Horizontal and Vertical Curves | Geomatics Engineering (Surveying) - Civil Engineering (CE) PDF Download

 Page 1


  
Curves
Introduction
Curves are generally used on highways and railways where 
it is necessary to change the direction of motion. This chap-
ter focuses on the elements of curves and methods for set-
ting out of curves on ground.
Curve may be circular, parabolic or spiral and is always 
tangential to the two straight directions.
Circular curves are of three types:
 1. Simple curves
 2. Compound curves
 3. Reverse curves
Simple Curves 
Simple curves consists of a single arc of a circle. It is tan-
gential to both the straight lines.
 1. Back tangent (AT
1
): The tangent before curve is 
called back tangent or fi rst tangent.
 2. Forward tangent (BT
2
): The tangent following the 
curve is called forward tangent or second tangent.
 3. Point of intersection, vertex (V): It is the point of 
intersection of two tangents AT
1
 and BT
2
.
? 
 
2
?
2
?
 
A
R 
O 
B 
D 
C
V ' 
(PI) V 
(PT) T 2 
(PC)   T 1 
R 
 4. Point of curve (T
1
): It is the beginning of the curve 
where the alignment changes from a tangent to a 
curve.
 5. Point of tangency (T
2
): It is the end of the curve 
where the alignment changes from a curve to a 
tangent.
 6. Defl ection angle or Deviation angle (?): The 
diff erence between the slopes of the two tangents.
Curves and
Triangulation Survey
Part III_Unit 12_Chapter 05.indd   1 5/31/2017   4:56:03 PM
Page 2


  
Curves
Introduction
Curves are generally used on highways and railways where 
it is necessary to change the direction of motion. This chap-
ter focuses on the elements of curves and methods for set-
ting out of curves on ground.
Curve may be circular, parabolic or spiral and is always 
tangential to the two straight directions.
Circular curves are of three types:
 1. Simple curves
 2. Compound curves
 3. Reverse curves
Simple Curves 
Simple curves consists of a single arc of a circle. It is tan-
gential to both the straight lines.
 1. Back tangent (AT
1
): The tangent before curve is 
called back tangent or fi rst tangent.
 2. Forward tangent (BT
2
): The tangent following the 
curve is called forward tangent or second tangent.
 3. Point of intersection, vertex (V): It is the point of 
intersection of two tangents AT
1
 and BT
2
.
? 
 
2
?
2
?
 
A
R 
O 
B 
D 
C
V ' 
(PI) V 
(PT) T 2 
(PC)   T 1 
R 
 4. Point of curve (T
1
): It is the beginning of the curve 
where the alignment changes from a tangent to a 
curve.
 5. Point of tangency (T
2
): It is the end of the curve 
where the alignment changes from a curve to a 
tangent.
 6. Defl ection angle or Deviation angle (?): The 
diff erence between the slopes of the two tangents.
Curves and
Triangulation Survey
Part III_Unit 12_Chapter 05.indd   1 5/31/2017   4:56:03 PM
      
 7. Tangent distance: It is the distance from PC to PI 
(also the distance from PI to PT)
T
1
 V = T
2
 V = R tan
?
2
·
  R is the radius of the curve.
 8. External distance or apex distance (E): The 
distance from vertex V to the centre of the curve C.
E = VC = R sec
?
2
1 -
?
?
?
?
?
?
 9. Length of curve (L): Distance from PC to PT,
L = T
1
CT
2
 = R ?  ? in radians.
 10. Length of long chord: It is the chord joining PC to 
PT.
TT R
12
2
2
= sin
?
 11. Mid-ordinate/versed sine: The distance from mid-
point of the long chord to mid-point of the curve.
CD R =-
?
?
?
?
?
?
1
2
cos
?
 12. Normal chord: A chord between two successive 
regular stations on a curve.
 13. Sub-chord: Any chord shorter than the normal chord.
 14. Right-hand curve: If the curve deflects to the right 
of the direction of the progress of survey, it is called 
the right-hand curve.
 15. Left-hand curve: If the curve deflects to the left of 
the direction of the progress of survey, it is called the 
left-hand curve.
Chainages
 Chainage of T
1
 = Chainage of V – Tangent length
 Chainage of T
2
 = Chainage of T
1
+ Length of Curve
Degree of a Curve (D)
 • The central angle subtended by a chord of a fixed length.
 • In India, the fixed length of chord is taken as 20 m to cal-
culate the degree of curve.
 360 degrees = 2pR
 D degrees = 20 m
 ? Therefore, R = 
1146
D
metres
Methods of Setting out a Curve
Linear Methods
In this method, only a chain or tape is used. Linear methods 
are used when:
 • A high degree of accuracy is not required.
 • The curve is short.
 1. By ordinates or offsets from the long chord:
  Mid-ordinate, 
OR R
L
R
0
2
2
=- -
?
?
?
?
?
?
O
x
 = RX RO
22 0
-- - ()
= 
XK X
R
() -
2
(approx)
  O
x
 = Ordinate at distance of ‘X’ from central ordinate.
  To set out the curve, the long chord is divided into an 
even number of equal parts. Offsets calculated at each 
point are then set out at each of these points.
 2. By successive bisection of arcs or chords:
 • Join tangent points T
1
, T
2 
and bisect them at D. 
Erect a perpendicular DC whose length is equal to 
versed sine of the curve.
CD RR R
L
=-
?
?
?
?
?
?
=- -
?
?
?
?
?
?
1
22
2
2
cos
?
 • Join T
1
C and T
2
C and bisect them at D
1
 and D
2
. 
Erect perpendiculars D
1
C
1
 and D
2
C
2
 which are 
equal to R 1
4
-
?
?
?
?
?
?
cos
?
 • By successive bisection of these chords, more 
points may be obtained and by joining all C, C
1
, 
C
2
, … points, curve is obtained.
T 2 
D 
C 2 
D
2
 
