1. TACHEOMETRY

2. INTRODUCTION
Tacheometry or tachemetry or telemetry is a branch of angular surveying in which the horizontal and Vertical distances of points are obtained by optical means as opposed to the ordinary slower process of measurements by tape or chain.
The method is very rapid and convenient.
It is best adapted in obstacles such as steep and broken ground, deep ravines, stretches of water or swamp, which make chaining difficult or impossible
The primary object of tacheometry is the preparation of contoured maps or plans requiring both the horizontal as well as vertical control. Also, on surveys of higher accuracy, it provides a check on distances measured with the tape.

3. INTRODUCTION
Tacheometry (from Greek, quick measure), is a system of rapid surveying, by which the positions, both horizontal and vertical, of points on the earth surface relatively to one another are determined without using a chain or tape or a separate leveling instrument.

4. TACHEOMETRY’S APPLICATIONS
The tacheometric methods of surveying are used with advantage over the direct methods of measurement of horizontal distances and differences in elevations. Some of the uses are:
Preparation of topographic maps which require both elevations and horizontal distances.
Survey work in difficult terrain where direct methods are inconvenient
Detail filling
Reconnaissance surveys for highways, railways, etc.
Checking of already measured distances
Hydrographic surveys and

5. INSTRUMENTS USED
TACHEOMETER

6. INSTRUMENTS USED
STADIA DIAPHRAGM

7. INSTRUMENTS USED
VERTICAL HOLDING
NORMAL HOLDING
STADIA ROD

8. Merit and Demerits of Vertical and Normal Holding
INSTRUMENTS USED
Merit and Demerits of Vertical and Normal Holding
It is a bit easy to ensure that the stuff is perfectly vertical.
A slight error in not keeping the staff vertical causes a serious error in computation of distance
In the case of an inclined sight, it is difficult to keep the staff perpendicular to the line of sight during high winds and in rough country
In the base of normal holding, the accuracy of the direction of the staff can be judged by the transitman even during high winds

9. INSTRUMENTS USED
SUBTENSE THEODOLITE
SUBTENSE BAR

10. INSTRUMENTS USED
now

11. METHODS OF TACHEOMETRY
STADIA METHOD
TANGENTIAL METHOD
OBSERVATION ARE MADE FOR VERTICAL ANGLE
STAFF INTERCEPTS OBTAINED BY CROSS-WIRES ONLY.
SIMILAR TO THE METHOD OF TRIGONOMETRILCAL LEVELLING
RANGE FINDING
TO FIND HORIZONTAL DISTANCE AND DIRECTION OF A LINE WITHOUT GOING TO THE FAR END OF THE LINE.
USING INSTRUMENT KNOWN AS RANGE FINDER.
ONLY HORIZONTAL DISTANCE CAN BE MEASURED.
FIXED HAIR METHOD
MOVABLE HAIR METHOD (SUBTENSE METHOD)
NON STADIA METHOD

12. METHODS OF TACHEOMETRY
STADIA METHOD
FIXED HAIR METHOD
MOVABLE HAIR METHOD (SUBTENSE METHOD)

13. FIXED HAIR METHOD Most prevalent for tacheometric surveying
 In this method, the telescope of the theodolite is equipped with two additional cross hairs, one above and the other below the main horizontal hair at equal distance. 

14. PRINCIPAL OF STADIA METHOD (HORIZONTAL SIGHT)
A tacheometer is temporarily adjusted on the station P with horizontal line of sight. Let a and b be the lower and the upper stadia hairs of the instrument and their actual vertical separation be designated as i. Let f be the focal length of the objective lens of the tacheometer and c be horizontal distance between the optical centre of the objective lens and the vertical axis of the instrument. Let the objective lens is focused to a staff held vertically at Q, say at horizontal distance D from the instrument station.

