One alignment design homework I had for my students was to calculate the installation coordinates for a simple alignment. This was a long time ago, before Excel was a thing. We were using scientific calculators back then; a lot of typing.

One of the submissions puzzled me. All the coordinates were almost correct. The length of the circular curve was off by half a metre, and everything followed from it, with interval point coordinates further and further away from my check calculated value.

Almost the entire calculation was based on the length of the circular arc, C.

The angle α was correctly calculated, but the length C was not quite there. What was wrong?

Then my penny dropped!

Next day, I asked my student to show me how he calculated C.

He started typing … 3.14 * R * Alfa / 180

– Wait, my friend? Why 3.14?

– Well, Prof, Pi? Duh!

– Pi?! That is the button in that corner there! Check this out. One button, 9 decimals Pie! For alignment design, that’s Pi. Never 3.14!

When the humans started building things, they did not rely much on science but on some common sense rules that didn’t require too much theoretical knowledge to be successfully applied.

As this wisdom of building things evolved into Engineering, it started to rely more and more on theory, on science. It became more rigorous, more exact and easily replicable.

When the first alignment traced railways were designed and built, that was done using log tables and smart approximations.

Then we got calculating machines and shortly after, computers. We invented Computer Aided Design.

Now we rely less on approximations because we can let the computer do its thing. We can be as exact as we need and want.

We no longer use cubic parabolas. We use clothoids. One reason for this is that we can so easily can.

We used vertical parabolas because we could draw them easily. After a while, we switched to vertical circles because some software could easily compute circular coordinates. Then, we realized that the vertical movement is on the parabolic trajectory, and we set our standard rules for vertical parabolas of constant %g. Now, some of us are still stuck with vertical circles because … radius … but never mind.

One of the main issues with our grandfathers’ approximations was the management of retaining its effect in the theoretical realm, avoiding it becoming an error, a potential hazard.

Now, our computers allow us to be precise; so we can forget about this approximation management. But in order to do that we need to know, to understand, what are the approximations we still use today, what things from our grandfathers’ dark art were not exactly right, what things they used were just smart tricks.

We need to know where they used 3.14!

And to understand that 3.14 is never really Pi!

But enough engineering philosophy; let’s return to the equivalent radius. Parlez vous Français?