C 1
C 
T
1 
D 1
 3. By offsets from the tangents: If the deflection angle 
and the radius of curvature are both small, the curves 
can be set out by offsets from the tangent. The offsets 
from tangent are of two types.
Part III_Unit 12_Chapter 05.indd   2 5/31/2017   4:56:04 PM
Page 3


  
Curves
Introduction
Curves are generally used on highways and railways where 
it is necessary to change the direction of motion. This chap-
ter focuses on the elements of curves and methods for set-
ting out of curves on ground.
Curve may be circular, parabolic or spiral and is always 
tangential to the two straight directions.
Circular curves are of three types:
 1. Simple curves
 2. Compound curves
 3. Reverse curves
Simple Curves 
Simple curves consists of a single arc of a circle. It is tan-
gential to both the straight lines.
 1. Back tangent (AT
1
): The tangent before curve is 
called back tangent or fi rst tangent.
 2. Forward tangent (BT
2
): The tangent following the 
curve is called forward tangent or second tangent.
 3. Point of intersection, vertex (V): It is the point of 
intersection of two tangents AT
1
 and BT
2
.
? 
 
2
?
2
?
 
A
R 
O 
B 
D 
C
V ' 
(PI) V 
(PT) T 2 
(PC)   T 1 
R 
 4. Point of curve (T
1
): It is the beginning of the curve 
where the alignment changes from a tangent to a 
curve.
 5. Point of tangency (T
2
): It is the end of the curve 
where the alignment changes from a curve to a 
tangent.
 6. Defl ection angle or Deviation angle (?): The 
diff erence between the slopes of the two tangents.
Curves and
Triangulation Survey
Part III_Unit 12_Chapter 05.indd   1 5/31/2017   4:56:03 PM
      
 7. Tangent distance: It is the distance from PC to PI 
(also the distance from PI to PT)
T
1
 V = T
2
 V = R tan
?
2
·
  R is the radius of the curve.
 8. External distance or apex distance (E): The 
distance from vertex V to the centre of the curve C.
E = VC = R sec
?
2
1 -
?
?
?
?
?
?
 9. Length of curve (L): Distance from PC to PT,
L = T
1
CT
2
 = R ?  ? in radians.
 10. Length of long chord: It is the chord joining PC to 
PT.
TT R
12
2
2
= sin
?
 11. Mid-ordinate/versed sine: The distance from mid-
point of the long chord to mid-point of the curve.
CD R =-
?
?
?
?
?
?
1
2
cos
?
 12. Normal chord: A chord between two successive 
regular stations on a curve.
 13. Sub-chord: Any chord shorter than the normal chord.
 14. Right-hand curve: If the curve deflects to the right 
of the direction of the progress of survey, it is called 
the right-hand curve.
 15. Left-hand curve: If the curve deflects to the left of 
the direction of the progress of survey, it is called the 
left-hand curve.
Chainages
 Chainage of T
1
 = Chainage of V – Tangent length
 Chainage of T
2
 = Chainage of T
1
+ Length of Curve
Degree of a Curve (D)
 • The central angle subtended by a chord of a fixed length.
 • In India, the fixed length of chord is taken as 20 m to cal-
culate the degree of curve.
 360 degrees = 2pR
 D degrees = 20 m
 ? Therefore, R = 
1146
D
metres
Methods of Setting out a Curve
Linear Methods
In this method, only a chain or tape is used. Linear methods 
are used when:
 • A high degree of accuracy is not required.
 • The curve is short.
 1. By ordinates or offsets from the long chord:
  Mid-ordinate, 
OR R
L
R
0
2
2
=- -
?
?
?
?
?
?
O
x
 = RX RO
22 0
-- - ()
= 
XK X
R
() -
2
(approx)
  O
x
 = Ordinate at distance of ‘X’ from central ordinate.
  To set out the curve, the long chord is divided into an 
even number of equal parts. Offsets calculated at each 
point are then set out at each of these points.
 2. By successive bisection of arcs or chords:
 • Join tangent points T
1
, T
2 
and bisect them at D. 
Erect a perpendicular DC whose length is equal to 
versed sine of the curve.
CD RR R
L
=-
?
?
?
?
?
?
=- -
?
?
?
?
?
?
1
22
2
2
cos
?
 • Join T
1
C and T
2
C and bisect them at D
1
 and D
2
. 
Erect perpendiculars D
1
C
1
 and D
2
C
2
 which are 
equal to R 1
4
-
?
?
?
?
?
?
cos
?
 • By successive bisection of these chords, more 
points may be obtained and by joining all C, C
1
, 
C
2
, … points, curve is obtained.
T 2 
D 
C 2 
D
2
 
