(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.