When I hear the name Bernoulli I think about some funny high school experiments of spraying coloured water on paper but also about an infinite loop curve the Roads Professor tormented us in Uni. I would have bet both things were discovered by the same Bernoulli.

But no. The more known one, D. Bernoulli, the fluid guy, is not the one that discovered the loop. That’s a different Bernoulli, I was surprised to find preparing this article.

Funny fact. This other Bernoulli, Jakob, is also the one that discovered e, Euler’s number. Why not Bernoulli’s number?

Interest-e-ing history this number has. But this blog is about railway track and not about banking and compound interest. Never mind e. Back on track, with this little roads detour.

The clothoid infinite loop

If you drive a car with constant speed and rotate the steering wheel at a constant rate, you will draw a clothoid.

This fundamental characteristic of the clothoid, to have a constant rate of change of curvature, made it the perfect transition curve for alignment design of road and railways.

The road and highway alignments sometime require “exotic” changes in geometry to fit the requirements of the site. One classic example is the highway interchange design, where the complex set of alignment junctions required becomes a challenge.

Below is a cloverleaf interchange – seen a long while ago as a revolutionary design, proven in time to be a traffic management nightmare.

The highlighted reversing loops are usually designed as clothoids arcs, with no intermediate circular curves.

This kind of clothoid loop curve creates a triangle in the curvature diagram.

This type of loop appears not only in cloverleaf interchanges (now obsolete) but for other quick direction changes required in highway and road design.

To drive through this loop, the driver rotates the steering wheel at a constant rate until reaches the middle of the loop and then, suddenly needs to start rotating the steering wheel back to straight, with a similar rate as before.

In theory this movement would be better if the driver would have an intermediate circular arc to allow the change. But that would mean a tighter transition on the same footprint.

Well, to solve this problem, the engineers of the past have looked at a different, very peculiar, curve.

Bernoulli’s lemniscate

“Lemniscate” is the mathematical name for any infinite ( ∞ ) or 8 shaped curve.

There is an entire family of curves called lemniscate, with various applications.

From all of them, the one defined by Bernoulli was of interest to the alignment designers of the past, because it was providing an elegant solution to the steering problem encountered designing the loop mentioned above. Bernoulli’s lemniscate looks like this:

On one hand, the curve defines the loop in full, with no requirement of intermediate elements.

On the other, the curvature diagram of this curve is not the sharp triangle of the mirrored clothoids. It looks like this (not to scale – the diagram is distorted for illustration and due to the incredible talent of the artist):

The rate of change of curvature RocC (steering rate?) has a smooth transition at the middle of the loop, better, at least in theory, than the mirrored clothoids.

As a bonus, beside these two advantages, there was a third.

The coordinates of this transition curve can be easily calculated. Its polar coordinates are easy:

r² = a² cos θ

which was a heavenly gift for the topo team, they like polar coordinates.

Similarly, the cartesian coordinates are easy to calculate, also. Much easier than the coordinates of two clothoids arcs, a computation nightmare for any designer before the computer age. If you dig deep enough through old road design documentation you might find various geometrical tricks to scale, distort, compound this curve in very elegant complex junction designs. Some of those tricks proved to be difficult to turn into software code.

Besides the computer, some other things have made irrelevant this curve. One is that the theoretical steering advantage was proven to be only theoretical (so I heard). Its odd curvature variation caused some problems, too.

Now this curve is lost in the history of road alignment design, rarely mentioned even in modern books – in some with less details than on Wikipedia.

Only a few countries still have it in their road design standards, as a fanatic vestige of the past.

Although lost in time, we could say that, on paper at least, this is the curve we use the most, far often than the clothoid, cubic parabola or any other transition curve. We draw a lemniscate in every 8 and ∞ symbol we use.

On paper, the lemniscate is forever the winner.