C 1
C 
T
1 
D 1
 3. By offsets from the tangents: If the deflection angle 
and the radius of curvature are both small, the curves 
can be set out by offsets from the tangent. The offsets 
from tangent are of two types.
Part III_Unit 12_Chapter 05.indd   2 5/31/2017   4:56:04 PM
    
 • Radial offsets
 • Perpendicular offsets
E 
O 
R 
Ox
T 1 
D 
D 
x 
A 
D
 
T 1 D 
V
E 
R 
O 
R 
  Radial offset:
OR xR
x
=+ -
22
 (Exact)
· ˜
x
R
2
2
(approx)
  Perpendicular offset:
OR Rx
x
=- -
22
(Exact)
· ˜
x
R
2
2
(approx)
 4. By offsets produced from the chords (deflection 
distances)
 • This method is very much useful for long curves 
and is generally used when a theodolite is not 
available.
C
 
D
 
C
1 
C
3 
C
1 
O
1 
B
1 
T
1 
A
1 
C
2 
O
2 
D
 
R
 
R
 
C
3 
C
2 
C
1 
O
3 
O
 
D
 
A
 
 • Assuming C
1
, C
2
, …, C
n
 sub-chord lengths and 
calculating O
1
, O
2
, …, O
n
.
 First offset, 
O
C
R
1
1
2
2
=
 Second offset, 
O
C
R
2
2
2
2
=
 O
CC C
R
n
nn n
=
+
-
()
1
2
 O
1
, O
2
,… O
n
 = 1st, 2nd, …, nth offset.
 • Great disadvantage in this method is that the error 
in fixing a point is carried forward.
Angular Methods 
In this method, an instrument such as a theodolite is used 
with or without a chain (or tape).
 1. Rankine’s method of deflection (tangential) angles:
  A deflection angle ( d) to any point on the curve is the 
angle at PC between the back tangent and the chord 
from the PC to that point.
 • d (in minutes) = 
1718 9 . C
R
 C is the length of the chord.
 R is the radius of the curve.
 • One theodolite (to measure angles), one chain or 
tape (to measure distances) are used in this method.
 • The deflection angle for any chord is equal to the 
deflection angle for the previous chord plus the 
tangential angle for that chord.
 ?
1
 = d
1
; ?
2
 = ?
1
 + d
2
; ?
n
 = ?
n–1
 + d
n
 2. Two theodolite method:
 • In this method, two theodolites are used one at PC 
and the other at P .T.
 • This method is used when the ground is unsuitable 
for chaining.
 • Only angular measurements are used (no chain or 
tape).
 • It is based on the principle that the angle between 
the tangent and the chord is equal to the angle 
which that chord subtends in the opposite segment.
 3. Tacheometric method:
 • Chaining is completely eliminated and the method 
is less accurate than Rankine’ s.
 • Setting of curve is done with stadia theodolite.
(a) Compound curves: A curve with two or more 
simple curves turn in the same direction and 
join at common tangent points.
(b) Reverse curves: Two curves turn in the oppo-
site directions. 
 [The characteristics, length and sight distance 
requirements of transition and vertical curves 
are discussed in transportation]
Triangulation Survey
Introduction
The triangulation is the system which consists of a num-
ber of inter-connected triangles in which length of one line 
and the angles of triangles are measured very precisely. 
Part III_Unit 12_Chapter 05.indd   3 5/31/2017   4:56:05 PM
Page 4


  
Curves
Introduction
Curves are generally used on highways and railways where 
it is necessary to change the direction of motion. This chap-
ter focuses on the elements of curves and methods for set-
ting out of curves on ground.
Curve may be circular, parabolic or spiral and is always 
tangential to the two straight directions.
Circular curves are of three types:
 1. Simple curves
 2. Compound curves
 3. Reverse curves
Simple Curves 
Simple curves consists of a single arc of a circle. It is tan-
gential to both the straight lines.
 1. Back tangent (AT
1
): The tangent before curve is 
called back tangent or fi rst tangent.
 2. Forward tangent (BT
2
): The tangent following the 
curve is called forward tangent or second tangent.
 3. Point of intersection, vertex (V): It is the point of 
intersection of two tangents AT
1
 and BT
2
.
? 
 
2
?
2
?
 
