There are basically three systems of tacheometric leveling:

•  Stadia hair system
•  Tangential system
•  Subtense bar system

THE STADIA HAIR METHOD:

 This is the most extensively used system of tacheometry, particularly for detailed work, such as those required in engineering surveys. Stadia hair system may be of two types:

•  The fixed hair method
• The movable hair method

FIXED HAIR METHOD:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system is fixed. The staff intercept `s`,therefore changes according to the distance `D` and vertical angle `theta`. when the staff intercept is more than the length of the staff, only half intercept is read, which is equal to the length of the difference between central stadia hair reading and the lower/upper stadia hair reading.

fig: Fixed hair method

This methods can be useful even when horizontal sights are not possible. For inclined sights, reading may be taken by holding the staff either vertical or normal to the line of sight.some important features of this method are:

• Distance between hairs is constant.
• The staff intercept varies.
• The apex angle is constant.
• The distance varies with the staff intercept.

Movable Hair Method:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system can be varied.The staff intercept,`s` is kept fixed. The stadia hairs can be moved vertically up and down by using micrometer screws. Movable hair method is not in common use due to difficulties in determining the stadia interval accurately. Some important features of this method are:

• Top and bottom hairs are movable.
• Staff intercept is kept constant.
• This method is rarely used.

There are basically three systems of tacheometric leveling:

•  Stadia hair system
•  Tangential system
•  Subtense bar system

THE STADIA HAIR METHOD:

 This is the most extensively used system of tacheometry, particularly for detailed work, such as those required in engineering surveys. Stadia hair system may be of two types:

•  The fixed hair method
• The movable hair method

FIXED HAIR METHOD:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system is fixed. The staff intercept `s`,therefore changes according to the distance `D` and vertical angle `theta`. when the staff intercept is more than the length of the staff, only half intercept is read, which is equal to the length of the difference between central stadia hair reading and the lower/upper stadia hair reading.

fig: Fixed hair method

This methods can be useful even when horizontal sights are not possible. For inclined sights, reading may be taken by holding the staff either vertical or normal to the line of sight.some important features of this method are:

• Distance between hairs is constant.
• The staff intercept varies.
• The apex angle is constant.
• The distance varies with the staff intercept.

Movable Hair Method:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system can be varied.The staff intercept,`s` is kept fixed. The stadia hairs can be moved vertically up and down by using micrometer screws. Movable hair method is not in common use due to difficulties in determining the stadia interval accurately. Some important features of this method are:

• Top and bottom hairs are movable.
• Staff intercept is kept constant.
• This method is rarely used.

Subtense bar is an instrument used for measuring horizontal distances in the areas where direct chaining becomes difficult due to undulation or other obstructions.

It consists of a metal tube of length varying from 3m to 4 m. Two disc 20 cm in diameter painted either black or red on one side and white on the other, each with a 7.5m white or black center are placed 3m apart. Red or black faces of discs are kept towards the theodolite.

At the center of the bar,an alidade perpendicular to the axis of the bar is attached. The bar is mounted on special tripod.

COMPUTATION OF SUBTENSE BAR DISTANCE:

Here, from triangle `OAC` and `OCB`,

`D*tan (theta/2)=S/2`

`D=S/2 cot(theta/2)`

Or, `D=S/(2tan(theta/2))`

Now, for an small angle `theta`,

`tan(theta/2)=theta/2`, where `theta` is in radian.

      `=1/2*theta/(206265)`, where `theta` is in second..

 Thus, we have,

 `D=S/(2*(1/2*theta/206265))`
Or,`=(S*206265)/theta`, Where `theta` is in second.

 EFFECT OF ANGULAR ERROR ON HORIZONTAL DISTANCE:

 From figure,

 `tan(theta/2)=((S/2)/D)`

 `S/2=D*tan(theta/2)=D*theta/2`

Or,`S=D*theta`.......(i)

 We know,

 `D=(S*206265)/theta`

 Let the negative error in `theta` be `delta theta` and the positive error in `D` be `delta D`then,

 `S=(D+delta D)(theta- delta theta)`......(ii)

 Or,`(D+delta D)/D=theta/(theta+delta theta)`

 On cross multiplication,

 `D theta-D delta theta-theta*D delta theta=D theta`

Or,`delta D(theta - delta theta)=D*delta theta`

Or,`delta D=(D delta theta)/(theta + delta theta)`

Similarly, if `delta theta` is positive error and `delta D` is the negative error.,

`delta D=(D delta theta)/(theta+delta theta)`

However,`delta theta` is too small as compared to `theta`,so,

`delta theta=(D delta theta)/theta`

Subtense bar is an instrument used for measuring horizontal distances in the areas where direct chaining becomes difficult due to undulation or other obstructions.

