How to find out the length of closing error?

Question

Examoogle · Accepted Answer

The correct answer is Option A: $\sqrt{(\Sigma L)^2 + (\Sigma D)^2}$. The closing error (or error of closure) in a closed traverse represents the linear distance between the starting point and the ending point when the traverse fails to close geometrically. It is obtained by combining the algebraic sum of latitudes ($\Sigma L$) and the algebraic sum of departures ($\Sigma D$) using the Pythagorean theorem. Therefore, the length of the closing error $e$ is given by $e = \sqrt{(\Sigma L)^2 + (\Sigma D)^2}$.

Step-by-Step Solution & Concepts:
The closing error (or error of closure) in a closed traverse represents the linear distance between the starting point and the ending point when the traverse fails to close geometrically. It is obtained by combining the algebraic sum of latitudes ($\Sigma L$) and the algebraic sum of departures ($\Sigma D$) using the Pythagorean theorem. Therefore, the length of the closing error $e$ is given by $e = \sqrt{(\Sigma L)^2 + (\Sigma D)^2}$.

Working Principle: In a mathematically closed traverse, the sum of latitudes (north-south displacements) and the sum of departures (east-west displacements) should both equal zero. Due to observational errors, $\Sigma L 
eq 0$ and $\Sigma D 
eq 0$. These two orthogonal sums act as perpendicular components of a right-angled triangle, where the hypotenuse represents the magnitude of the closing error.

Key Formulas: $e = \sqrt{(\Sigma L)^2 + (\Sigma D)^2}$ — Length of Closing Error; $	heta = 	an^{-1}\left(\left|\frac{\Sigma D}{\Sigma L}ight|ight)$ — Direction (Reduced Bearing) of Closing Error; 	ext{Relative Precision} = \frac{e}{P}$ — Relative Error of Closure (where $P$ is the perimeter)

Key Points: Latitude ($L$) is the orthographic projection of a traverse line on the north-south meridian ($L = l \cos 	heta$).; Departure ($D$) is the orthographic projection of a traverse line on the east-west axis ($D = l \sin 	heta$).; For a closed loop traverse, ideally $\Sigma L = 0$ and $\Sigma D = 0$.; Bowditch's rule or Transit rule is commonly used to balance and adjust the closing error.

Additional Info: Option B incorrectly uses subtraction inside the square root.. Options C and D miss squaring the component sums before taking the square root.. Bowditch's Rule assumes linear errors are proportional to $\sqrt{l}$ and angular errors are inversely proportional to $\sqrt{l}$.

Answer: A is correct — The length of closing error is the hypotenuse formed by the total latitude error and departure error, given by $\sqrt{(\Sigma L)^2 + (\Sigma D)^2}$.

Exam Tip: Always determine the correct quadrant of the closing error based on the individual signs of $\Sigma L$ and $\Sigma D$ when calculating Whole Circle Bearing (WCB).

Examoogle · Answer

Option B: $\sqrt{(\Sigma L)^2 - (\Sigma D)^2}$

Examoogle · Answer

Option C: $\sqrt{\Sigma L + \Sigma D}$

Examoogle · Answer

Option D: $\sqrt{\Sigma L - \Sigma D}$

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