A
R 
O 
B 
D 
C
V ' 
(PI) V 
(PT) T 2 
(PC)   T 1 
R 
 4. Point of curve (T
1
): It is the beginning of the curve 
where the alignment changes from a tangent to a 
curve.
 5. Point of tangency (T
2
): It is the end of the curve 
where the alignment changes from a curve to a 
tangent.
 6. Defl ection angle or Deviation angle (?): The 
diff erence between the slopes of the two tangents.
Curves and
Triangulation Survey
Part III_Unit 12_Chapter 05.indd   1 5/31/2017   4:56:03 PM
      
 7. Tangent distance: It is the distance from PC to PI 
(also the distance from PI to PT)
T
1
 V = T
2
 V = R tan
?
2
·
  R is the radius of the curve.
 8. External distance or apex distance (E): The 
distance from vertex V to the centre of the curve C.
E = VC = R sec
?
2
1 -
?
?
?
?
?
?
 9. Length of curve (L): Distance from PC to PT,
L = T
1
CT
2
 = R ?  ? in radians.
 10. Length of long chord: It is the chord joining PC to 
PT.
TT R
12
2
2
= sin
?
 11. Mid-ordinate/versed sine: The distance from mid-
point of the long chord to mid-point of the curve.
CD R =-
?
?
?
?
?
?
1
2
cos
?
 12. Normal chord: A chord between two successive 
regular stations on a curve.
 13. Sub-chord: Any chord shorter than the normal chord.
 14. Right-hand curve: If the curve deflects to the right 
of the direction of the progress of survey, it is called 
the right-hand curve.
 15. Left-hand curve: If the curve deflects to the left of 
the direction of the progress of survey, it is called the 
left-hand curve.
Chainages
 Chainage of T
1
 = Chainage of V – Tangent length
 Chainage of T
2
 = Chainage of T
1
+ Length of Curve
Degree of a Curve (D)
 • The central angle subtended by a chord of a fixed length.
 • In India, the fixed length of chord is taken as 20 m to cal-
culate the degree of curve.
 360 degrees = 2pR
 D degrees = 20 m
 ? Therefore, R = 
1146
D
metres
Methods of Setting out a Curve
Linear Methods
In this method, only a chain or tape is used. Linear methods 
are used when:
 • A high degree of accuracy is not required.
 • The curve is short.
 1. By ordinates or offsets from the long chord:
  Mid-ordinate, 
OR R
L
R
0
2
2
=- -
?
?
?
?
?
?
O
x
 = RX RO
22 0
-- - ()
= 
XK X
R
() -
2
(approx)
  O
x
 = Ordinate at distance of ‘X’ from central ordinate.
  To set out the curve, the long chord is divided into an 
even number of equal parts. Offsets calculated at each 
point are then set out at each of these points.
 2. By successive bisection of arcs or chords:
 • Join tangent points T
1
, T
2 
and bisect them at D. 
Erect a perpendicular DC whose length is equal to 
versed sine of the curve.
CD RR R
L
=-
?
?
?
?
?
?
=- -
?
?
?
?
?
?
1
22
2
2
cos
?
 • Join T
1
C and T
2
C and bisect them at D
1
 and D
2
. 
Erect perpendiculars D
1
C
1
 and D
2
C
2
 which are 
equal to R 1
4
-
?
?
?
?
?
?
cos
?
 • By successive bisection of these chords, more 
points may be obtained and by joining all C, C
1
, 
C
2
, … points, curve is obtained.
T 2 
D 
C 2 
D
2
 
C 1
C 
T
1 
D 1
 3. By offsets from the tangents: If the deflection angle 
and the radius of curvature are both small, the curves 
can be set out by offsets from the tangent. The offsets 
from tangent are of two types.
Part III_Unit 12_Chapter 05.indd   2 5/31/2017   4:56:04 PM
    
 • Radial offsets
 • Perpendicular offsets
E 
O 
R 
Ox
T 1 
D 
D 
x 
A 
D
 
T 1 D 
V
E 
R 
O 
R 
  Radial offset:
OR xR
x
=+ -
22
 (Exact)
· ˜
x
R
2
2
(approx)
  Perpendicular offset:
OR Rx
x
=- -
22
(Exact)
· ˜
x
R
2
2
(approx)
 4. By offsets produced from the chords (deflection 
distances)
 • This method is very much useful for long curves 
and is generally used when a theodolite is not 
available.
C
 
