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Tacheometric Surveying 
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Tacheometry 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.
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.
Uses of Tacheometry
The tacheometric methods of surveying are used with an advantage over the direct methods of measurement of horizontal distances and differences in elevations.
Some of the uses are:
Instrument
Different types of stadia hairs are shown below
Methods of Tachometric Survey
Various methods of tacheometry survey are based on the principle that the horizontal distance between an instrument Station “A” and a staff station “B” depending on the angle subtended at point “A” by a known distance at point “B” and the vertical angle from point “B” to point “A” respectively.
This principle is used in different methods in different ways.
Mainly there are two methods of tachometry survey:
1. Stadia System of Tacheometry
In the stadia system, the horizontal distance to the staff Station from the instrument station and the elevation of the staff station concerning the line of sight of the instrument is obtained with only one observation from the instrument Station.
In the stadia method, there are mainly two systems of surveying.
(i) Fixed Hair Method
(a) Principle of Stadia Hair Method
This constant k entirely depends upon the magnitude of the angle β
(b) Horizontal Line of Sight
Consider the figure, in which O is the optical center of the objective of an external focusing telescope.
ab = i = interval b/w the stadia hairs (stadia interval)
AB = s = staff intercept,
f = focal length of the objective
f_{1} = horizontal distance of the staff from the optical center of the objective
f_{2} = horizontal distance of the crosswires from O.
d = distance of the vertical axis of the instrument from O.
D = horizontal distance of the staff from the vertical axis of the instruments.
M = center of the instrument, corresponding to the vertical axis.
Horizontal distance between the axis and the staff is
D = (f/i)s + (f + d) = k.s + C
Above equation is known as the distance equation. In order to get the horizontal distance, therefore, the staff intercept s is to be found by subtracting the staff readings corresponding to the top and bottom stadia hairs.
(c) Inclined Line of Sight (staff held vertical)
MC = L = k.S.Cosθ + C
Horizontal Distance
MQ’ = D = L.Cosθ
Vertical Distance
CQ’ = V = L.Sinθ
Elevation of the staff station for the angle of elevation:
If the line of sight has an angle of elevation θ, as shown in the figure, we have
Elevation of staff station = Elevation of instrument station + h + V – r.
Elevation of the staff station for the angle of depression:
Elevation of Q = Elevation of P + h – V  r
(d) Inclined Line of Sight (staff held normal to the sight)
Case(a): Line of Sight at an angle of elevation Θ
Let AB = s = staff intercept;
CQ = r = axial hair reading
With the same notations as in the last case, we have
MC = L = Ks + C
The horizontal distance between P and Q is given by
D = MC' + C'Q' = LcosΘ + rsinΘ
=(k s + C)cosΘ + rsinΘ
Similarly, V = lsinΘ = (k s + C) sinΘ
Case(b): Line of Sight at an angle of depressionθ
Figure shows the line of sight depressed downwards,
MC = L = k s + C
D = MQ' = MC'  Q'C'
= L cosΘ  r sinΘ
D = (k s + C)cosΘ  r sin Θ
V = L sin Θ = (k s + C) sin Θ
Elevation of Q = Elevation of P + h  V  r cosΘ
(ii) Movable Hair Method (Subtense Method)
In the movable Hair method of tacheometric surveying, the instrument used for taking observations consist of a telescope fitted with stadia hairs which can be moved and fixed at any distance from the central hair (within the limits of the diaphragm).
The staff used with this instrument consists of two targets (marks) at a fixed distance apart (say 3.4 mm).
The Stadia interval which is variable for the different positions of the staff is measured, and the horizontal distance from the instrument station to the staff station is computed.
Note: Out of the two methods mentioned above of tacheometric surveying, the “fixed hair method “is widely employed.
2. Tangential System of Tacheometric Surveying
Case 1: When both angles are above the horizontal line of sight
V = D tanθ_{1}
V + s = D tanθ_{2}
Thus, s = D ( tanθ_{2}  tanθ_{1})
Case 2: When both angles are below the horizontal line of sight
Thus, s = D ( tanθ_{1}  tanθ_{2})
Or D = s/(tanθ_{2}  tanθ_{1})
and V = (s/tanθ_{1}  tanθ_{2}))tanθ_{1}
Therefore R.L. of Q = (R.L. of P + h)  V – r
where h is the height of the instrument, r is the staff reading corresponding to the lower vane.
Case 3: When one angle is above and one angle is below the horizontal line of sight
Thus, s = D (tanθ_{2} + tanθ_{1})
Or, D = S/(tanθ_{2} + tanθ_{1})
and, V = (s/(tanθ_{2} + tanθ_{1}))tanθ_{1}
Therefore R.L. of Q = (R.L. of P + h)  V – r
where h is the height of the instrument, r is the staff reading corresponding to the lower vane.
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