15. PRINCIPAL OF STADIA METHOD (CONT.)
By the laws of optics, the images of readings at A and B of the staff will appear along the stadia hairs at a and b respectively. Let the staff interval i.e., the difference between the readings at A and B be designated by s. Similar triangle between the object and image will form with vertex at the focus of the objective lens (F). Let the horizontal distance of the staff from F be d. Then, from the similar Ds ABF and a'b' F,

16. PRINCIPAL OF STADIA METHOD (CONT.)
as a’ b’ = ab = i. The ratio (f /i) is a constant for a particular instrument and is known as stadia interval factor, also instrument constant. It is denoted by K and thus  
d = Ks

17. PRINCIPAL OF STADIA METHOD (CONT.)
The horizontal distance (D) between the center of the instrument and the station point (Q) at which the staff is held is d + f + c. If C is substituted for (f + c), then the horizontal distance D from the center of the instrument to the staff is given by the equation
K = 𝒇 𝒊
D = Ks + C
C = f + c
The distance C is called the stadia constant. Equation above is known as the stadia equation for a line of sight perpendicular to the staff intercept.

18. PRINCIPAL OF STADIA METHOD (HORIZONTAL SIGHT) SUMMARY
D = Ks + C
K = 𝒇 𝒊
C = f + c
f =Focal length
D =Distance between vertical axis of tacheometer and the staff
C =Distance between vertical axis of tacheometer and the optical center
i =Stadia interval
s = Staff intercept(A-B)

19. EXAMPLE 1 K = 𝒇 𝒊 C = f + c D = Ks + C How to find the i ??
The stadia readings with horizontal sight on a vertical staff held 50 m(D) from a tacheometer were m(B) and m(A). The focal length of the object glass was 25 cm(f). The distance between the object glass and the vertical axis of the tacheometer was 15 cm(c). Calculate the stadia interval,i. Given , D = 50 m Upper stadia, A = m Lower stadia, B = m s = A-B = = m f = 25 cm C = f + c = = 40 cm  0.4 m
K = 𝒇 𝒊
EQUATON 1.1
C = f + c
EQUATON 1.2
D = Ks + C
EQUATON 1.3
How to find the i ??

20. EXAMPLE 1 (SOLUTION) 𝐷= 𝑓 𝑖 𝑠+(𝑓+𝑐) 50= 𝟎.𝟐𝟓 𝒊 (𝟎.𝟒𝟗𝟓)+(𝟎.𝟒)
D = 50 m Upper stadia, A = m Lower stadia, B = m s = A-B =
= m f = 25 cm C = f + c = = 40 cm  0.4 m
𝐷= 𝑓 𝑖 𝑠+(𝑓+𝑐)
50= 𝟎.𝟐𝟓 𝒊 (𝟎.𝟒𝟗𝟓)+(𝟎.𝟒)
i = 2.49 x 10 −3 𝑚 = 2.49 mm

21. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS)
It is usual that the line of sight of the tacheometer is inclined to the horizontal. Thus, it is frequently required to reduce the inclined observations into horizontal distance and difference in elevation.

22. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS)
Let us consider a tacheometer (having constants K and C) is temporarily adjusted on a station, say P . The instrument is sighted to a staff held vertically, say at Q. Thus, it is required to find the horizontal distance PP1 (= H) and the difference in elevation P1Q.

23. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS)
Let A, R and B be the staff points whose images are formed respectively at the upper, middle and lower cross hairs of the tacheometer. The line of sight, corresponding to the middle cross hair, is inclined at an angle of elevation q and thus, the staff with a line perpendicular to the line of sight. 

24. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS)
D(OR) = C + K.s cos q
H = D cos q
The distance D (= OR) is C + K. s cos  q  (from Equation 1.3). But the distance OO1 is the horizontal distance H, which equals OR cos  q. Therefore the horizontal distance H is given by the equation. 
H = (C + K.s cos q) cos q)
H = (C cos q + K.𝐬 𝒄𝒐𝒔 𝟐 q )

25. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS)
D(OR) = C + K.s cos q
V = D sin q
in which K is the stadia interval factor (f / i), s is the stadia interval, C is the stadia constant (f + c), and  q  is the vertical angle of the line of sight read on the vertical circle of the transit.
The distance RO1, which equals OR sin  q, is the vertical distance between the telescope axis and the middle cross-hair reading. Thus V is given by the equation
V = (C + K.s cos q) sin q)
V = (C sin q + K.s cos q sin q)
Sin2q = 2 cos q sin q
V = C sin q + 1/2( K.s sin2q)

26. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS)
Difference in Elevation = (h+V-r)
Thus, the difference in elevation between P and Q is (h + V - r), where h is the height of the instrument at P and r is the staff reading corresponding to the middle hair.
Elevation of staff station for angle of depression = h – V - r

27. EXAMPLE 2
In order to carry out tacheometric surveying, following observations were taken through a tacheometer set up at station P at a height 1.235m. Compute the horizontal distance(D) of S from P and reduced level of station at S if R.L. of station P is m

28. V = c sin q + 1/2( k.s sin2q) h = 1.235 m r = 2.275 m
EXAMPLE 2 (SOLUTION)
To obtain distance (V,H), First ,we need to obtain the K value
FIND : a) DISTANCE(D)
b) R.L Station S
D = K s cos2  q +  C.cos  q
 R.L. of station S = R.L. of P + h + V - r
V = c sin q + 1/2( k.s sin2q)
h = m
r = m
R.L P = m

29. K= 100 1.02 EXAMPLE 2 (SOLUTION) SOLVING EQ. 1 & 2 K = 98.04 C = 0.98
Since the staff station P and Q are at known distances and observations are taken at horizontal line of sight,
from D = K.s + C, we get
100 = K C Equation 1
200 = K C Equation 2
SOLVING EQ. 1 & 2
where K and C are the stadia interval factor and stadia constant of the instrument. K=
K = 98.04
C = 0.98

30. EXAMPLE 2 (SOLUTION) K = 98.04 C = 0.98 D = K s cos2 q + C.cos q
Now, S is the intercept of A’ and B’ A’
R(r) B’
s = = m
K = 98.04
C = 0.98
q = 5˚ 24’ 40’’
D = K s cos2  q +  C.cos  q

31. V = c sin q + 1/2( k.s sin2q) R.L P = 262. 575m h = 1.235 m V = 20.203
EXAMPLE 2 (SOLUTION)
q = 5˚ 24’ 40’’
 R.L. of station S = R.L. of P + h + V - r
K = 98.04
V = c sin q + 1/2( k.s sin2q)
C = 0.98
s = 2.185
 R.L. of station S = R.L. of P + h + V - r
R.L P = m
h = m
 R.L. of station S
= m
V =
r = m

32. Example

33. Example

34. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS-STAFF NORMAL)
Let ,
AB = s = staff intercept
CE = h = central hair reading
= angle of elevation
OC = L = inclined distance
Drop perpendicular CF’ to horizontal OF. Then,
L = Ks + C
OF’ = (Ks + C) cos q
But, D = OF’ + F’F
= (Ks + C) cos q + h sin q
Elevation of the staff station, V = OC sin q
= L sin q
= (Ks + C) sin q
Elevation of staff station = HI + V – h cos q

35. PRINCIPAL OF STADIA METHOD (INCLINED SIGHTS-STAFF NORMAL)
Let ,
AB = s = staff intercept
CE = h = central hair reading
= angle of elevation
OC = L = inclined distance
L = Ks + C
OF’ = L cos q = (Ks + C) cos q
Now,
D = OF’ – FF’ = OF’ – EE’
= (Ks + C) cos q - h sin q
Elevation of staff station, V = OC sin q
= L sin q
= (Ks + C) sin q
Elevation of staff station = HI – V – h cos q

36. The stadia method of surveying is particularly useful for following cases:
In differential leveling, the backsight and foresight distances are balanced conveniently if the level is equipped with stadia hairs.
In profile leveling and cross sectioning, stadia is a convenient means of finding distances from level to points on which rod readings are taken.
In rough trigonometric, or indirect, leveling with the transit, the stadia method is more rapid than any other method.
For traverse surveying of low relative accuracy, where only horizontal angles and distances are required, the stadia method is a useful rapid method.
On surveys of low relative accuracy - particularly topographic surveys-where both the relative location of points in a horizontal plane and the elevation of these points are desired, stadia is useful. The horizontal angles, vertical angles, and the stadia interval are observed, as each point is sighted; these three observations define the location of the point sighted.

37. ACCURACY
The degree of accuracy that can be achieved by a tacheometer depends by:
Power and optical precision of the instrument
Vision of an observer
Inclination line of collimation
Position of the sun
Length of sight

38. *This note has been amended from the original note prepared by Puan Nur Fitriah Isa of JTKA.