D
 
C
1 
C
3 
C
1 
O
1 
B
1 
T
1 
A
1 
C
2 
O
2 
D
 
R
 
R
 
C
3 
C
2 
C
1 
O
3 
O
 
D
 
A
 
 • Assuming C
1
, C
2
, …, C
n
 sub-chord lengths and 
calculating O
1
, O
2
, …, O
n
.
 First offset, 
O
C
R
1
1
2
2
=
 Second offset, 
O
C
R
2
2
2
2
=
 O
CC C
R
n
nn n
=
+
-
()
1
2
 O
1
, O
2
,… O
n
 = 1st, 2nd, …, nth offset.
 • Great disadvantage in this method is that the error 
in fixing a point is carried forward.
Angular Methods 
In this method, an instrument such as a theodolite is used 
with or without a chain (or tape).
 1. Rankine’s method of deflection (tangential) angles:
  A deflection angle ( d) to any point on the curve is the 
angle at PC between the back tangent and the chord 
from the PC to that point.
 • d (in minutes) = 
1718 9 . C
R
 C is the length of the chord.
 R is the radius of the curve.
 • One theodolite (to measure angles), one chain or 
tape (to measure distances) are used in this method.
 • The deflection angle for any chord is equal to the 
deflection angle for the previous chord plus the 
tangential angle for that chord.
 ?
1
 = d
1
; ?
2
 = ?
1
 + d
2
; ?
n
 = ?
n–1
 + d
n
 2. Two theodolite method:
 • In this method, two theodolites are used one at PC 
and the other at P .T.
 • This method is used when the ground is unsuitable 
for chaining.
 • Only angular measurements are used (no chain or 
tape).
 • It is based on the principle that the angle between 
the tangent and the chord is equal to the angle 
which that chord subtends in the opposite segment.
 3. Tacheometric method:
 • Chaining is completely eliminated and the method 
is less accurate than Rankine’ s.
 • Setting of curve is done with stadia theodolite.
(a) Compound curves: A curve with two or more 
simple curves turn in the same direction and 
join at common tangent points.
(b) Reverse curves: Two curves turn in the oppo-
site directions. 
 [The characteristics, length and sight distance 
requirements of transition and vertical curves 
are discussed in transportation]
Triangulation Survey
Introduction
The triangulation is the system which consists of a num-
ber of inter-connected triangles in which length of one line 
and the angles of triangles are measured very precisely. 
Part III_Unit 12_Chapter 05.indd   3 5/31/2017   4:56:05 PM
      
This chapter aims at the establishment of geodetic survey 
using triangulation.
Geodetic Surveying 
 • To determine precisely the relative or absolute position 
on the earths’ surface. 
 • The stations at which the astronomical observations 
for azimuth and longitude are also made are known as 
Laplace stations.
Objects of Geodetic Triangulation
 1. To provide the most accurate system of horizontal 
control points.
 2. To assist in the determination of the size and shape of 
the earth.
Classification of Triangulation System 
(Based on Accuracy)
First Order or Primary Triangulation
This is of the highest order and is employed either to deter-
mine the earths figure or to furnish the most precise control 
points.
General specifications:
 1. Average triangle closure < 1 second
 2. Maximum triangle closure > /  3 seconds
 3. Length of base line—5.5 km
 4. Probable error in astronomic azimuth—0.5 seconds.
Second Order or Secondary Triangulation
The stations are fixed at close intervals so that the sizes of the 
triangles formed are smaller than the primary triangulation.
General specifications: 
 1. Average triangle closure—3 seconds
 2. Maximum triangle closure—8 seconds
 3. Length of base line—1.5–5 km
 4. Probable error in astronomic azimuth—2.0 seconds
Third-Order or Tertiary Triangulation
General specifications:
 1. Average triangle closure—6 seconds 
 2. Maximum triangle closure—12 seconds
 3. Length of base line—0.5–3 km
 4. Probable error in astronomic azimuth—5 seconds
Triangulation Figures or Systems
 1. Single chain of triangles: This figure is used where 
a narrow strip of terrain is to be covered. Though 
it is rapid and economical, it is not so accurate for 
primary work since the number of conditions to be 
fulfilled in the figure adjustment is relatively small.
 2. Double chain of triangles: It is used to cover larger 
area.
 3. Centred figures: These are used to cover area and 
give very satisfactory results in flat country. Centred 
figures may be quadrilaterals, pentagons or hexagons 
with central stations.
 4. Quadrilaterals: Quadrilateral with four corner 
stations and observed diagonal forms the best figures. 
They are best suited for hilly country and most 
accurate.
Criteria for Selection of the Figure
 1. The figure should be such that the computations can 
be done through two independent routes.
 2. The figure should be such that at least one, and 
preferably both routes should be well-conditioned.
 3. All the lines in a figure should be of comparable 
length. Very long lines should be avoided.
 4. The figure should be such that least work may secure 
maximum progress.
 5. Complex figures should not involve more than about 
twelve conditions.
 • In very extensive survey, the primary triangula-
tion laid in two series of chains usually in N–S 
and E–W respectively is filled by secondary and 
tertiary triangulation figures. This is known as the 
grid iron system and is adopted for France, Spain, 
Austria and India.
 • In another system called central system, which 
extends outward in all directions from base line 
and covered by a network of primary triangulation 
is adopted for United Kingdom.
Well-conditioned Triangle
 • The shape of the triangle should be such that any error in 
the measurement of angle shall have a minimum effect 
upon the lengths of the calculated side. Such a triangle is 
called well-conditioned triangle with base angles equal 
to 56°14'. Form practical considerations, an equilateral 
triangle is the most suitable. However triangle with an 
angle < 30° and >120° should be avoided.
 • The accuracy attained in each figure depends on 
 ¦ The magnitude of the angles in each individual triangle 
and 
 ¦ The arrangement of triangles.
Strength of Figure
 • The strength of figure is to be considered in triangulation 
as the computations can be maintained within a desired 
degree of precision.
 • The geodetic survey has developed a very rapid and con-
venient method of evaluating the strength of triangulation 
figure and is based on an expression for the square of the 
Part III_Unit 12_Chapter 05.indd   4 5/31/2017   4:56:05 PM
Page 5