It consists of a metal tube of length varying from 3m to 4 m. Two disc 20 cm in diameter painted either black or red on one side and white on the other, each with a 7.5m white or black center are placed 3m apart. Red or black faces of discs are kept towards the theodolite.

At the center of the bar,an alidade perpendicular to the axis of the bar is attached. The bar is mounted on special tripod.

COMPUTATION OF SUBTENSE BAR DISTANCE:

Here, from triangle `OAC` and `OCB`,

`D*tan (theta/2)=S/2`

`D=S/2 cot(theta/2)`

Or, `D=S/(2tan(theta/2))`

Now, for an small angle `theta`,

`tan(theta/2)=theta/2`, where `theta` is in radian.

      `=1/2*theta/(206265)`, where `theta` is in second..

 Thus, we have,

 `D=S/(2*(1/2*theta/206265))`
Or,`=(S*206265)/theta`, Where `theta` is in second.

 EFFECT OF ANGULAR ERROR ON HORIZONTAL DISTANCE:

 From figure,

 `tan(theta/2)=((S/2)/D)`

 `S/2=D*tan(theta/2)=D*theta/2`

Or,`S=D*theta`.......(i)

 We know,

 `D=(S*206265)/theta`

 Let the negative error in `theta` be `delta theta` and the positive error in `D` be `delta D`then,

 `S=(D+delta D)(theta- delta theta)`......(ii)

 Or,`(D+delta D)/D=theta/(theta+delta theta)`

 On cross multiplication,

 `D theta-D delta theta-theta*D delta theta=D theta`

Or,`delta D(theta - delta theta)=D*delta theta`

Or,`delta D=(D delta theta)/(theta + delta theta)`

Similarly, if `delta theta` is positive error and `delta D` is the negative error.,

`delta D=(D delta theta)/(theta+delta theta)`

However,`delta theta` is too small as compared to `theta`,so,

`delta theta=(D delta theta)/theta`

Let us consider the external focusing type of telescope and let us consider the line of sight to be horizontal. Let,

`f=`focal length of the object glass.

`i=`stadia hair interval `=ab`

 `S=`Staff intercept

  `=AB`

`c=`siatance from O to the vertical axis of the instrument

`d=`distance from O to the staff

`d'=`distance from O to the plane of the diaphragm

`D=`horizontal axis from the vertical axis to the staff.

Now, from triangle AOB and aOb, we have,

`d/d'=u/v=S/i`.......(i)

Again, from lens formula, we have,

`1/f=1/u+1/v`

Multiplying both sides by `uf`, we get,

`(uf)/f=(uf)/u+(uf)/v`

or, `u=(uf)/v+f`....(ii)

Now, from (i) and (ii), we get,

`u=f+f/(i/S)`

 `=f+(fS)/i`

Adding a constant `c` on both sides, we get,

`u+c=(fS)/i+(f+c)`

Again from figure,

`D=u+c`

Thus,`D=(fS)/i=(f+c)

Also, let `K=f/i` and `C=f+c`,

then the above expression becomes,

`D=K*S+C`.....(iii)

This is the standard equation used to explain the principle of stadia hair system.

Where `K` is the multiplying constant and `C` is the additive constant.

Now, 

for analytic lens, `K=100` and `C=0`

for external focusing telescope, `C=0.3` to `0.61`

for internal focusing telescope, `C=0.08` to `0.2`

Let us consider the external focusing type of telescope and let us consider the line of sight to be horizontal. Let,

`f=`focal length of the object glass.

`i=`stadia hair interval `=ab`

 `S=`Staff intercept

  `=AB`

`c=`siatance from O to the vertical axis of the instrument

`d=`distance from O to the staff

`d'=`distance from O to the plane of the diaphragm

`D=`horizontal axis from the vertical axis to the staff.

Now, from triangle AOB and aOb, we have,

`d/d'=u/v=S/i`.......(i)

Again, from lens formula, we have,

`1/f=1/u+1/v`

Multiplying both sides by `uf`, we get,

`(uf)/f=(uf)/u+(uf)/v`

or, `u=(uf)/v+f`....(ii)

Now, from (i) and (ii), we get,

`u=f+f/(i/S)`

 `=f+(fS)/i`

Adding a constant `c` on both sides, we get,

`u+c=(fS)/i+(f+c)`

Again from figure,

`D=u+c`

Thus,`D=(fS)/i=(f+c)

Also, let `K=f/i` and `C=f+c`,

then the above expression becomes,

`D=K*S+C`.....(iii)

This is the standard equation used to explain the principle of stadia hair system.

