SURVEY-II
(TACHEOMETRY)
Definition of Tacheometry
Tacheometry is a rapid and efficient way of indirectly
measuring distances and elevation differences. The accuracy
attainable with stadia is suitable for lower order trigonometric
leveling, locating topographic details for mapping, measuring
lengths of back sights and fore sights in differential leveling, and
making quick checks of measurements made by higher order
methods.
Tacheometry
• The word tachymetry is derived from the Greek takhus metron
meaning ‘swift measurement’.
• It is a branch of surveying where height and distances between
ground marks are obtained by optical means only.
• An example of tachymetry method is the stadia method. The distance
between marks can be obtained without using a tape. The
tachymeter is any theodolite adapted, or fitted with an optical device
Cross
to enable measurement to be made optically.
Hair
reticle
i=
Stadia
Interval
Two Types of Stadia Hair
Principles of Stadia
Distance “H” = 100*i
With the line of sight horizontal and directed towards a
graduated rod held vertically at a point
i = 1/100*H
Standard symbols used in stadia measurements and their definitions are as follows:
f = focal length of lens
i = spacing between stadia hairs
f/i=stadia interval factor, usually 100 and denoted by K
Methods of Tacheometric Measurements
• The stadia System
• The tangential System
• Measurements by means of special instruments
Fixed Hair Method/ Stadia Method
Movable Hair Method/ Sub-tense Method
The intercept on the levelling staff is kept constant and
the distance between the stadia hairs are variable
Fixed Hair Method
• Stadia hairs are kept at fixed
interval
• The intercepts of the levelling staff
varies, depending upon the
horizontal distance between the
instrument station and the staff.
• The intercept used in the
computation is calculated by
subtracting the lower stadia
reading from the upper stadia
reading.
The Stadia System
Diaphragm
Fixed Hair Method
Principle of Stadia Tacheometric Method
OC D D1 D2
f
=
=
=
=
= k (Constant)
AB S
S1 S2
i
f =Focal Length
i = Stadia Interval
Where, k is multiplying factor
f
i
is stadia interval factor
Constant for a given instrument
The Stadia System
Fixed Hair Method
Principle of Stadia Tacheometric Method
D
1 Cot β/2
=
S
2
k Constant
Depend on β
Hence, D = Sk
The Stadia System
Fixed Hair Method
Principle of Stadia Tacheometric Method
If β = 34՛ 22 ՛ ՛
K=?
The Stadia System
Fixed Hair Method
The Stadia System
 In actual practice, observations may be made with either horizontal
line of sight or with inclined line of sight.
 In the later case the staff may be kept either vertically or normal to
the line of sight.
First
the distance-elevation formulae for the horizontal sights
should be derived
Fixed Hair Method
The Stadia System
Horizontal Sights
M = Instrument Centre
O = Optical Centre
C = MF = d+f = Additive Constant
S = Staff Intercept
Consider the figure, in which O is the optical centre of the objective of an
external focusing telescope.
Let A, C, and B = the points cut by the three lines of sight corresponding to
three wires.
b, c, and a = top, axial and bottom hairs of the diaphragm.
ab = i = interval b/w the stadia hairs (stadia interval)
AB = s = staff intercept;
f = focal length of the objective
Fixed Hair Method
Horizontal Sights
M = Instrument Centre
O = Optical Centre
C = MF = d+f = Additive Constant
S = Staff Intercept
u = horizontal distance of the staff from the optical centre of the objective
v = horizontal distance of the cross-wires 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 = Centre of the instrument, corresponding to the vertical axis.
The Stadia System
Fixed Hair Method
Horizontal Sights
From the Figure
v
i
=
u
S
ui
Or v =
S
……… (1)
The Stadia System
Lens Formula
A lens formula may be defined as the formula which gives
the relationship between the distance of image (v),
distance of object (u), and the focal length (f) of the lens. It
may be written as:
Where, v = Distance of image from optical centre of lens
u = Distance of object from optical centre of lens
and f = Focal length of lens
The lens formula is applicable both in convex lenses and
concave lenses.
The Stadia System
Fixed Hair Method
Horizontal Sights
From the Figure
v
i
=
u
S
ui
Or v =
S
……… (1)
S
) ……… (2)
u = f (1 +
i
The Stadia System
Fixed Hair Method
The Stadia System
Horizontal Sights
M = Instrument Centre
O = Optical Centre
C = MF = d+f = Additive Constant
S = Staff Intercept
Now, from the figure, the horizontal distance from the centre of the
instrument (M) to the stadia rod => D = u + d
Put values from equation (2)
Fixed Hair Method
Horizontal Sights
ui
……… (1)
Or v =
S
S
) ……… (2)
u = f (1 +
i
C = MF = d+f = Additive Constant
D = kS + C ……… (3)
The Stadia System
The Stadia System
Calculation of Constant k & C
D = kS + C ……… (4)
D1 = kS1 + C ……… (5)
D2 = kS2 + C ……… (6)
k=
D1 - D
……… (7)
S1 - S
C=
DS1 – D1S
S1 - S
……… (8)
Fixed Hair Method
The Stadia System
First
the distance-elevation formulae for the horizontal sights
should be derived
Inclined line of sight
Staff
Vertical
Staff
Normal
Fixed Hair Method
The Stadia System
Inclined line of Sights but Staff Vertical
M = Instrument Centre
O = Optical Centre
C = MF = d+f = Additive Constant
S = Staff Intercept <
< BCB՛ = 90 – (90 – θ) = θ
< ACA՛ = < BCB՛ = θ
< CA՛F = 90 – β/2
< CA՛A = 180 – (90 - β/2) = 90 + β/2
In spherical astronomy, the parallactic angle is the angle between the great circle through a celestial object and the
zenith, and the hour circle of the object. Despite its name, this angle is unrelated with parallax. The parallactic angle is
zero when the object crosses the meridian.
Fixed Hair Method
Inclined line of Sights but Staff Vertical
A՛C + B՛C = (AC + BC) Cosθ) = S Cosθ ………
l = (f/i) S Cosθ = KS Cos θ
……… (2)
L = KS Cos θ + c
D = KS Cos² θ + c Cosθ
V = ½ KS Sin2θ + c Sinθ
V = D tanθ
(1)
The Stadia System
Fixed Hair Method
Inclined line of Sights but Staff Normal
The Stadia System
Fixed Hair Method
Inclined line of Sights but Staff Normal
Solve it yourself as 1st Assignment.
The Stadia System
???
PRINCIPLES OF OPTICAL DISTANCE MEASUREMENT
Isosceles triangle
Distance AB = ½ (Cd) x Cot α/2
Whereby
If distance AB = D, distance Cd = S , so D = distance between two point
S = base line
D = ½ S Cot α/2
α = paralactic angle
Fixed Hair Method
Horizontal Sights
f1
f2
b
i
c
a
A
.
O
C
M
d
D
B
s