  
Curves
Introduction
Curves are generally used on highways and railways where 
it is necessary to change the direction of motion. This chap-
ter focuses on the elements of curves and methods for set-
ting out of curves on ground.
Curve may be circular, parabolic or spiral and is always 
tangential to the two straight directions.
Circular curves are of three types:
 1. Simple curves
 2. Compound curves
 3. Reverse curves
Simple Curves 
Simple curves consists of a single arc of a circle. It is tan-
gential to both the straight lines.
 1. Back tangent (AT
1
): The tangent before curve is 
called back tangent or fi rst tangent.
 2. Forward tangent (BT
2
): The tangent following the 
curve is called forward tangent or second tangent.
 3. Point of intersection, vertex (V): It is the point of 
intersection of two tangents AT
1
 and BT
2
.
? 
 
2
?
2
?
 
A
R 
O 
B 
D 
C
V ' 
(PI) V 
(PT) T 2 
(PC)   T 1 
R 
 4. Point of curve (T
1
): It is the beginning of the curve 
where the alignment changes from a tangent to a 
curve.
 5. Point of tangency (T
2
): It is the end of the curve 
where the alignment changes from a curve to a 
tangent.
 6. Defl ection angle or Deviation angle (?): The 
diff erence between the slopes of the two tangents.
Curves and
Triangulation Survey
Part III_Unit 12_Chapter 05.indd   1 5/31/2017   4:56:03 PM
      
 7. Tangent distance: It is the distance from PC to PI 
(also the distance from PI to PT)
T
1
 V = T
2
 V = R tan
?
2
·
  R is the radius of the curve.
 8. External distance or apex distance (E): The 
distance from vertex V to the centre of the curve C.
E = VC = R sec
?
2
1 -
?
?
?
?
?
?
 9. Length of curve (L): Distance from PC to PT,
L = T
1
CT
2
 = R ?  ? in radians.
 10. Length of long chord: It is the chord joining PC to 
PT.
TT R
12
2
2
= sin
?
 11. Mid-ordinate/versed sine: The distance from mid-
point of the long chord to mid-point of the curve.
CD R =-
?
?
?
?
?
?
1
2
cos
?
 12. Normal chord: A chord between two successive 
regular stations on a curve.
 13. Sub-chord: Any chord shorter than the normal chord.
 14. Right-hand curve: If the curve deflects to the right 
of the direction of the progress of survey, it is called 
the right-hand curve.
 15. Left-hand curve: If the curve deflects to the left of 
the direction of the progress of survey, it is called the 
left-hand curve.
Chainages
 Chainage of T
1
 = Chainage of V – Tangent length
 Chainage of T
2
 = Chainage of T
1
+ Length of Curve
Degree of a Curve (D)
 • The central angle subtended by a chord of a fixed length.
 • In India, the fixed length of chord is taken as 20 m to cal-
culate the degree of curve.
 360 degrees = 2pR
 D degrees = 20 m
 ? Therefore, R = 
1146
D
metres
Methods of Setting out a Curve
Linear Methods
In this method, only a chain or tape is used. Linear methods 
are used when:
 • A high degree of accuracy is not required.
 • The curve is short.
 1. By ordinates or offsets from the long chord:
  Mid-ordinate, 
OR R
L
R
0
2
2
=- -
?
?
?
?
?
?
O
x
 = RX RO
22 0
-- - ()
= 
XK X
R
() -
2
(approx)
  O
x
 = Ordinate at distance of ‘X’ from central ordinate.
  To set out the curve, the long chord is divided into an 
even number of equal parts. Offsets calculated at each 
point are then set out at each of these points.
 2. By successive bisection of arcs or chords:
 • Join tangent points T
1
, T
2 
and bisect them at D. 
Erect a perpendicular DC whose length is equal to 
versed sine of the curve.
CD RR R
L
=-
?
?
?
?
?
?
=- -
?
?
?
?
?
?
1
22
2
2
cos
?
 • Join T
1
C and T
2
C and bisect them at D
1
 and D
2
. 
Erect perpendiculars D
1
C
1
 and D
2
C
2
 which are 
equal to R 1
4
-
?
?
?
?
?
?
cos
?
 • By successive bisection of these chords, more 
points may be obtained and by joining all C, C
1
, 
C
2
, … points, curve is obtained.
T 2 
D 
C 2 
D
2
 
C 1
C 
T
1 
D 1
 3. By offsets from the tangents: If the deflection angle 
and the radius of curvature are both small, the curves 
can be set out by offsets from the tangent. The offsets 
from tangent are of two types.
Part III_Unit 12_Chapter 05.indd   2 5/31/2017   4:56:04 PM
    
