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TACHEOMETRY

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What is tacheometry??
Easy and cheap method of collecting much topographic data.
Tachymetry (or tacheometry) also called “stadia surveying” in countries like England and the United States
means “fast measurement”; rapid and efficient way of indirectly measuring distances and elevation differences
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Figure 1 shows the set-up of a tachymetric measurement.
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Tacheometry
Concept
Determine distances indirectly using triangle geometry
Methods
Stadia
Establish constant angle and measure length of opposite side
Length increases with distance from angle vertex
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Stadia System
The theodolite/auto level is directed at the level staff and the distance is measured by reading the top and bottom stadia hairs on the telescope view.
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Stadia System
In the first type the distance between the two. There are two types of instruments used for stadia surveying.
Stadia hairs in the theodolite telescope is fixed.
In the second type of equipment the distance between the stadia hairs is variable, being measured by means of a micrometer.
The most common method used involves the fixed hair tacheometer, or theodolite.
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Electronic Tacheometry:
Uses a total station which contains an EDM, able to read distance by reflecting off a prism.
Subtense Bar system:
An accurate theodolite, reading to 1" of arc, is directed at a staff, two pointings being made and the small subtended angle measured
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Equipment
Measurement can be taken with theodolites, transits and levels and stadia rods
While in the past, distances were measured by the “surveyor’s chain”, this can be done easier and faster using a telescope equipped with stadia hairlines in combination with a stadia rod (auto level and staff)
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Tacheometry: Stadia L2 d2  L1 d1
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Stadia Readings
Upper Hair
Lower Hair
Middle Hair
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Stadia Principles C d c f A b b' i S a a' F B D
A,B rod intercepts
a, b stadia hairs
S = rod intercept
F = principal focus of objective lens
f = focal length
i = stadia hair spacing
c = distance from instrument center to objective lens center
C = stadia constant
K = f/i = stadia interval factor
d = distance from focal point to rod
D = distance from instrument center to rod
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Stadia Equations
From similar triangles
Horizontal sights
Inclined sights
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13. Tacheometry: SubtenseL 2 d2 L 1 d1
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Subtense Equation
Derive equation for computing distance by subtense  L d
What value would you choose for L?
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The notes below shows the calculation of the distance (D) from the centre of the fixed hair tacheometer to a target.
staff A
Centre of instrument b
Object lens S F i x X o c f a U V B D
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From the diagram, triangles AOB , a O b are similar
Also if OF = f = focal length of object lens
Then =
(lens equation) and multiply both sides by (U f)
U = f + f 1 1 1 U V f U V AB
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AB is obtained by subtracting the reading given on the staff by the lower stadia hair from the top one and is usually denoted by s (staff intercept), and ab the distance apart of the stadia lines is denoted by i.
This value i is fixed, known and constant for a particular instrument.
U = s + f
D = s + ( f + c ) f i f i
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The reduction of this formula would be simplified considerably if the term (f / i) is made some convenient figure, and if the term (f + c) can be made to vanish.
D = C.S + k
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19. Constant determination
In practice, the multiplicative constant generally equals 100 and the additive constant equals zero.
This is certainly the case with modern instruments by may not always be so with older Theodolites.
The values are usually given by the makers but this is not always the case.
It is sometimes necessary to measure them in an old or unfamiliar instrument.
The simplest way, both for external and internal focusing instruments, is to regard the basic formula as being a linear one of the form:
D = C.S + k
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20. anallatic lens the additive constant k = 0
On a fairly level site chain out a line 100 to 120m long, setting pegs at 25 to 30 meter intervals.
Set at up at one end and determine two distances using tacheometer or theodolite, one short and one long. hence C and K may be determined.
I.E D1 (known) = C.S1 (known) + k
D2 (known) = C.S2 (known) + k Where the instrument designed with an
anallatic lens the additive constant k = 0
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For example:
Distance Readings Intervals
(m) upper Stadia Centre Lower Stadia upper lower total
D =C.S + k
30.00 = * C + k
90.00 = * C + k
therefore C = 100 & K = 0
Any combination of equations gives the same result, showing that the telescope is anallatic over this range, to all intents and purposes.
