Using Simpson's Rule in different transition curves

Transition curves are route elements in modern road and rail transport structures that are as important as straight and curved ones. To prevent sudden changes in centrifugal force, it is necessary to apply a transition curve due to the effect of motion on a sharp curve. Over the years, clothoid practice has become widespread in many countries in the world. However, different transition curves were needed because the application of clothoid at high speeds causes problems in the safety and comfort of the road. In this context; sinusoid and fourth order parabola transition curves were used to solve the problems related to road dynamics caused by the clothoid for high speed vehicles. Compared to the simple mathematical analysis of the clothoid, the calculation of the coordinates of the sinusoidal and fourth order parabola transition curves involves complex mathematical structures. In this study, by presenting the basic mathematical properties of the clothoid, sinusoid and fourth order parabola, it has been shown that the coordinates of the sinusoidal and fourth order parabola transition curves are calculated by numerical integration with an accuracy and precision without using a computer program.