(Quickly but nostalgically written, remembering the good old days when Taylor was not yet known as the name of a beautiful singer but as the laborious math trick used to solve rather painful Mathematical Analysis problems …)
The deflection angle (θ) of the Bloss transition is:
The rectangular equations (x and y) of the curve are based on the following integrals:
The cosine and sine can be written as polynomial expansions, using the Taylor (Maclaurin) series:
Replacing θ and integrating the significant terms we get (don’t trust me with these; check before use. This is the first integration I have done this year. As you can see in the comments, I was kind of clumsy with this – although corrected twice, it might not be entirely correct yet):
For quick checks only the first term is retained for x and only the first two are retained for y, allowing to roughly estimate the coordinates:
This is how the Bloss simplified equation is defined, for example, in the Network Rail‘s Track Design Handbook – NR/L3/TRK/2049.
A similar calculation procedure is used to define the rectangular coordinates of the Clothoid, starting from the Clothoid’s deflection angle, θ = l²/(2RL).
In this case, simplifying the coordinates and retaining only the first term for x and y defines a different transition curve – the cubic parabola.
A railway track blog 
Hi Marc. Tricky. What I know is the curvature variation of the bloss used for compound or reverse curves shoud have the same shape as the one for simple curve (straight to circular). Only in the German standard the reverse is split in two blosses – one for each curve – but I don’t see why.
The equation definition for the bloss curve used for compound curves should start from the curvature shape but I don’t know (now) the details of it and I don’t remember any book or paper detailing it. I’ll think about it …
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Hi Constantin, thank you for the article. What about Bloss transition with a beginning curvature, e.g. connecting two arcs (different radii) with a Bloss transition? Do you have any idea how this could be integrated in the formulae?
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Thanks a lot, I really appreciate your effort.
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Hi Robert. Yes you are right, the divisor of the fourth term should be 44 – I corrected that. The equations are providing good precision for normal alignment applications. The example you used to check this is extreme and obviously pushes the limits, requiring more terms to provide precise enough coordinates. I added more terms now, but, as I mentioned before, this is just a theoretical article and proves to be a clumsy integration. Hence the reader should do the math him(her)self or, better, should use a good design software, able to provide accurate design coordinates for any application.
PS: I had to edit your comment to remove the table – it makes my browser unable to load the comment.
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Constantin, I looked at some other sources and it seems there is a mistake in (7). In the forth component of the formula you have 40 but it should be 44.
Could you add 1 or 2 more components to the series? Last components still give a several milliliter values, see the example. One more for y and two more for x would be enough.
R= 500 L= 200
[edited by CC to remove table]
X= 200 2.286 2.000 0.444 0.016 199.286 199.271
Y= 20 8.000 0.213 0.291 0.133 0.021 11.965 11.965
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Thank you for your answer and again for your article. Waiting for more… 😉
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Thank you for your comment and check, Robert.
In the design process we don’t need to worry about these equations as now all the commercial design software can calculate accurately the Bloss coordinates. (9) or similar equations for Clothoid can be used to quickly evaluate the footprint of the transition relative to existing elements – rarely used now, again due to the flexibility of the design software, able to easily produce the accurate geometry of the transition.
Both for Bloss and Clothoid, “l” is used to define not only X and Y, but also the deflection angle and the corresponding cant.
(7) and (9) are now corrected (second term of the y equation is negative).
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Hi Constantin. The equation (8) can only be used for large radii and short transitions. I put R = 500 and L = 200 and according to the software I use the x at the end is 199.271. Also the y value coming from the equation (9) is (after correction of the formula as there should be minus sign instead of plus) 12.000 while the value from software is 11.965.
Some may say this is accurate enough but if you want to calculate it by your own then at least 1 mm of accuracy should be maintained. So for very rough calcs (8) and (9) can be used but for precise calcs I think we should stick to (6) and (7). But how to calculate current x and y if you don’t know the current l? 🙂
It would be very helpful to bind (6) and (7) into one equation y(x) and remove l (small L).
Please correct signs in (7) and (9). Thank you for the article. 🙂
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