Prologue

One of the main giants on whose shoulders stood proudly Isaac Newton, is the French mathematician Renatus Cartesius.

He was the first to label the unknowns in equations by the letters X and Y and also he defined the annotation of powers as superscripted labels X² . Believe it or not, that was a giant leap for Mathematics. They were using geometrical equivalence before this. That’s why even now we speak of squares and cubes.

In his book La Géométrie, published in 1637, he defined the orthogonal system of coordinates, later named Cartesian in his honour. Thanks to him we have so many Xs and Ys.

Thanks to him we can write m² and we can calculate the versine based on the chord length and radius, and basically any geometrical representation that is defined by a formula draws the origins from his work.

Thanks to him we have the nightmare called Calculus. This great man is the father of the analytic geometry.

Under the name René Descartes, he is also the father of modern philosophy.

We know him from the saying Cogito Ergo Sum – I think, therefore I am.

You probably know this saying is completed by another: Dubito Ergo Cogito – I doubt, therefore I think.

On these statements have been built schools of thought and so many branches of modern philosophy.

The ability to doubt, to step back from the things we believe to be true, and to check their validity, to try to see things from different perspective, are all essential skills of critical thinking and precious tools in engineering.

What I found out over the few years of my life as an educator is that this set of skills is quite difficult to teach. People want certainty, people want ready cooked facts, people want to believe more than they want to think.

The equivalent radius – en français

I presented in a previous article the proof of the formula for equivalent radius, based on the British Railway Track book.

There are a few other ways of demonstrating the formula, but I didn’t find anywhere (yet) a demonstration with a proper drawing.

So when I started to look again into these things, I thought I know a book where for sure I’ll find a drawing demonstration – Jean Alias has to have one.

Well, my dear friends, this is the only thing I found in La Voie Ferre.

The section 6.2 is called The radius of the turnouts placed on curves. No drawing.

I’ll leave you to deal with the translation – one easy way is to paste the link of this image in google translate. The beast is now smart enough to read the snapshot.

Let me translate in my own way.

If we take an arc of radius R₁ tangent to a straight and measure a versine f (from the French word fleche = versine) exactly at the tangent point, relative to a chord 2l, we get a drawing as the sketch below.

Now, if we imagine we bend the straight to a curve or radius R, then the circle R₁ becomes now a circle of radius R₂.

We can write for the versine f₂ the following sum

If we convert the versine to the formula we all well know, but in this case based on the half-chord length l, we can write the equation below

And we can then divide both sides of the equality by 2/l² ; we have now the formula we found in the book

If we do a bit more math, we get this

And in the end we get the equivalent radius formula

When I learned about placing turnouts on curves, using this formula for bending was called the versine method. That because it is based on the versine sum we started with, f₂ = f₁ +F, or the bent radius versine equals the sum of the turnout radius on straight and the versine of the mainline curve.

The sum becomes difference if contraflexure bending is applied, f₂ = f₁ – F.

Later I found out that this method is called by some the Swiss bending, although I don’t think it fits the Swiss stereotype. None of the books I have on railway geometry refers to this method as being Swiss.

The other calculation, the trigonometrical method of bending, is called by some also using a nation reference, in that case fitting so well the stereotype.

Hairsplitting exercise

Everything seems to be all right with this versine bending formula. Except perhaps those question marks that point to where one more approximation comes from, beside the approximation brought by the versine formula.

Let’s do a theoretical exercise and imagine circular turnout that on straight has a 300m centerline radius and a gauge is 1432mm.

If we bend it inside (similar flexure) using the above formula we get the following radii – for Centerline, Inside and Outside rail respectively.

One thing we can notice straight away is that the difference between the outside and inside radius of the bended turnout route is not 1432mm but 924mm. Strange, isn’t it?

The gauge at the origin of the unit is 1432mm on both routes, but at the crossing intersection point, on the bended turnout route, due to the difference in equivalent radii, we get 1428mm.

We have a 4mm gauge narrowing. That’s a pleasant surprise, I was expecting more.

Furthermore, we also get a difference between the IP to turnout origin distance of 30mm. That is just the diameter of a two-pound UK coin. Only 0.1% of the total length. Practically nothing.

If we measure the difference in crossing angle at the intersection point, or if we use some derivation to get that angle using analytic geometry (honouring Cartesius), we will get a difference in crossing angle of 1 minute of degree, or about 65 seconds. Not much! We take ten breaths, and they pass.

If we do the same exercise for contra-flexure, we get again 1428mm gauge (is there a pattern here?). Intersection point to origin is only 9mm shorter and the crossing angle is sharper by 85 seconds.

I note here for the careful reader that some of these figures might very well be wrong. Don’t trust everything you read online! Check this for yourself!

And you probably realized that due to the difference in radii between the crossing defining rail and the other, outer rail, we will have for this theoretical exercise a bearing change on the centerline and on the outside rail, corresponding to the crossing intersection point. Some other 80ish seconds, depending on where and how are measured.

Yes, I know that this is not how it is done. It is just a theoretical exercise. For practical purposes it just proves that it matters what is used as reference for bending. Centerline, outer or inner rail are not interchangeable references when using the versine based bending. And, obviously, beside the reference string for bending, there are other things to consider.

Epilogue

If the Cartesian coordinates system was invented only in 1637 by René Descartes, what did the people use before?

Were they totally uncoordinated?

Well, no, obviously.

Since ancient times they used polar systems of coordinates, defined by a pole or centre, and a direction or polar axis. Relative to these you can define a distance or radius to pole and an angle to the polar axe.

All the sun dials are basic applications of it. A lot of the geometry and trigonometry we know today was developed long before Descartes, for polar calculations.

Even though in design engineering we use the cartesian system, all the topographical data is measured in a polar system of coordinates. And when our designs are traced back on site, a polar system is used again.

Which makes you wonder, why don’t we design directly in polar coordinates?

Can you imagine how easy it is to trace a circular curve in polar coordinates?!

Just set the pole to the centre of the curve, and the rest is easy.

Did you know that there is a special transition curve, natively defined in polar coordinates?

I wonder if we can define turnout bending using some polar coordinate conversion tricks?