How to find out angle of closure ?

Question

Examoogle · Accepted Answer

The correct answer is Option A: $	an	heta = \frac{\Sigma D}{\Sigma L}$. The angle of closure (or direction of closing error) in a closed traverse is the angle $	heta$ whose tangent is equal to the ratio of the total error in departures ($\Sigma D$) to the total error in latitudes ($\Sigma L$). It defines the angular bearing of the closing error line.

Step-by-Step Solution & Concepts:
The angle of closure (or direction of closing error) in a closed traverse is the angle $	heta$ whose tangent is equal to the ratio of the total error in departures ($\Sigma D$) to the total error in latitudes ($\Sigma L$). It defines the angular bearing of the closing error line.

Working Principle: When a closed traverse is plotted using field observations, the total sum of departures ($\Sigma D$) and total sum of latitudes ($\Sigma L$) should theoretically equal zero. Due to observational errors, $\Sigma D 
eq 0$ and $\Sigma L 
eq 0$. The resultant closing error $e$ is given by $e = \sqrt{(\Sigma D)^2 + (\Sigma L)^2}$, and its direction $	heta$ relative to the meridian is determined using $	an	heta = \frac{\Sigma D}{\Sigma L}$.

Key Formulas: $	an	heta = \frac{\Sigma D}{\Sigma L}$ — Angle of closure (direction of closing error); $e = \sqrt{(\Sigma D)^2 + (\Sigma L)^2}$ — Linear closing error; 	ext{Relative Closing Error} = \frac{e}{	ext{Perimeter}}$ — Precision of the traverse

Key Points: Latitude ($L$) is the projection of a traverse line on the North-South meridian ($L = l \cos	heta$).; Departure ($D$) is the projection of a traverse line on the East-West perpendicular ($D = l \sin	heta$).; For a perfectly closed traverse, $\Sigma L = 0$ and $\Sigma D = 0$.; The quadrant of $	heta$ depends on the algebraic signs of $\Sigma D$ and $\Sigma L$.

Applications: Determining the direction of error vector in total station and theodolite traversing.; Applying corrections using Bowditch's or Transit rule to adjust closed traverses.

Additional Info: Option B ($\tan\theta = \frac{D}{L}$) gives the bearing of an individual line, not the closing error of a traverse.. Option C ($\tan\theta = \frac{L}{D}$) represents the cotangent of the line's bearing angle.. Option D ($\tan\theta = \frac{\Sigma L}{\Sigma D}$) has the numerator and denominator inverted.

Answer: A is correct — The angle of closure $	heta$ is calculated using $	an	heta = \frac{\Sigma D}{\Sigma L}$, where $\Sigma D$ is total departure error and $\Sigma L$ is total latitude error.

Exam Tip: Remember that $	an	heta = \frac{	ext{East-West error}}{	ext{North-South error}} = \frac{\Sigma D}{\Sigma L}$. This is mathematically consistent with $	an	heta = \frac{y}{x}$ on a Cartesian plane where Departure corresponds to the X-axis and Latitude to the Y-axis.

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Option B: $	an	heta = \frac{D}{L}$

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Option C: $	an	heta = \frac{L}{D}$

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Option D: $	an	heta = \frac{\Sigma L}{\Sigma D}$

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