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.

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.

TACHEOMETER

 A tacheometer is a transit theodolite having a stadia telescope fitted with two horizontal hairs called Stadia hairs(stadia lines) in addition to the usual cross hairs. The stadia diaphragm has three horizontal hairs: a central horizontal hair, and upper and lower stadia hairs. The upper and lower stadia hairs are equidistant from the central horizontal hairs.

fig:  Tacheometer

 Characteristics of tacheometer:

•  The value of multyplying constant should be 100.
•  The value of additive constant should be zero.
•  The telescope is fitted with analytic lens.
•  The magnification of telescope should be 20 to 80 diameter.
• Magnifying power of eye piece is kept high.

 Stadia rods/Levelling staffs:

 The stadia rod is about 3m to 5 m long. For short sights, an ordinary levelling staffs may be used. For long sights, special staff called stadia rods is generally used.

TACHEOMETER

 A tacheometer is a transit theodolite having a stadia telescope fitted with two horizontal hairs called Stadia hairs(stadia lines) in addition to the usual cross hairs. The stadia diaphragm has three horizontal hairs: a central horizontal hair, and upper and lower stadia hairs. The upper and lower stadia hairs are equidistant from the central horizontal hairs.

fig:  Tacheometer

 Characteristics of tacheometer:

•  The value of multyplying constant should be 100.
•  The value of additive constant should be zero.
•  The telescope is fitted with analytic lens.
•  The magnification of telescope should be 20 to 80 diameter.
• Magnifying power of eye piece is kept high.

 Stadia rods/Levelling staffs:

 The stadia rod is about 3m to 5 m long. For short sights, an ordinary levelling staffs may be used. For long sights, special staff called stadia rods is generally 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.

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.

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`

The stadia constant can be determined by:

• Laboratory measurement method
• Field measurement method
• Semi field method

Laboratory measurement method:

In this method,

• The stadia intercept (i) can be measured from the diaphragm by means of using vernier caliper.
• The distance `c` between the optical center and the vertical axis of the tacheometer can also be measured.
• The focal length can be determined using lens formula as:

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

Thus, after measuring `f`,`i` and `c`, the constants `K` and `C` can be calculated as:

`K=f/i` and `C=f+c`

Field measurement method:

In this method,

A fairly leveled ground is selected and the tacheometer is set up at point `A` and another point, say `A_4` is also selected at a known distance, say 200m.The pegs are then driven at a regular intervals,say `50m` and let the points be `A_1`,`A_2`, and `A_3`.Now, holding the staff at `A_1`,`A_2`, and `A_3`, the staff intercept `S_1`,`S_2`,`S_3`, and `S_4` are taken.

Let,

`A A_1=D_1`

`A A_2=D_2`

`A  A_3=D_3`

`A A_4=D_4`

Now, we know that,

`D_1=K*S_1+C`.....(i)

`D_2=K*S_2+C`.....(ii)

Thus, `D_2-D_1=K*S_2-K*S_1`

Or,`K=(D_2-D_1)/(S_2-S_1)`....(iii)

Where, `D_1`,`D_2`, `S_1`, `S_2` are known and hence `K` can be determined.

Now, from (i), we get,

`C=D_1-K*S_1`

Or, `C=D_1-(D_2-D_1)/(S_2-S_1)*S_1`

Or, `C=((D_1*S_2)-(D_2S_1))/(S_2-S_1)`

Similarly,

`k=(D_4-D_3)/(S_4-S_3)` and

`C=(D_3*S_4-D_4-S_3)/(S_4-S_3)`.....(iv)

Thus, the mean value of `K` and ` C` can be determined from the above equation.

Semi-Field Method:

In this method, the value of `K` is measured from field measurement method and the value of `C=f+c` is measured by means of measuring `f` and `c` in lab.

The stadia constant can be determined by:

• Laboratory measurement method
• Field measurement method
• Semi field method

Laboratory measurement method:

In this method,

• The stadia intercept (i) can be measured from the diaphragm by means of using vernier caliper.
• The distance `c` between the optical center and the vertical axis of the tacheometer can also be measured.
• The focal length can be determined using lens formula as:

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

Thus, after measuring `f`,`i` and `c`, the constants `K` and `C` can be calculated as:

`K=f/i` and `C=f+c`

Field measurement method:

In this method,

A fairly leveled ground is selected and the tacheometer is set up at point `A` and another point, say `A_4` is also selected at a known distance, say 200m.The pegs are then driven at a regular intervals,say `50m` and let the points be `A_1`,`A_2`, and `A_3`.Now, holding the staff at `A_1`,`A_2`, and `A_3`, the staff intercept `S_1`,`S_2`,`S_3`, and `S_4` are taken.

Let,

`A A_1=D_1`

`A A_2=D_2`

`A  A_3=D_3`

`A A_4=D_4`

Now, we know that,

`D_1=K*S_1+C`.....(i)

`D_2=K*S_2+C`.....(ii)

Thus, `D_2-D_1=K*S_2-K*S_1`

Or,`K=(D_2-D_1)/(S_2-S_1)`....(iii)

Where, `D_1`,`D_2`, `S_1`, `S_2` are known and hence `K` can be determined.

Now, from (i), we get,

`C=D_1-K*S_1`

Or, `C=D_1-(D_2-D_1)/(S_2-S_1)*S_1`

Or, `C=((D_1*S_2)-(D_2S_1))/(S_2-S_1)`

Similarly,

`k=(D_4-D_3)/(S_4-S_3)` and

`C=(D_3*S_4-D_4-S_3)/(S_4-S_3)`.....(iv)

Thus, the mean value of `K` and ` C` can be determined from the above equation.

Semi-Field Method:

In this method, the value of `K` is measured from field measurement method and the value of `C=f+c` is measured by means of measuring `f` and `c` in lab.

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.

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.

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`

Instrumental errors:

1. Permanent adjustment not perfect.
2. Graduation of staff not uniform.
3. The value of multiplying constant not correct.
4. Error due to imperfect adjustment of the plate.
5. Idex erroe i.e, error due to imperfect adjustment of vertical circle.
6. Error due to horizontal axis not being perpendicular to the vertical axis.

Observational errors:

1. Error due to incorrect centering or levelling.
2. Error due to Staff not being perpendicular.
3. Error due to imperfect focusing.i.e, parallex error causing incorrect readings.
4. Error due to taking very long sights.
5. Error due to slip.
6. Error due to inaccurate bisection to targets.
7. Mistakes.

Natural Errors:

1. High winds.
2. Very high temperature causing expansion.
3. Error due to glare of sun.
4. Poor visibility.
5. Error due to unequal settlement of tripod legs.
6. Error due to unequal refraction.

Instrumental errors:

1. Permanent adjustment not perfect.
2. Graduation of staff not uniform.
3. The value of multiplying constant not correct.
4. Error due to imperfect adjustment of the plate.
5. Idex erroe i.e, error due to imperfect adjustment of vertical circle.
6. Error due to horizontal axis not being perpendicular to the vertical axis.

Observational errors:

1. Error due to incorrect centering or levelling.
2. Error due to Staff not being perpendicular.
3. Error due to imperfect focusing.i.e, parallex error causing incorrect readings.
4. Error due to taking very long sights.
5. Error due to slip.
6. Error due to inaccurate bisection to targets.
7. Mistakes.

Natural Errors:

1. High winds.
2. Very high temperature causing expansion.
3. Error due to glare of sun.
4. Poor visibility.
5. Error due to unequal settlement of tripod legs.
6. Error due to unequal refraction.