Advantages of Transition Curve: (i) Chance of overturning of the vehicles and derailment of the trains gets significantly reduced.
(ii) It allows the speedy turning of the vehicle as compared to take the turn by stopping the vehicle to dead-stop or slowing to dead-slow.
(iii) Discomfort of the passengers and damage to the goods is lessened.
(iv) Wear & tear / damage of the vehicle especially its moving parts gets significantly reduced.
Basic Requirements of Transition Curves: (i) It must be tangential to the straight path.
(ii) It must meet the circular curve tangentially.
(iii) It should have zero curvature at its beginning i.e. the point where it initiates from the straight path.
(iv) Its curvature must be equal to that of the circular curve when it ends to circular curve.
(v) Rate of change of curvature of the Transition Curve must be in proportion to the increase in Super-elevation.
(vi) Length of the Transition Curve must be such that that the full elevation of the Super-elevation is attained at its junction with the circular curve.
Types of Transition Curves: There are three types of Transition Curve: (i) Cubic Spiral (ii) Cubic Parabola and (iii) Lemniscate.
First two types of Transition Curve are best suited and applied to Railways, and third one to Highways and Motorways.
(i) Cubic Spiral Transition Curve: Its equation is: y = l 3/ 6RL where:
y is the perpendicular offset from the tangent
l is the distance measured along the curve only
R is the radius of the circular curve
L is the total (and usually the straight) horizontal distance between the starting and ending points of the Transition Curve.
This method is most widely used due to its ease. It is used to set out the Transition Curve by Rectangular Coordinate Method.
(ii) Cubic Parabola Transition Curve: Its equation is: y = X3/ 6RL where:
y is the perpendicular offset from the tangent
X is the distance measured along the tangent corresponding to the value of y of offset from the tangential line
R is the radius of the circular curve
L is the total (and usually the straight) horizontal distance between the starting and ending points of the Transition Curve.
(iii) Lemniscate Transition Curve: Its equation is: P = k Sin2 where:
P is the Polar Distance of a point
k is a constant of this equation
is the Polar Deflection Angle
Minimum Radius of Curvature: Its equation is: y = Mx3 where x is the same as given in Cubic Parabola Transition Curve, and M = [1/6RL]. Differentiate this equation to w.r.to x, we get: dy/dx = 3Mx2 = tan  x =  tan/3M _______
Differentiating the above equal again w.r.t. x  d2y/ dx2= 6Mx = 6M  tan/3M
=  36M2tan/3M = 12Mtan

To make r minimum i.e. rMin, 12 M Sin Cos5 must be minimum. Hence to obtain this condition, differentiate it .
Cos6 – 5Sin2Cos4 = 0  Cos4(Cos2 – 5Sin2) = 0
Cos2 – 5Sin2 = 0 / Cos4  Cos2 – 5Sin2 = 0 …… (ii)
Dividing the eq. (ii) by Cos2, we get: 1 – 5 tan2 = 0
Tan = 1 / 5  = 24.08o Substituting the value of  in eq. (i), we get:
rMin = 1 / 12 M Sin(24.08o) Cos(24.08o) = 1.39  RL Hence the radius of curvature of the cubic parabola decreases from a value of infinity when  = 0,
to a minimum value of r = 1.39  RL when  = 24.08o.

Vertical Curve: Constructing the roads and railways in undulating topography or hilly or mountainous area in the valleys with ridges, curves in the vertical plane are required to provide for safety and comfort of the passengers and to reduce the earthwork (amount of cutting the ground in the ridges and mounds and filling in depressions. The gradient in the said areas rises followed by the falling gradient. Hence this smooth rising and falling curve formed in the vertical plane allows a vehicle to undergo this Vertical Transition Curve easily. Such curves also provide the clear visibility while climbing on crest and trough on the road and also provide the drivers safe and sufficient time to handle any dangerous situation. These curves are usually parabolic which are easy to set out in the field. It is very essential for the vertical curves to maintain a constant rate of change of gradient and this is only possible if the curves are parabolic.
General equation of parabola in x – y plane is given as: y = ax2 + bx …. (i)
To define the slope of the curve, the eq. (i) is required to be differentiated:
dy/dx = 2ax + b ……… (ii)
Rate of change of this slope is obtained by differentiating the eq. (ii) further:
d2y/dx2 = 2a ……… (iii) When the R.H.S of the eq. (iii) is constant, the rate of change of gradient of parabola along a vertical plane is also constant.
Types of Vertical Curves: (i) Summit or Crest Curve (ii) Sag or Trough Curve
Summit or Crest Curves: This type of vertical curve is further sub-divided into three types of curves, based on the gradient g which is both positive and negative between the two straight paths. When a vertical curve has its convexity upward, it is called Summit or Crest Curve. Depending on the field conditions, three types of sub-divisions of Summit or Crest Curve are shown as follows:
Setting Out - Deflection Angle METHOD :-
It Is Also Called RANKINE’S METHOD.

 Let AB & BC be two tangent intersecting at B, the deflection angle be (shown in fig.)

the tangent length is calculated & tangent pt. are marked.
Let
= first pt. on the curve.
= length of cord.
.
R = radius of curve

Procedure:-

fix one end of tape at A, measure off 'c' meters, and swing tape until it aligns with.
. Turn theodolite a further Φ°. Fix one end of tape at B, measure off 'c' meters, and swing tape until that point on the tape crosses the line of sight.

Precautions to take:
if many pegs must be placed.

 

The final reading, to the
other tangent point, should
equal L.
Precautions to take:

Calculate the angle Φ to seconds, or errors will be considerable
if many pegs must be placed.

 

The final reading, to the
other tangent point, should
equal L.
CURVE SURVEYING 
The need for Transition Curves: Circular curves are limited in road designs due to the forces which act on a vehicle as they travel around a bend. Transition curves are used to exert those forces gradually and uniformly thus ensuring the safety of passenger. Transition curves have much more complex formulas and are more difficult to set out on site than the circular curves due to varying radius.

vehicle. Any vehicle leaving a straight section of road and entering a circular curve of radius r will immediately experience the full radial force P. If the radius is too small and the
Use of Transition Curves: Transition curves are used to join to straight paths in one of two ways: (i) Composite Curve, and (ii) Wholly Transitional Curve
Composite Curve: Previous Figure is the figure of Composite Curve: Here, the transition curves of equal length are used on either side of a central circular arc of radius R.
Wholly Transitional Curve: A wholly transitional curve comprises the two transitional curves of equal length with . and therefore the radial force is constantly changing too. There is only one point Tc (Common Tangent Point) at which P is a maximum. This means wholly transitional curves are safer than the composite curves.