Types of curves | Horizontal and Vertical Curves in Surveying | Road Survey -lceted LCETED INSTITUTE FOR CIVIL ENGINEERS

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Apr 13, 2022

Types of curves | Horizontal and Vertical Curves in Surveying | Road Survey

During the survey of the alignment of a project involving roads or railways,  the direction of the line may change due to some unavoidable circumstances.  The angle of the change in direction is known as the deflection angle. For it to be possible for a vehicle to run easily along the road or railway track, the two straight lines (the original line and the deflected line) are connected by an arc (Fig. below) which is known as the curve of the road or track.

Curve of Road

Curve of Road


When the curve is provided in the horizontal plane, it is known as a horizontal curve.

Again, the nature of the ground may not be uniform along the alignment of any project and may consist of different gradients (for instance, a rising gradient may be followed by a falling gradient and vice versa). In such a case,  a parabolic curved path is provided in the vertical plane in order to connect the gradients for easy movement of the vehicles.

This curve is known as a Vertical curve. The following are the different  forms of curves:


Types of curves



Degree of a Curve

Degree of a Curve

1. Degree of Curve

The angle a unit chord of 30 m length subtends at the centre of the circle  formed by the curve is known as the degree of the curve. It is designated as  D (Fig. above).

A curve may be designated according to either the radius or the degree of  the curve.

When the unit chord subtends an angle of 1°, it is called a one-degree  curve, when the angle is 2°, a two-degree curve, and so on.

It may be calculated that the radius of a one-degree curve is 1,719 m.


2. Relation between Radius and Degree of Curve

Radius and Degree of a Curve

Radius and Degree of a Curve


Let AB be the unit chord of 30 m, O the centre, R the radius and D the degree of the curve (Fig. above).

Radius and Degree of a Curve

3. Superelevations

When a particle moves in a circular path, a force (known as centrifugal force) acts upon it and tends to push it away from the centre.

Similarly, when a vehicle suddenly moves from a straight to a curved path,  the centrifugal force tends to push the vehicle away

from the road or track.  This is because there is no component force to counterbalance this centrifugal force.

To counterbalance the centrifugal force, the outer edge of the road or rail is raised to some height (with respect to the inner edge), so that the sine component of the weight of the vehicle (W sin θ) may counterbalance the overturning force, The height through which the outer edge of the road or rail is raised is known as superelevation or cant.




In Fig. below, P is the centrifugal force, W sin θ is the component of the weight of the vehicle, and h is the superelevation given to the road or rail.  For equilibrium,



Where, b = width of the road in metres

G = distance between centres of rails (gauge) in metres

R = radius of the curve in metres

g = acceleration due to gravity = 9.8 m/s2

V = speed of the vehicle in metres per second

h = superelevation in metres.


4. Centrifugal Ratio

The ratio between the centrifugal force and the weight of the vehicle is known as the centrifugal ratio.

Centrifugal Ratio



The following are the different types of horizontal curves:

1. Simple Circular Curve

When a curve consists of a single arc with a constant radius connecting the two tangents, it is said to be a circular curve (Fig. below).

Circular Curve

Circular Curve


2. Compound Curve

When a curve consists of two or more arcs with different radii, it is called a compound curve. Such a curve lies on the same side of a common tangent and the centres of the different arcs lie on the same side of their respective tangents (Fig. below).

Compound Curve

Compound Curve


3. Reverse Curve

A reverse curve consists of two arcs bending in opposite directions. Their centres lie on opposite sides of the curve. Their radii may be either equal or different, and they have one common tangent (Fig. below).

Reverse Curve

Reverse Curve


4. Transition Curve

A curve of the variable radius is known as a transition curve. It is also called a  spiral curve or easement curve. In railways, such a curve is provided on both sides of a circular curve to minimise superelevation. Excessive superelevation may cause wear and tear of the rail section and discomfort to passengers (Fig. below).


Transition Curve

Transition Curve


5. Lemniscate Curve

A lemniscate curve is similar to a transition curve and is generally adopted in city roads where the deflection angle is large. In Fig. 10.9, OPD shows the shape of such a curve. The curve is designed by taking a major axis OD, minor axis PP′, with origin O, and axes OA and OB. OP(ρ) is known as the polar ray and α as the polar angle.

Lemniscate Curve

Lemniscate Curve





When two different gradients meet at a point along a road surface, they form a sharp point at the apex. Unless this apex point is rounded off to form a  smooth curve, no vehicle can move along that portion of the road. So, for the smooth and safe running of vehicles, the meeting point of the gradients is rounded off to form a smooth curve in a vertical plane. This curve is known as a vertical curve.

Generally, the parabolic curves are preferred as it is easy to work out the minimum sight distance in their case. The minimum sight distance is an important factor to be considered while calculating the length of the vertical curve.



The gradient is expressed in two ways:

a)   As a percentage, e.g. 1%, 1.5%, etc.

b)  As 1 in n, where n is the horizontal distance and 1 represents vertical distance, e.g. 1 in 100, 1 in 200, etc.

Again, the gradient may be ‘rise’ or ‘fall’. An up gradient is known as  ‘rise’ and is denoted by a positive sign. A down gradient is known as ‘fall’  and is indicated by a negative sign.


Rate of Change of Grade

The characteristic of a parabolic curve is that the gradient changes from point to point but the rate of change in grade remains constant. Hence, for finding the length of the vertical curve, the rate of change of grade should be an essential consideration as this factor remains constant throughout the length of the vertical curve.

Generally, the recommended rate of change of grade is 0.1% per 30 m at summits and 0.05% per 30 m at sags.


Length of Vertical Curve

The length of the vertical curve is calculated by considering the sight distance. To provide minimum sight distance, a certain permissible rate of  change of grade is determined and the length of the vertical curve is  calculated as follows:

Length of Vertical Curve

Length of Vertical Curve


Example: Let us find the length of the vertical curve connecting two grades +0.5% and –0.4% where the rate of change of grade is 0.1%.

Length of vertical curve = (0.5-(-0.4)x30)/0.1 = ((0.5+0.4)x30x10)/1 = 0.9 x 30 x 10 = 270 m

Length of vertical curve


Types of Vertical Curves

The following are the different types of vertical curves that may occur.

(a) Summit Curve Figure below figure shows a summit curve where an up gradient is followed by a down gradient.

Summit Curves

Summit Curves

The figure below shows a summit curve where a down gradient is followed by another down gradient.

(b) Sag Curve: below figure shows a sag curve where a down gradient is followed by an up gradient.

Sag Curves

The figure below shows a sag curve where an up gradient is followed by another up the gradient.

Sag Curves

Sag Curves

The vertical curve may be set out by the following two methods:

·       The tangent correction method

·       The chord gradient method


Why Curve is Provided?

Having a straight highway or railroad in a country is practically feasible or impossible. Some changes in the direction of their alignment are required for terrain, culture, feature or other unavoidable reasons.

Such direction change can not be sharp but should be gradual, it is necessary to introduce curves between straight lines.

Following a regular curved path is called a railway or highway alignment curve.


What are the two types of curves used in road surveys?

There are two types of curves provided mainly

a)   Horizontal Curve

b)  Vertical Curve


What are the Types of Horizontal Curve?

Simple Curve

Compound Curve

Reverse Curve

Transition or Spiral Curve

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