Where `K` is the multiplying constant and `C` is the additive constant.

Now, 

for analytic lens, `K=100` and `C=0`

for external focusing telescope, `C=0.3` to `0.61`

for internal focusing telescope, `C=0.08` to `0.2`

• Reconnaissance
• Establishment of tacheometric closed traverse as per field requirement ensuring each instrument station should command a wide area. If the area is too large,link traverse and addition of triangles may also be prederred.

• A tacheometer is set up at the starting station. It is centered and levelled accurately. Height of instruments is Measured.
• The instrument is oriented with reference to any predetermined station (BS) by taking it's bearing.Horizontal angles of all nearby details are recorded in a field book. Index map and reference sketches are also prepared in field.

• A back sight reading is taken on a nearby benchmark if available.If not,fly levelling should be done to know the RL of the instrument station. After completing the loop of fly levelling between bench mark and any nearby traverse station, loop levelling is conducted along the traverse to know the RL of all stations.
• Similarly other stations are connected and above steps are repeated.
• Horizontal distance, vertical distance, bearings are computed from field book. RLs of all points are calculated in a separate field book for the purpose of plotting. The points are plotted in the maps with suitable scale and Rls of the respective points are noted by taking the distances and RLs from the prepared table.The contour lines may be drawn by the method of interpolation.

• Reconnaissance
• Establishment of tacheometric closed traverse as per field requirement ensuring each instrument station should command a wide area. If the area is too large,link traverse and addition of triangles may also be prederred.

• A tacheometer is set up at the starting station. It is centered and levelled accurately. Height of instruments is Measured.
• The instrument is oriented with reference to any predetermined station (BS) by taking it's bearing.Horizontal angles of all nearby details are recorded in a field book. Index map and reference sketches are also prepared in field.

• A back sight reading is taken on a nearby benchmark if available.If not,fly levelling should be done to know the RL of the instrument station. After completing the loop of fly levelling between bench mark and any nearby traverse station, loop levelling is conducted along the traverse to know the RL of all stations.
• Similarly other stations are connected and above steps are repeated.
• Horizontal distance, vertical distance, bearings are computed from field book. RLs of all points are calculated in a separate field book for the purpose of plotting. The points are plotted in the maps with suitable scale and Rls of the respective points are noted by taking the distances and RLs from the prepared table.The contour lines may be drawn by the method of interpolation.

STATEMENT: In a isosceles triangle, the ratio of the perpendiculars from the vertex on their bases is constant.

Let `ABC` and `AB'C'` be two isosceles triangles whose bases are `BC` and `B'C'` and their vertex is at A. If `AO` and `AO'` are perpendiculars to their respective bases, then

`(AO)/(BC)=(AO')/(B'C')=1/2*cot(alpha/2)`

        `=Constant`

    Where, `alpha =`apex angle.

STATEMENT: In a isosceles triangle, the ratio of the perpendiculars from the vertex on their bases is constant.

Let `ABC` and `AB'C'` be two isosceles triangles whose bases are `BC` and `B'C'` and their vertex is at A. If `AO` and `AO'` are perpendiculars to their respective bases, then

`(AO)/(BC)=(AO')/(B'C')=1/2*cot(alpha/2)`

        `=Constant`

    Where, `alpha =`apex angle.

The stadia method of traversing is useful in following situations:

1. In differential levelling, the backsight and foresight distances are balanced conveniently if the leval is equipped with stadia hairs.
2. In profile levelling and cross sectioning, stadia is convenient means of finding distance from level to points on which rod readings are taken.
3. In rough trignometric, or indirect levelling with the transit, the stadia method is more rapid than any other method.
4. For traverse surveying of low relative accuracy, where only horizontal angles and distances are required, the stadia method is a useful rapid method.
5. On surveys of low relative accuracy-particularly topographic surveys where both the relative locations of points in a horizontal plane and the elevation of these points are desired, stadia are useful.

The stadia method of traversing is useful in following situations:

1. In differential levelling, the backsight and foresight distances are balanced conveniently if the leval is equipped with stadia hairs.
2. In profile levelling and cross sectioning, stadia is convenient means of finding distance from level to points on which rod readings are taken.
3. In rough trignometric, or indirect levelling with the transit, the stadia method is more rapid than any other method.
4. For traverse surveying of low relative accuracy, where only horizontal angles and distances are required, the stadia method is a useful rapid method.
5. On surveys of low relative accuracy-particularly topographic surveys where both the relative locations of points in a horizontal plane and the elevation of these points are desired, stadia are useful.

