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What is the formula for finding vertical distance if the staff is held vertical and line of sight is inclined?

$V = k s \sin 2 \theta 2 + c \sin \theta$

$V = k s \sin 2 \theta 2 + c \cos \theta$

$V = k s cos^{2} \theta + c \sin \theta$

$V = k s cos^{2} \theta + c \cos \theta$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilAdvanced Survey

Option A

$V = k s \sin 2 \theta 2 + c \sin \theta$

Quick Summary:

In tacheometry, when the line of sight is inclined at an angle $\theta$ to the horizontal and the leveling staff is held vertically, the inclined distance along the line of sight is given by $L = k s \cos\theta + c$ . The vertical component $V$ is obtained by multiplying this inclined distance by $\sin\theta$ , which yields $V = k s \cos\theta \sin\theta + c \sin\theta$ . Using the trigonometric identity $\sin 2\theta = 2\sin\theta\cos\theta$ , this reduces to $V = k s \frac{\sin 2\theta}{2} + c \sin\theta$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$V = k s \frac{\sin 2\theta}{2} + c \sin\theta$ — Vertical distance formula for inclined line of sight with staff held vertical

$D = k s \cos²\theta + c \cos\theta$ — Horizontal distance formula for inclined line of sight with staff held vertical

$k = \frac{f}{i}$ — Tacheometric multiplying constant (typically 100)

$c = f + d$ — Tacheometric additive constant (typically 0 for anallatic lens)

⚙️ Working Principle

The stadia method relies on solving similar isosceles triangles formed by the stadia hairs and the staff intercept. When the line of sight is inclined, the normal staff intercept is approximated by $s' = s \cos\theta$ . Multiplying the normal intercept by the multiplying constant $k$ gives the inclined slope length, whose vertical projection yields the vertical component $V$ through trigonometric resolution.

📌 Key Points

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The vertical component $V$ represents the vertical height from the horizontal instrument axis to the middle cross-hair reading on the staff.
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For an anallatic lens system, $c = 0$ , simplifying the formula to $V = \frac{k s \sin 2\theta}{2}$ .
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If the angle is an angle of elevation, Reduced Level (RL) of point = $\text{RL of Instrument Axis} + V - h$ , where $h$ is the central hair reading.
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If the angle is an angle of depression, Reduced Level (RL) of point = $\text{RL of Instrument Axis} - V - h$ .

✅ Advantages

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Enables accurate elevation determination without needing physical horizontal distance chaining.
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Allows rapid survey of hilly and steep terrain where leveling with traditional instruments is difficult.

❌ Disadvantages / Limitations

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Slight inaccuracies in vertical angle measurement $\theta$ lead to significant errors in calculated vertical distance.
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Refraction effects near the ground surface can affect staff intercept readings.

🛠️ Applications / Uses

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Contour mapping and topographic surveys in rough, hilly, or undulating terrain.
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Location survey
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📊 Diagram / Illustration

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