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Highway Mathematics: The Science Behind Highway Construction

Have you ever wondered why highways have graceful curves and bends? Aesthetics is secondary. The main reason for these curves is to ensure that drivers avoid obstacles like buildings, bodies of water, or other natural features that are too costly or impractical to move. Curves also help drivers to travel dafe by reducing vehicle speeds.

Beautiful mathematics lie behind the construction of curvy highways. In particular, a tool known as Fresnel integrals are used in calculating and designing transition curves (also known as spirals) to smoothly connect sections of a highway with different curvatures.

Fresnel integrals are named after the French physicist and civil engineer, Augustin-Jean Fresnel, who pioneered the study of optics and did much to establish the wave theory of light. They were originally used in the calculation of the electromagnetic field intensity in situations where light bends around opaque objects. As it turned out, these integrals were also found to be useful for designing highways with curves and bends.

Augustin-Jean Fresnel (1788-1827)

Fresnel was born in Broglie, Normandy in 1788. His father was a successful architect and his mother was the daughter of an influential lawyer. A sickly child, Fresnel had difficulty with words and communication and lagged behind other students in “classical education.” But he showed remarkable proficiency in science and mathematics. By a stroke of luck, he was sent to accompany his older brother, Louis, to Caen to study math. In 1804, 16-year old Fresnel and his brothers were selected to attend Ecole Polytechnic in Paris, the best of all French engineering universities. It was there that Fresnel flourished in geometry, graphic arts and technical drawing.

After completing his studies, Fresnel was assigned by the government to repair canals and highways at a time when Napoleon wanted a new network of roads to be constructed for his troops to easily travel the length and breath of France. He worked hard at his projects but his real passion was in light and optics where he famously argued that Newton was wrong in viewing light as a “swarm of tiny corpuscles.” Instead, he proposed that wave travelled like a wave, a highly unorthodox view at the time and one that cost him much delays in getting his research published. But Fresnel persevered and continued to work on experiments that gradually convinced several prominent French scientists that he was on to something.

Fresnel Integrals in Highway Design

Think of a highway as a series of transitions between straight lines. The smoother the transitions, the smoother the drive as this avoids sudden manoeuvres and jerky displacements due centrifugal force. The technical challenge is how to create this smooth transition. The answer lies in Euler’s spiral, also known as a clothoid or Conru spiral (I will use the term clothoid from now on).

The clothoid has a beautiful property: its curvature increases linearly with distance from zero (at the tangent end between a straight section and an arc) to the curvature of the arc where they meet. A road that incorporates this property allows the driver to adjust his speed as he enters the curve, thus minimizing discomfort and also ensuring safety around the bend. Fresnel integrals provide engineers with the mathematical tool to calculate the exact road curvature necessary to ensure road safety. In the next section, I will flesh out a bit more details on how this tool works (non-technical readers can skip this section altogether).

Diving into the Math

Fresnel integrals admit the following power series expansions that converge for all x:

\displaystyle S(x) = \int_{0}^{x} sin^2(t) dt = \sum_{0}^{\infty} (-1)^n \frac{x^{4n+3}} {{(2n+1)!}{(4n+3)}}

\displaystyle C(x) = \int_{0}^{x} cos^2(t) dt = \sum_{0}^{\infty} (-1)^n \frac{x^{4n+1}} {{(2n)!}{(4n+1)}}

The clothoid is the curve generated by a parametric plot of S(x) against C(x). It is symmetric around the origin (see the above diagram).

Let t denote the curve length measured from the origin. The length of the entire curve is:

\displaystyle L = \int_{0}^{t_0} dt = t_0 = \sqrt {dx^2 + dy^2}

where

\displaystyle dx = C'(x) = cos(t^2)dt

\displaystyle dy= S'(x) = sin(t^2)dt

The vector {cos(t^2), sin(t^2)} expresses the unit tangent along the spiral. Thus, if curvature (\kappa ) is defined as the change in the direction of the tangent vector over the change in arclength, by the chain rule, the angle of the tangent vector (call this \theta ) can be derived as equal to:

\displaystyle \theta = t^2

The radius (R ) of the arc is:

\displaystyle R = \frac {dt}{d \theta}

Therefore, \kappa (which is also the inverse of the radius) varies linearly with distance or arclength:

\displaystyle \kappa =\frac {1}{R} = \frac {d \theta}{dt} = 2t

and the rate of change of curvature with distance t is a constant equal to 2.

These features of the Euler spiral make it useful as a transition curve in highway construction. Since more bendy roads have greater curvature, wider arcs are needed to counteract this effect. Finally, the last expression implies that if a vehicle follows the spiral at constant speed, it will also have a constant rate of angular acceleration.

Further Reading:

Wikipedia: https://en.wikipedia.org/wiki/Fresnel_integral

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