Good roads for a smooth drive
In order to allow a smooth driving, any road should meet at least two requirements: no discontinuities in its curvature and its tangent.
The lack of continuity of its tangent - envolving the first derivative of the function representing the track of a vehicle - are easily visible in many roads as, for example, in the picture below.
This "90º angle curve" is acceptable in a road where traffic circulates at low speed but never, for example, in a motorway. As is easy to see, in a curve of this type the tangent has a sudden change at the point where the two line segments meet, making it impossible to circulate safely along this path at high speed.
Now, think about the problem about the continuity of the curvature function. In a road that is made up of only line segments and circular arcs, the value of the curvature changes abruptly from zero (straight parts) to some constant value (circular parts). These discontinuities at the junction points are a serious problem because, apart from being very uncomfortable for the driver and passengers, they may also be the cause of accidents due to the abrupt change of the centrifugal acceleration when the vehicle starts the circular paths. Note that the value of the acceleration, like the curvature, is associated with the second derivative.
To avoid this problem, roads are designed with transition curves between straight and circular paths. The curvature of these curves must change gradually from zero to the value of the curvature of the circular part to which one needs to join the straight segment. The most used transition curve is a clothoid (also called Cornu spiral).
As its graph shows, the curvature of the clothoid varies linearly from zero to any constant and is therefore the right curve for a smooth join between a straight path and a circular path. There are several possible configurations that respect the continuity of curvature. The most common are:
1) "straight line → clothoid → circle"
2) "straight line → clothoid → circle → clothoid →straight line"
3) "straight line → clothoid → clothoid → straight line"
Look at pictures below and check if the continuity of curvature function is preserved in all road sections.