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.

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.

Tacheometry is the branch of angular surveying in which both horizontal and vertical distances between the stations of observation and staff positions are determined from instrumental observation (i.e, by angular observation with a tacheometer) without necessity of chaining. 

A tacheometer is a transit theodolite having a stadia telescope fitted with two horizontal hairs called Stadia hairs in addition to the usual cross hairs.

Purpose Of Tacheometric Survey:

The primary objective of tacheometry is the preparation of contoured plans and traversing. It is assumed to be rapid and accurate method in rough country and has thus been widely used by engineers/surveyors in location surveys for railways, canals, resorvoirs etc. For surveys of high accuracy, tacheometry provides a good check on the distances measured with tapes/chains.

USES OF Tacheometry:

Tacheometry is used for:

1. Preparation of a topographic maps where both horizontal and vertical distances are required to be measured.
2. Survey work in difficult terain where direct methods of measurements are inconvinient.
3. Reconnaissance survey for highways and railways.
4. Establishment of secondary control points.

It is best suited when obstacles such as steep and broken ground, deep ravines, and stretches of water or swamps are met with.

ADVANTAGES OF TACHEOMETRY:

1. Both the horizontal distances and difference of elevation are determined indirectly in tacheometry surveying.
2. Tacheometric methods can be used in terrain, where direct methods are inconvenient.
3. There is considerable saving in time and moneywith the use of tacheometric methods.
4. It can be widely used for reconnaissance surveys of routes, for hydrographic surveying and for filling in details in a traverse.

Tacheometry is the branch of angular surveying in which both horizontal and vertical distances between the stations of observation and staff positions are determined from instrumental observation (i.e, by angular observation with a tacheometer) without necessity of chaining. 

A tacheometer is a transit theodolite having a stadia telescope fitted with two horizontal hairs called Stadia hairs in addition to the usual cross hairs.

Purpose Of Tacheometric Survey:

The primary objective of tacheometry is the preparation of contoured plans and traversing. It is assumed to be rapid and accurate method in rough country and has thus been widely used by engineers/surveyors in location surveys for railways, canals, resorvoirs etc. For surveys of high accuracy, tacheometry provides a good check on the distances measured with tapes/chains.

USES OF Tacheometry:

Tacheometry is used for:

1. Preparation of a topographic maps where both horizontal and vertical distances are required to be measured.
2. Survey work in difficult terain where direct methods of measurements are inconvinient.
3. Reconnaissance survey for highways and railways.
4. Establishment of secondary control points.

It is best suited when obstacles such as steep and broken ground, deep ravines, and stretches of water or swamps are met with.

ADVANTAGES OF TACHEOMETRY:

1. Both the horizontal distances and difference of elevation are determined indirectly in tacheometry surveying.
2. Tacheometric methods can be used in terrain, where direct methods are inconvenient.
3. There is considerable saving in time and moneywith the use of tacheometric methods.
4. It can be widely used for reconnaissance surveys of routes, for hydrographic surveying and for filling in details in a traverse.

• Permissible error in a single horizontal distance should be within 1 in 500, and vertical distance should not exceed 0.1m.
• In open traverse, average error should not exceed 1 in 600 to 1 in 850.
• For sights upto 120m, an accuracy of `1/3000` to `1/10000` is must.
• Closing error in stadia traverse > `0.055 root(P)`, `P` is the perimeter of the traverse.
• Error in closing for elevation varies from `0.08 root(K)` on level ground to `0.25 root(K)` on hilly ground, where `K=` distance in km.

• Permissible error in a single horizontal distance should be within 1 in 500, and vertical distance should not exceed 0.1m.
• In open traverse, average error should not exceed 1 in 600 to 1 in 850.
• For sights upto 120m, an accuracy of `1/3000` to `1/10000` is must.
• Closing error in stadia traverse > `0.055 root(P)`, `P` is the perimeter of the traverse.
• Error in closing for elevation varies from `0.08 root(K)` on level ground to `0.25 root(K)` on hilly ground, where `K=` distance in km.

The method of tangential tacheometry can be used:

• When staff is held much away from the instrument making it difficult to read it.
• When the diaphragm doesnot have stadia hairs.

In tangential tacheometry, horizontal and vertical distance from the instruments to the staff position are computed from the observed vertical angles to two targets fixed at a distance `S` on the staff.