 • Radial offsets
 • Perpendicular offsets
E 
O 
R 
Ox
T 1 
D 
D 
x 
A 
D
 
T 1 D 
V
E 
R 
O 
R 
  Radial offset:
OR xR
x
=+ -
22
 (Exact)
· ˜
x
R
2
2
(approx)
  Perpendicular offset:
OR Rx
x
=- -
22
(Exact)
· ˜
x
R
2
2
(approx)
 4. By offsets produced from the chords (deflection 
distances)
 • This method is very much useful for long curves 
and is generally used when a theodolite is not 
available.
C
 
D
 
C
1 
C
3 
C
1 
O
1 
B
1 
T
1 
A
1 
C
2 
O
2 
D
 
R
 
R
 
C
3 
C
2 
C
1 
O
3 
O
 
D
 
A
 
 • Assuming C
1
, C
2
, …, C
n
 sub-chord lengths and 
calculating O
1
, O
2
, …, O
n
.
 First offset, 
O
C
R
1
1
2
2
=
 Second offset, 
O
C
R
2
2
2
2
=
 O
CC C
R
n
nn n
=
+
-
()
1
2
 O
1
, O
2
,… O
n
 = 1st, 2nd, …, nth offset.
 • Great disadvantage in this method is that the error 
in fixing a point is carried forward.
Angular Methods 
In this method, an instrument such as a theodolite is used 
with or without a chain (or tape).
 1. Rankine’s method of deflection (tangential) angles:
  A deflection angle ( d) to any point on the curve is the 
angle at PC between the back tangent and the chord 
from the PC to that point.
 • d (in minutes) = 
1718 9 . C
R
 C is the length of the chord.
 R is the radius of the curve.
 • One theodolite (to measure angles), one chain or 
tape (to measure distances) are used in this method.
 • The deflection angle for any chord is equal to the 
deflection angle for the previous chord plus the 
tangential angle for that chord.
 ?
1
 = d
1
; ?
2
 = ?
1
 + d
2
; ?
n
 = ?
n–1
 + d
n
 2. Two theodolite method:
 • In this method, two theodolites are used one at PC 
and the other at P .T.
 • This method is used when the ground is unsuitable 
for chaining.
 • Only angular measurements are used (no chain or 
tape).
 • It is based on the principle that the angle between 
the tangent and the chord is equal to the angle 
which that chord subtends in the opposite segment.
 3. Tacheometric method:
 • Chaining is completely eliminated and the method 
is less accurate than Rankine’ s.
 • Setting of curve is done with stadia theodolite.
(a) Compound curves: A curve with two or more 
simple curves turn in the same direction and 
join at common tangent points.
(b) Reverse curves: Two curves turn in the oppo-
site directions. 
 [The characteristics, length and sight distance 
requirements of transition and vertical curves 
are discussed in transportation]
Triangulation Survey
Introduction
The triangulation is the system which consists of a num-
ber of inter-connected triangles in which length of one line 
and the angles of triangles are measured very precisely. 
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This chapter aims at the establishment of geodetic survey 
using triangulation.
Geodetic Surveying 
 • To determine precisely the relative or absolute position 
on the earths’ surface. 
 • The stations at which the astronomical observations 
for azimuth and longitude are also made are known as 
Laplace stations.
Objects of Geodetic Triangulation
 1. To provide the most accurate system of horizontal 
control points.
 2. To assist in the determination of the size and shape of 
the earth.
Classification of Triangulation System 
(Based on Accuracy)
First Order or Primary Triangulation
This is of the highest order and is employed either to deter-
mine the earths figure or to furnish the most precise control 
points.
General specifications:
 1. Average triangle closure < 1 second
 2. Maximum triangle closure > /  3 seconds
 3. Length of base line—5.5 km
 4. Probable error in astronomic azimuth—0.5 seconds.
Second Order or Secondary Triangulation
The stations are fixed at close intervals so that the sizes of the 
triangles formed are smaller than the primary triangulation.
General specifications: 
 1. Average triangle closure—3 seconds
 2. Maximum triangle closure—8 seconds
 3. Length of base line—1.5–5 km
 4. Probable error in astronomic azimuth—2.0 seconds
Third-Order or Tertiary Triangulation
General specifications:
 1. Average triangle closure—6 seconds 
 2. Maximum triangle closure—12 seconds
 3. Length of base line—0.5–3 km
 4. Probable error in astronomic azimuth—5 seconds
Triangulation Figures or Systems
 1. Single chain of triangles: This figure is used where 
a narrow strip of terrain is to be covered. Though 
it is rapid and economical, it is not so accurate for 
primary work since the number of conditions to be 
fulfilled in the figure adjustment is relatively small.
 2. Double chain of triangles: It is used to cover larger 
area.
 3. Centred figures: These are used to cover area and 
give very satisfactory results in flat country. Centred 
figures may be quadrilaterals, pentagons or hexagons 
with central stations.
 4. Quadrilaterals: Quadrilateral with four corner 
stations and observed diagonal forms the best figures. 
They are best suited for hilly country and most 
accurate.
Criteria for Selection of the Figure
 1. The figure should be such that the computations can 
be done through two independent routes.
 2. The figure should be such that at least one, and 
preferably both routes should be well-conditioned.
 3. All the lines in a figure should be of comparable 
length. Very long lines should be avoided.
 4. The figure should be such that least work may secure 
maximum progress.
 5. Complex figures should not involve more than about 
twelve conditions.
 • In very extensive survey, the primary triangula-
tion laid in two series of chains usually in N–S 
and E–W respectively is filled by secondary and 
tertiary triangulation figures. This is known as the 
grid iron system and is adopted for France, Spain, 
Austria and India.
 • In another system called central system, which 
extends outward in all directions from base line 
and covered by a network of primary triangulation 
is adopted for United Kingdom.
Well-conditioned Triangle
 • The shape of the triangle should be such that any error in 
the measurement of angle shall have a minimum effect 
upon the lengths of the calculated side. Such a triangle is 
called well-conditioned triangle with base angles equal 
to 56°14'. Form practical considerations, an equilateral 
triangle is the most suitable. However triangle with an 
angle < 30° and >120° should be avoided.
 • The accuracy attained in each figure depends on 
 ¦ The magnitude of the angles in each individual triangle 
and 
 ¦ The arrangement of triangles.
Strength of Figure
 • The strength of figure is to be considered in triangulation 
as the computations can be maintained within a desired 
degree of precision.
 • The geodetic survey has developed a very rapid and con-
venient method of evaluating the strength of triangulation 
figure and is based on an expression for the square of the 
Part III_Unit 12_Chapter 05.indd   4 5/31/2017   4:56:05 PM
    
