1. CHAPTER II
HORIZONTAL CURVES
When a highway changes horizontal direction, making the point where it changes
direction a point of intersection between two straight lines is not feasible. The change in
direction would be too abrupt for the safety of modem, high-speed vehicles. It is therefore
necessary to interpose a curve between the straight lines. The straight lines of a road are
called tangents because the lines are tangent to the curves used to change direction.
Curves are provided in the line of communication like roads, railways, canals etc. to
bring about the change of direction gradually. Horizontal curves are curves that are used
to connect straight line called tangent. The curves employed normally are circular.
Although spiral curves may be used to provide gradual transitions to or form the circular
curves.
Classification of Circular Curves
Circular curves can be classified as
 Sample circular curves
 Compound Circular curves
 Reverse Circular Curves
 Spiral Curves
1. SIMPLE. The simple curve is an arc of a circle. The radius of the circle determines the
sharpness or flatness of the curve.
2. COMPOUND. Frequently, the terrain will require the use of the compound curve. This
curve normally consists of two simple curves joined together and curving in the same
direction.
3. REVERSE. A reverse curve consists of two simple curves joined together, but curving
in opposite direction. For safety reasons, the use of this curve should be avoided when
possible.
4. SPIRAL. The spiral is a curve that has a varying radius. It is used on railroads and
most modem highways.

2. CURVE Designation
Curves are designated either by their radius (R) or their degree of curve
A. Simple Circular Curves
A Simple Curve is a circular are joining two intersecting tangents. The radius of the circle
determines the sharpness or flatness of the curve.

3. Elements of the simple circular curve
1. Vertex (V)- the point of intersection (PI) of two intersecting tangents.
2. Point of curvature (PC) the point of tangency where the curve leaves the tangent.
3. Point of tangency (PT)- the point of tangency where the curve meat the other
tangent.
4. Tangent distance (T) - the distance from the vertex to the JPC or PT.
5. Intersection angle (I) or D- the angle by which the forward tangent deflects from
the back tangent
6. Radius (R) - the radius of the circle of which the curve is made.
7. External distance (E) is the distance from the vertex to the mid point of the
circular curve.
8. Long chord (C)- is the distance from the mid point of the long chord to the
midpoint of the circular curve.
9. Middle Ordinate (M) -is the distance from the mid point of the long chord to the
mid point of the circular curve.
10. Degree of curve (D)-
Curves are designated either by their radius (R), or their
degree of curve (D0
)
The degree of curvature (D0
) is defined as the angle subtended at the center of the circle
by an arc of 20mts. (Curve definition)
Chord basis- D is the central angle subtended at the chord length of 20m.

4. 11. Length of curve (L) length of the curve from PC to PT.
Note: the sharpness of circular curve may be described by its radius or by the degree of
curve (D0) subtending & standard arc length as defined above.
Important Relationship in circular curve
1 Radius of curve (R)
a) Arc basis
0
360 360*20 1145.916
0
20 2 2
D
D M
RR R
= ⇒ = =
∏ ∏
b) Chord basis
10
sin
2
sin
2
D
R
R
R M
D
=
=
2. Tangent distance (T) - take AOV
tan
2
tan
2
D T
R
DT R
=
=
3. Length of long chord (C) - take AD)
2sin
2
2 sin
2
c
D
R
DCl R
=
=
4. The external distance (E)- consider AOV
( )
cos
2
sec 1
2
RD
E R
DE R
=
+
⇒ = −
5. The Middle ordinate (M) - consider AOD
( )
_
2
1 cos
2
R MDCos
R
DM R
=
= −
6. The length of the curve (L)
i. by arc definition - the actual length of the curve

5. ii. by the chord definition - the total length as measured along the chords of
an inscribed polygon.
20
D D
L
⇒ = ⇒
7. Sub - Angle (d) is the central angle. Less than degree of curve (D0
) subtended by an
arc length less than 20m or chord length less than 20m
- by arc definition
20 l
D d
= ⇒ - sub-arc
- Chord definition
2sin
2
2sin
2
cld
R
cl
dl
R
=
=
-Sub chord
1
2sin
2
cl
d
R
−  
⇒ =  ÷
 