Depending upon the vertical angles, the following three cases may arises:

• Both vertical angles may be elevation angle.
• Both vertical angles may be depression angles.
• One elevation angle and other depression angle.

CASE(i):

Let,

`S`=Distance between targets

`V`=Vertical distance between lowest target(staff reading) and axis of instruments.

`h`=Height of lower staff reading above the staff station

Now, from triangle `OAP'` and `OBP'`,we get,

`V+S=D*tan alpha_1` and

`V=D*tan alpha_2`

Thus,`V+S-V=D*(tan alpha_1-tan alpha_2)`

Or, `D=S/(tan alpha_1-tan alpha_2)`,

Where `S=D*tan alpha_1-tan alpha_2`

Thus, RL of staff station= RL of instrument axis+V-h

CASE(ii)

Here,

`V-S=D*tan alpha_2` and

`V=D*tan alpha_1`

Thus,`V-(V-S)=D(tan alpha_1-tan alpha_2)`

Or, `D=S/(tan alpha_1-tan alpha_2)`,

Where `S=D*tan alpha_1-tan alpha_2`

Also,

`V=(S*tan alpha_1)/(tan alpha_1-tan alpha_2`

Thus, RL of staff station=RL of instrument axis-V-h

CASE (iii):

Here,

`S-V=D*tan alpha_1` and

`V=D*tan alpha_2`

Thus,`(S-V)+V=D(tan alpha_1+tan alpha_2)`

Or, `D=S/(tan alpha_1+tan alpha_2)`,

Where `S=D*tan alpha_1- tan alpha_2`

Also,

`V=(S*tan alpha_2)/(tan alpha_1+tan alpha_2`

Thus, RL of staff station=RL of instrument axis-V-h

DISADVANTAGES OF TANGENTIAL METHOD:

Tangential Method has the following disadvantages:

1. It is rigorous method which requires more time that means it is slow method.
2. It involves more computations for reducing distances and elevations.
3. Minimum two vertical angles are required to be observed for computations.
4. During observing vertical angles, instruments may be unnoticeable disturbed.
5. Staff (targets) is assumed to be perfectly vertical.

The method of tangential tacheometry can be used:

• When staff is held much away from the instrument making it difficult to read it.
• When the diaphragm doesnot have stadia hairs.

In tangential tacheometry, horizontal and vertical distance from the instruments to the staff position are computed from the observed vertical angles to two targets fixed at a distance `S` on the staff.

Depending upon the vertical angles, the following three cases may arises:

• Both vertical angles may be elevation angle.
• Both vertical angles may be depression angles.
• One elevation angle and other depression angle.

CASE(i):

Let,

`S`=Distance between targets

`V`=Vertical distance between lowest target(staff reading) and axis of instruments.

`h`=Height of lower staff reading above the staff station

Now, from triangle `OAP'` and `OBP'`,we get,

`V+S=D*tan alpha_1` and

`V=D*tan alpha_2`

Thus,`V+S-V=D*(tan alpha_1-tan alpha_2)`

Or, `D=S/(tan alpha_1-tan alpha_2)`,

Where `S=D*tan alpha_1-tan alpha_2`

Thus, RL of staff station= RL of instrument axis+V-h

CASE(ii)

Here,

`V-S=D*tan alpha_2` and

`V=D*tan alpha_1`

Thus,`V-(V-S)=D(tan alpha_1-tan alpha_2)`

Or, `D=S/(tan alpha_1-tan alpha_2)`,

Where `S=D*tan alpha_1-tan alpha_2`

Also,

`V=(S*tan alpha_1)/(tan alpha_1-tan alpha_2`

Thus, RL of staff station=RL of instrument axis-V-h

CASE (iii):

Here,

`S-V=D*tan alpha_1` and

`V=D*tan alpha_2`

Thus,`(S-V)+V=D(tan alpha_1+tan alpha_2)`

Or, `D=S/(tan alpha_1+tan alpha_2)`,

Where `S=D*tan alpha_1- tan alpha_2`

Also,

`V=(S*tan alpha_2)/(tan alpha_1+tan alpha_2`

Thus, RL of staff station=RL of instrument axis-V-h

DISADVANTAGES OF TANGENTIAL METHOD:

Tangential Method has the following disadvantages:

1. It is rigorous method which requires more time that means it is slow method.
2. It involves more computations for reducing distances and elevations.
3. Minimum two vertical angles are required to be observed for computations.
4. During observing vertical angles, instruments may be unnoticeable disturbed.
5. Staff (targets) is assumed to be perfectly vertical.