probable error (L
2
), that would occur in the sixth place of 
the logarithm of any side, 
Ld R
22
4
3
=
R = 
DC
D
A AB A
-
++ ?
?
?
?
S dd dd
22
 Where 
  d =  Probable error of an observed Direction (in 
seconds)
  D = Number of directions observed (forward/back)
  d
A 
=  Difference per second in he sixth place of loga-
rithm of the sine of the distance angle A of each 
triangle.
  d
B
 = Same as d
A
 but for the distance angle B.
  C =  Number of angles and side conditions to be satis-
fied in the net from the known line to the side in 
equation. 
  C = (n' – s' + 1) + (n – 2s + 3)
  n = Total number of lines
  n' = Number of lines observed in both directions.
  s = Total number of stations.
  s' = Number of occupied stations.
  (n' – s' + 1) = Number of angle conditions.
  (n – 2s + 3) = Number of side conditions.
 • The relative strength of figure can be computed in terms 
of factor R.
 • Lower the value of R, stronger the figure.
 • Value of R computed for the strongest chain of trian-
gles is called R
1
 and that for the second strongest chain 
R
2
. Generally, strength of a figure is almost equal to the 
strength of the strongest chain. Therefore R
1
 is a measure 
of the strength of figure.
 • For angles measured with the same precision, the strength 
of figure depends upon:
 ¦ Number of directions observed.
 ¦ The number of geometrical conditions imposed by the 
shape of the figures, together with the number of sta-
tions occupied in the field.
 ¦ The size of distance angles used in computation.
Signals and Towers
Tower: A tower is a structure erected over a station for the 
support of the instrument and observing party and is pro-
vided when station or the signal or both are to be elevated.
 • A signal is a device erected to define the exact position of 
an observed station.
Non-luminous Signals
Diameter of signal in cm = 1.3D to 1.9D
Height of signal in cm = 13.3D
Where, D = distance (length of sight) for non-luminous 
signals (km).
Luminous or Sun Signals
Used when length of sight distance > 30 kms.
The heliotrope and heliograph and special instruments 
used as sun signals. The heliotrope consists of a plane mir-
ror to reflect the sun’s rays and a line of sight to enable the 
attendant to direct the reflected rays towards the observing 
stations. Another form of heliotrope is ‘galton sun signal’.
Phase of Signals It is the error of bisection which arises, 
when the signal is partly in light and partly in shade. The 
observer needs only illuminated portion and bisects it. It 
is thus apparent displacement of the signal. Thus the phase 
correction is necessary.
S 
(a) 
(b) 
90
o 
– ( ) ß - a
2
1
 
a 
(Plan)
 
S
1 
A
 
90
o- 
ED
 
 
1
?
 
Signal
 
S
 
A
 
S
1 
ß
 
90
o
( ) ß - a -
2
1
 
a
 
ß
 
2
?
 
F 
E
 
B
 
a 
90 – a
  
C 
 1. When observation is made on the bright portion,
  Phase correction,
ß
a
=
r
D
cos
2
2
radians
  Where
    a = Angle which the direction of sun makes with 
line of sight.
   r = Radius of the signal.
   D = Distance of sight.
 2. When the observation is made on the bright line,
ß
a
=
r
D
cos
2
radians
  The phase correction is applied algebraically to the 
observed angle, according to the relative position of 
the sun and the signal.
Total Station There are three methods of measuring dis-
tance between any two given points.
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