Case I:When the line of sight is inclined and staff is held vertical:
a) when the line of sight is inclined upwards and staff is held vertical:let `theta` is the angle of elevation of the line of sight. Let us draw an intercept `A'B'` through `C` Perpendicular to `OC`.
Now,`ACA'=theta` and `A AC'` may be taken as `90 degree`.
From triangle `A A'C`, we have,
`cos(theta)=(A'C)/(AC)`
Thus, `A'B'=2*A'C` and `AB=2*AC`
so,`A'B'=S'=AB*cos(theta)=Scos(theta)`
Now,
`D=K*S'+C`
Or,`D=K*Scos(theta)+C`......(i)
Again, from triangle GCF,
`cos(theta)=(GF)/(GC)=H/D`
Or, `H=D*cos(theta)`
Or, `H=K*S*cos^2theta+C*costheta`.....(ii)
From (ii), the horizontal diatance between the staff held vertical and instrument station can be computed.
For determining the elevation difference between P and Q, it is necessary to determine the value of `V` i.e, `FC`, which is the difference of levels between the co....
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Case I:When the line of sight is inclined and staff is held vertical:
a) when the line of sight is inclined upwards and staff is held vertical:
let `theta` is the angle of elevation of the line of sight. Let us draw an intercept `A'B'` through `C` Perpendicular to `OC`.
Now,`ACA'=theta` and `A AC'` may be taken as `90 degree`.
From triangle `A A'C`, we have,
`cos(theta)=(A'C)/(AC)`
Thus, `A'B'=2*A'C` and `AB=2*AC`
so,`A'B'=S'=AB*cos(theta)=Scos(theta)`
Now,
`D=K*S'+C`
Or,`D=K*Scos(theta)+C`......(i)
Again, from triangle GCF,
`cos(theta)=(GF)/(GC)=H/D`
Or, `H=D*cos(theta)`
Or, `H=K*S*cos^2theta+C*costheta`.....(ii)
From (ii), the horizontal diatance between the staff held vertical and instrument station can be computed.
For determining the elevation difference between P and Q, it is necessary to determine the value of `V` i.e, `FC`, which is the difference of levels between the collimation plane and the central hair reading on the staff.
Again, from triangle CFG,
`sin(theta)=(CF)/(GC)=V/D`
Or, `V=D*Sin(theta)`
Or, `V=K*Scos(theta)*sin(theta)+C*sin(theta)`
Or, `V=K*S/2Sin2theta+Csintheta`
Let,`h=QC`, the central hair reading, then the level difference between G and Q for an angle of elevation is given by
`FQ=C-h`
If HI be the height of instruments, the reduced level is given by:
RL of `Q=HI+V-h`
Thus, if the RLs of instrument station and height of instrument are known,then
RL of Q=RL of instrument station +HI+V-h
b) when the line of sight is inclined downwards and staff is held vertical:
In this case, we can derive the expression of H and V following the above procedures.
`H=K*S*cos^2theta+C*cos(theta)`.......(iii)
`V=K*S/2Sin2theta+Csintheta`........(iv)
Similarly, If RL of instrument axis is known,
Elevation of Q=elevation of instrument axis (P')-V-h
Again, If the RL of instrument station and height of instrument is known,
RL of Q=RL of instrument station(P)+ height of instrument-V-h
Thus,combining both cases i.e (a) and (b),
RL of staff station (Q)=RL of instrument axis `+-V-h`
RL of staff station (Q)=RL of instrument station `+-V-h`
where positive sigh is used for angle of elevation and negative sign is used for angle of depression.
Case(ii): When the line of sight is inclined and the staff is held normal to the line of sight:
a)When the line of sight is inclined upwards and the staff is held normal to the line of sight i.e,angle of elevation:
Let `theta` is the angle of elevation of the line of sight.
`AB` is the slope intercept `S`.
`CQ` is the central hair reading `h`.
Now, from right angled triangles `CQC'`,
`sin theta=(C'C)/(CQ)=(C'C)/h`
Or, `C'C=h*sin theta`
Also,
`cos(theta)=(C'Q)/(CQ)=(C'Q)/h`
Or, `C'Q=h*cos theta`
From the basic principle of stadia system,
`D=K*S+C`
We know,`H=GF'+FF'`
Again, from triangle GCF',we have,
`cos(theta)=(GF)/(GC)=(GF)/D`
Or, `H=GF'+CC'`
Or, `H=D*cos theta+CC'`
Or, `H=D*cos theta+h*sin theta`
Or, `H=(K*S+C) cos theta+h*sin(theta)`
Similarly,
`V=F'C=D*sin theta`
Or,`V=(K*S+C)sin theta`
Now, for angle of elevation,
`FQ=FC'=D sin theta`
Or,`FQ=V-h*cos theta`
If HI is the height of instruments above datum,
RL of Q`=HI+V-h*cos theta`
Or,RL of Q= RL of instrument station`+HI+(K*S+C)sin(theta)-h cos theta`
b) When the line of sight is inclined downwards and staff is held normal to the line of sight i.e, angle of depression.
Let `theta` is the angle of depression of the line of sight.
We can derive,
`H=(K*S+C)cos(theta)-h*sin theta`
`V=(K*S+C)sin theta`
Now,for angle of depression,we have,
`FQ=V+h*cos theta`
and RL of Q`=HI-V-h cos theta`
Thus,combining both cases,
RL of staff station (Q)=RL of instrument station `(+-)(KS+C)sin theta-h costheta`
where positive sigh is used for angle of elevation and negative sign is used for angle of depression.
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