the length of the sides of the traverse in consistent units. Compute the area of your traverse using this relation and compare the result against that found by method of coordinates.