Solving Traverse Closing Error: Length and Bearing

A closed traverse should theoretically end at its starting point. When it doesn't, a closing error occurs, represented by the resultant of the sum of latitudes and departures.

Error Magnitude Calculation

The length of the closing error ($E$) is calculated using the Pythagorean theorem based on the total error in latitude ($\sum \text{Lat}$) and the total error in departure ($\sum \text{Dep}$).

Given:

• Sum of Latitudes ($\sum \text{Lat}$) = 1.39 m
• Sum of Departures ($\sum \text{Dep}$) = 2.17 m

The formula for the length of the closing error is:

$ E = \sqrt{(\sum \text{Lat})^2 + (\sum \text{Dep})^2} $

Substituting the given values:

$ E = \sqrt{(1.39 \text{ m})^2 + (2.17 \text{ m})^2} $

$ E = \sqrt{1.9321 \text{ m}^2 + 4.7089 \text{ m}^2} $

$ E = \sqrt{6.641 \text{ m}^2} $

$ E \approx 2.577 \text{ m} $

Rounding to two decimal places, the length is approximately 2.57 m.

Error Bearing Calculation

The Whole Circle Bearing (WCB) of the closing error is determined using the arctangent function. Since the sum of latitudes is positive (error towards North) and the sum of departures is positive (error towards East), the bearing is in the first quadrant.

The formula is:

$ \text{WCB} = \arctan\left(\frac{\sum \text{Dep}}{\sum \text{Lat}}\right) $

Substituting the values:

$ \text{WCB} = \arctan\left(\frac{2.17 \text{ m}}{1.39 \text{ m}}\right) $

$ \text{WCB} = \arctan(1.56115...) $

$ \text{WCB} \approx 57.37^\circ $

Rounding to the nearest degree, the WCB is approximately $57^\circ$.

Conclusion

The calculated length of the closing error is 2.57 m and its Whole Circle Bearing is approximately $57^\circ$.