Two straight lines AB &BC intersect at chain age 10+020, the intersecting angle being
400
. It is desired to connect these two tangents by a simple C/ curve of 40. Calculate the
elements of the curve & determine the chain age of the tangent pt. (arc basis)
Solution
20D
L
D
0
20
lD
d =
( )2 sin
2
dlc R=

6. ( )
( )
1145.916
286.5
4
tan 195.98
2
2 sin 195.98
2
sec 1 18.39
2
1 cos 17.28
2
20*40
200
4
10 020
0 104.28
9 915.72
0 200.00
10 115.72
R m
DT R m
DC R
DE R m
DM R M
L m
chainage of PE
T
PC
L
PT m
= =
= =
= =
= − =
= − =
= =
= +
− = − +
= +
+ = + +
= +
Assume that the chainage of a curve is 18+750,
I 750&D=100 compute the curve
a) Using arc definition
b) Using arc chord definition
Methods of setting out horizontal curves
Deflection Angle Method (rankine’s)
The method requires one theodolite and a tape or two theodolites.
The formula used for the deflection angles is derived as follows.
The angle formed between the back tangent and a line from PC to a point on the curve is
the deflection angle to the point. This deflection angle, measured at the PC between the
tangent and the line to the point, is one-half the central angle subtended between the PC
and the point.

7. From the geometry of circles, the angle b/n a tangent to a circular curve and a chord
drown from that point of tangency to some other point on the curve equals one-half of the
central angle subtended by the chord. Thus in fig. above the central angle d1 b/n the Pc
and the first station  on the curves d1, and the deflection angle at PC is 1
/2 d1. Similarly,
the central angle b/n the PC and the second station  on the curve is d1+ D. and the
deflection angle is 1
/2 (d1 +D).
The deflection angle to PT = 1
/2 (d1+D+D+… +d2) = D
/2 (This provides a check on the
calculation.
Central Angle & chord to first curve station
Let the difference in stationing b/n PC & the first full station on the curve is C1. Then by
direct proportioning.
1 1
20
d c
D
= The actual chord length b/n PC & the first full station on the curve
is used to
1
1 1 1
1
1 121 sin 2 sin
20 2 2
Cc D
d d c R d
R
= = ⇒ =
Central Angle & Chord from last curve station to PT
If C2
1
denotes the difference in stationing b/n the last station on the curve & the PT, then
the central angle d2 b/n these two points is
1
2
2
20
c d
d =
The chord length C2 b/n the last curve station and the PT is
2 2
1
2 sin
2
C R d=
Central Angle And Chord Between Any Two Curve Points
Let C1 be the difference in stationing b/n any two points on the curve, d be the central
angle b/n the two points, and D be the degree of curve, either by chord or arc definition.
Then
1
20
C D
d =
The actual chord distance C between the same two points, either by chord or arc
definition, is given by the relationship.

8. C=2R sin 1
/2 d.
FIELD PROCEDURES TO LAYOUT CIRCULAR CURVE BY DEFLECTION
ANGLE METHOD.
Case I when the theodolite is set at the PC.
I. Set the theodolite at the PC with angle reading zero. On the initial tangent
Points 1, 2, 3… etc are full stations to be established on the curve.
II. To locate point 1, turn the telescope until the angle reading is one half of d1,
so that the line of sight is directed along the 1st
sub-chord A1. Then when rod
is held at the proper chord length and the tape is swing about A as a center
until the rod is along the line of sight, point 1 on the curve is located.
III. To locate point 2, turn the telescope till the angle reading equals the
proceeding deflection plus D/2. Then the rod is held at the proper chord length
for the full station and the tape is swing about point 1 as center until the rod is
on the line of sight, point 2 on the curve is located.
IV. The remaining points on the curve are located in a similar manner. As a check
the defection should be equal to D/2 when point PT is located
Case II when the theodolite is set at PT.
When the theodolite is set at the PT with the telescope at its normal position, the
instrument is properly oriented either by sighting on the PI angle reading D/2, or by
sighting the PC angle reading zero, the same curve note can be used whether deflection
are from PC or PT.