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

A

V=kssinтБб2╬╕2+csinтБб╬╕V = k s \sin 2 \theta 2 + c \sin \thetaV=kssin2╬╕2+csin╬╕

B

V=kssinтБб2╬╕2+ccosтБб╬╕V = k s \sin 2 \theta 2 + c \cos \thetaV=kssin2╬╕2+ccos╬╕

C

V=kscos2╬╕+csinтБб╬╕V = k s cos^{2} \theta + c \sin \thetaV=kscos2╬╕+csin╬╕

D

V=kscos2╬╕+ccosтБб╬╕V = k s cos^{2} \theta + c \cos \thetaV=kscos2╬╕+ccos╬╕

Correct Answer

тЪЩя╕П TE тАв Technical Concept & PrincipleCivilAdvanced Survey
Option A

V=kssinтБб2╬╕2+csinтБб╬╕V = k s \sin 2 \theta 2 + c \sin \thetaV=kssin2╬╕2+csin╬╕

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=kscosтБб╬╕+cL = k s \cos\theta + cL=kscos╬╕+c. The vertical component VVV is obtained by multiplying this inclined distance by sinтБб╬╕\sin\thetasin╬╕, which yields V=kscosтБб╬╕sinтБб╬╕+csinтБб╬╕V = k s \cos\theta \sin\theta + c \sin\thetaV=kscos╬╕sin╬╕+csin╬╕. Using the trigonometric identity sinтБб2╬╕=2sinтБб╬╕cosтБб╬╕\sin 2\theta = 2\sin\theta\cos\thetasin2╬╕=2sin╬╕cos╬╕, this reduces to V=kssinтБб2╬╕2+csinтБб╬╕V = k s \frac{\sin 2\theta}{2} + c \sin\thetaV=ks2sin2╬╕тАЛ+csin╬╕.

тЪЩя╕ПTETechnical SolutionConcept & Principle
ЁЯТб Explanation

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=kscosтБб╬╕+cL = k s \cos\theta + cL=kscos╬╕+c. The vertical component VVV is obtained by multiplying this inclined distance by sinтБб╬╕\sin\thetasin╬╕, which yields V=kscosтБб╬╕sinтБб╬╕+csinтБб╬╕V = k s \cos\theta \sin\theta + c \sin\thetaV=kscos╬╕sin╬╕+csin╬╕. Using the trigonometric identity sinтБб2╬╕=2sinтБб╬╕cosтБб╬╕\sin 2\theta = 2\sin\theta\cos\thetasin2╬╕=2sin╬╕cos╬╕, this reduces to V=kssinтБб2╬╕2+csinтБб╬╕V = k s \frac{\sin 2\theta}{2} + c \sin\thetaV=ks2sin2╬╕тАЛ+csin╬╕.

ЁЯФв Key Formulas

V=kssinтБб2╬╕2+csinтБб╬╕V = k s \frac{\sin 2\theta}{2} + c \sin\thetaV=ks2sin2╬╕тАЛ+csin╬╕ тАФ Vertical distance formula for inclined line of sight with staff held vertical

D=kscosтБб2╬╕+ccosтБб╬╕D = k s \cos┬▓\theta + c \cos\thetaD=kscos2╬╕+ccos╬╕ тАФ Horizontal distance formula for inclined line of sight with staff held vertical

k=fik = \frac{f}{i}k=ifтАЛ тАФ Tacheometric multiplying constant (typically 100)

c=f+dc = f + dc=f+d тАФ Tacheometric additive constant (typically 0 for anallatic lens)

тЪЩя╕П 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 sтА▓=scosтБб╬╕s' = s \cos\thetasтА▓=scos╬╕. Multiplying the normal intercept by the multiplying constant kkk gives the inclined slope length, whose vertical projection yields the vertical component VVV through trigonometric resolution.

ЁЯУМ Key Points
  • тЦ╕

    The vertical component VVV represents the vertical height from the horizontal instrument axis to the middle cross-hair reading on the staff.

  • тЦ╕

    For an anallatic lens system, c=0c = 0c=0, simplifying the formula to V=kssinтБб2╬╕2V = \frac{k s \sin 2\theta}{2}V=2kssin2╬╕тАЛ.

  • тЦ╕

    If the angle is an angle of elevation, Reduced Level (RL) of point = RL┬аof┬аInstrument┬аAxis+VтИТh\text{RL of Instrument Axis} + V - hRL┬аof┬аInstrument┬аAxis+VтИТh, where hhh is the central hair reading.

  • тЦ╕

    If the angle is an angle of depression, Reduced Level (RL) of point = RL┬аof┬аInstrument┬аAxisтИТVтИТh\text{RL of Instrument Axis} - V - hRL┬аof┬аInstrument┬аAxisтИТVтИТh.

тЬЕ Advantages
  • тЦ╕

    Enables accurate elevation determination without needing physical horizontal distance chaining.

  • тЦ╕

    Allows rapid survey of hilly and steep terrain where leveling with traditional instruments is difficult.

тЭМ Disadvantages / Limitations
  • тЦ╕

    Slight inaccuracies in vertical angle measurement ╬╕\theta╬╕ lead to significant errors in calculated vertical distance.

  • тЦ╕

    Refraction effects near the ground surface can affect staff intercept readings.

ЁЯЫая╕П Applications / Uses
  • тЦ╕

    Contour mapping and topographic surveys in rough, hilly, or undulating terrain.

  • тЦ╕

    Location survey

  • тЦ╕
ЁЯУК Diagram / Illustration
Vertical Distance Formula (Staff Vertical)V = k ┬╖ s ┬╖ cos(╬╕) ┬╖ sin(╬╕) + c ┬╖ sin(╬╕)Using trigonometric identity: 2┬╖sin(╬╕)┬╖cos(╬╕) = sin(2╬╕)V = k ┬╖ s ┬╖ sin(2╬╕)2+ c ┬╖ sin(╬╕)k = multiplying constant (100), c = additive constant (0), s =staff intercept

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