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Case of inclined sights
Vertical elevation angle: ө S h L V B ө ∆L hi A D
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L = C S cos Ө + K ,
D = L cos Ө
Then ;
D = CS cos2 Ө + K cos Ө ;
V = L sin Ө = ( C S cos Ө + K ) sin Ө
= 1/2 C S sin 2Ө + K sin Ө ;
∆L = h i + V – h = R.L. of B - R.L. of A ;
Where : h is the mid hair reading
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Vertical depression angle: ө hi V A S ∆L h B D
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D = CS cos2 Ө + K cos Ө ;
= 1/2 C S sin 2Ө + K sin Ө ;
∆L = - h i + V + h = R.L. of A - R.L. of B ;
Where : h is the mid hair reading ;
Ө may be elevation or depression
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Example
From point D three points A, B and C have been observed as follows:
If the reduced level of D is m. , hi = 1.40 m. and the tacheometeric constant = , it is required to:
I ) find the horizontal distance to the staff points and their reduced levels.
II) find the distance AB , BC , and CA.
Stadia readings
Vertical angles
bearing
Staff points
(1.10,1.65,2.20)
5º 12΄
85º 30΄ A
(2.30,2.95,3.60)
125º 10΄ B
(1.45,2.15,2.85)
9º 30΄
104º 30΄ C
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27. ERT 247 GEOMATICS ENGINEERINGH1 D ө1 H3 ө2 B H2 C
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Solution
For line DA
S1 = 2.20 – 1.10 = m
H1 = 100 x 1.10 x Cos2 (+5o 12’) = m
V1 = x tan (+5o 12’) = m
R.L.of A = – 1.65 = m.
For line DB
S2 = 3.60 – 2.30 = 1.30 m.
H2 = 100 x 1.30 x Cos2 (+00.00) = 130 m.
V2 = 130 x tan (+00.00) = m.
R.L. of B = – 2.95 = m.
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For line DC
S3 = 2.85 – 1.45 = 1.40 m.
H3 = 100 x 1.40 x Cos2 (+9o 30’) = m.
V3 = tan (+9o 30’) = m.
R.L. of C = – 2.15 = m.
θ1 = 104o 30’ – 85o 30’ = 19o 00’
θ2 = 125o 10’ – 104o 30’ = 20o 40’
θ = 19o 00’ + 20o 40’ = 39o 40’
From Triangle DAC
AC =
AC = m
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From Triangle DCB
BC=
BC= m
From Triangle DAB
AB=
AB= m
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Tangential system
Horizontal line of sight : S Ө Ө S D D
D = S / tan Ө
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Inclined line of sight : Ө1 Ө1 Ө2 Ө2 S D D
D = S / ( tan Ө2 – tan Ө1 )
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Subtense bar system
1 m
1 m
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For distance up to 80 m
Subtense bar
theodolite α
2 m
D = cot( α / 2 )
plan
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For distance 80 – 160 m α1 α2
D1 = cot (α1/2)
D2 = cot (α2/2)
D = D1 + D2
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For distance 160 – 350 m
Auxiliary
base x α β
900
Theodolite 1
Theodolite 2
x/2 β α
x/2
X = ( 2D )1/2 ;
X = cot ( α/2 ) , D = X cot β , D = X/2 cot β/2
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For distance 350 – 800 m x α β2 β1
x/2 β1 β2 D1 D2
X = 0.7( 2D )1/2 ;
X = cot ( α/2 ) , D = X ( cot β1 + cot β2 ) ,
D = X/2 [ cot (β1/2) + cot (β2/2) ]
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38. Electronic Tacheometry
The stadia procedure is used less and less often these days, more commonly geomatic engineers use a combination theodolite-EDM known in jargon as a total station.
Often these instruments are connected to a field computer which stores readings and facilitates the processing of the data electronically.
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This instrumentation has facilitated the development of this method of detail and contour surveying into a very slick operation.
It is now possible to produce plans of large areas that previously would have taken weeks, in a matter of days.
The math's behind the operation is very simple, it is in effect the same as the stadia formulae with the term for the distance replaced by the measured slope distance.
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reflector D Hr V A Ө HI B S
S = D cos Ө
R.L.of point A = R.L.of point B + HI + V - Hr
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41. Tacheometry Field Procedure
Set up the instrument at a reference point
Read upper, middle, and lower hairs.
Release the rodperson for movement to the next point.
Read and record the horizontal angle (azimuth).
Read and record the vertical angle (zenith).
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Error Sources
There are 4 main sources of error:
Staff Readings
Tilt of the Staff
Vertical Angle
Horizontal Angle
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THANK YOU
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