Yep, that is exactly what I do, and it always works quite successfully.
Rich
I made several standard templates years ago for the track standards I am using.
I layout out small changes in direction using a minimum center radius ellipse formula, or “tangent method”.
Info from the 1968 NMRA data sheets:
The bent stick method published by the NMRA in 1952 has the stick on the correct side of both lines…
I know, what a silly idea, printed pages.
Sheldon
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Sheldon, I was surprised to learn that NMRA had written an explanation at such an early date. I am not familiar with the terms used there, so it is difficult to understand. However, from a quick look, the most important term “cubic function” is missing. This “cubic function” is important.
I will briefly explain the history of easements (also known as transition curves) based on Wikipedia and surveying textbooks. Since I am adapting it, there may be some mistakes. Please forgive me.
In the early 19th century, the need for curves with smoothly changing curvature became increasingly clear, and it was not until 1845 that the transition curve appeared. The clothoid curve studied by Leonhard Euler met the requirements, but its application in reality required complex calculations.
William Rankin proposed several approximation formulas in 1862, including a cubic polynomial. In 1880, the cubic spiral (not really clear what this is), and a more practical version of the cubic parabola were developed, and around 1890 various books explaining the concept were published.
Then, Japan’s government railways soon adopted it. There was also a lot of construction work done to add transition curves to existing tracks to increase speed. This is the same in every country.
Meanwhile, in the United States, the A.R.E.A. spiral (American Railway Engineering Association ten-chord spiral, 1919) was adopted. It was devised and decided on by the AREA spiral subcommittee under the conditions that it should be as close to a clothoid curve as possible, have a simple formula, and be easy to install with sufficient accuracy for practical use, and as a result, it is said to be a curve that is very close to a cubic spiral.
Then, as time passed, the demand for ultra-high speed and advances in computing technology meant that clothoid curves began to be used again, which is roughly the story.
From the above, it can be said that knowledge of cubic functions that is useful to modelers is lacking in North America. It is coincidental that the deflection of a cantilever beam is a cubic curve, and it is unclear to what extent the modelers understood it.
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Read sheet D3c in more detail. About 2/3rd down the page, paragraph (not well spaced in the printing) begins “In the prototype, an easement is laid out as a cubic spiral, which is a curve of uniformly increasing sharpness.” Cubic Spiral = cubic function in my geometry class 51 years ago.
I posted this info for the benefit of others. Don’t take this the wrong way, but I have little interest in debating things I learned 5 decades ago.
Yes amazing, the NMRA was thinking about this and providing modelers with this info in 1946 and 1952. And they provided it to me as part of my membership package in 1968.
They clearly spent considerable time adapting this information for easy use by modelers as well.
Sheldon
Sheldon, I don’t understand this “cubic spiral” that the AREA is so obsessed with. I think a track engineer was involved in the NMRA documents you showed.
“Cubic parabola” is a “cubic function,” right?
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Yes.
Keihan, the interurban railway where I worked, had no transition curves when it opened in 1910. According to a former colleague, a track engineer, they aren’t necessary when traveling at speeds below 20 mph. Currently, in addition to the Keihan, Shinkansen, JR and Hankyu trains run between Osaka and Kyoto, competing for customers. For this reason, Keihan has been working hard to improve the shape of its tracks.
Take a look at the two photos of railway bridges below. The tracks are laid out in an extraordinary way. The rails are shifted to one side of the ties, and although the ties are lined up in a straight line, the rails form an S-bend.
This means that the transition curves were set up even if it was just for a few inches. As a rollingstock engineer with a sane mind, I thought our track engineers were crazy.
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at least the NMRA doc suggests using Bezier Curve which is the intersection lines drawn between corresponding points allong the tangents to the straight and curves being joined, red lines in this distorted image
This drawing was in the Nov. 1956 issue of Railway Model Hobby magazine (TMS, Japan), and the original source was apparently a 1940 RMC magazine. This is not a clothoid curve, but an involute curve. It is the shape of the teeth of a gear. Its only purpose is to smoothly connect points. Bezier curve is also useful in drawing software, but it is difficult to draw it on the ground.
seems that a lot of geometric terms (cubic functions, cubic parabolas, clothoid curve) along with technical looking daigrams are being bandied about without much useful nor practical purpose.
I realized that the length, L, of the easement track is specified in the NMRA docs, along with the offset. It’s not an arbitrary length of track between 2 points. And that length determines 2 specific points the easement is between
it seems the easiest way to use the “bent stick” method is to
- properly determine the length of the “stick”
- pin it midway along the line between where the constant curve would end and tangent line,
- bend it to meet the tangent line and the curve, and
- draw a line along the bent stick
… OR just let the flex track bend out a little and eyeball it
the angle on the curve is not obvious although 180 * (L/2) / (Pi * Rad) is a good approximation. Similarly the point on the Tangent track also depends on a portion of the easement that is not straight, and :L/2 is a good approximation.
in the past i’ve strggled to geometrically graph an easment and only realized because of this thread that Bezier curves work and are what the NMRA describes. I find it interesting and useful to work the detiailed geometry for layout planning to determine if trackage will fit in those few tight spots where such detail matters
so i might generate a graph with an easment to know approx where the curved flex track should meet the tangent track and make sure i’m beyond that point when laying track
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Well, on the one hand, this discussion is extremely informative, but I would flunk the geometry test for sure. A good job by some very knowledgeable people explaining the geometry of the easement.
Rich
Rich
Yes, there is a long complex history of applying mathematics to all sorts of irregular curves.
We could debate the details of them for the next three years. Who cares, I sure don’t.
Again, I studied this stuff 50 years ago learning how to be an engineering draftsman.
The Bezier Curve, or tangent parabola formula, as we were taught, provides a perfect civil engineering formula to the problem of small changes in direction for railways and highways.
The elegant thing about the Bezier curve is that it has no fixed radius, it is two easements back to back.
Any reasonable increasing spiral formula within the operational requirements of railroad equipment is acceptable and has likely been used by some civil engineer somewhere building some railroad.
As a person who both designs mechanical things and then actually builds them or supervises their construction, there is a point where practicality supersedes theory.
What someone can draw on paper and reduce to a math equation only goes so far when you are in the field with earth moving equipment, rail, ties, a transit and a 100’ chain.
Have a nice rest of the discussion, my easements work great - for 50 years of model layout construction so far.
Sheldon
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Interesting topic. I have to redo all of my curved track b/c while I have 24" curves, it’s not uniform throughout the curve. My mentor suggested replacing the ME flex track with Atlas 9" sections of 24" curved track. I think this will create a more uniform curve. While the locos do go around the curves, gaining consistency is critical. What a shame to remove all this track, wiring, and ballast, but I know the long-term results are worth it. The lay person won’t notice a difference.
HO or N? I would never use sectional track as my main track method. I cringe when I here people say Unitrack.
Just learn the proper methods to layout and install flex track. It is all in the layout and prep.
Sheldon
did you draw a centerline for the curve using a trammel or simply an anchored string?
An old Naval Architect I once knew told me a story about the early days of CADD when they could not get the CADD programs to plot the hull curves on the ships they were designing…
They had to leave that part out and draw it in by hand. Later software engineers were able to teach the CADD program to think and adjust - in other words just fudge it.
No matter how hard we try, it is still an analog world we live in.
Sheldon
Drew a center line.
But the question is how did you draw the center line?
Oh, I used a dremel. Sorry for not understanding the question. Perhaps the homemade dremel I made wasn’t secured correctly.
presumably you mean trammel.
you would need a uniform line even with sectional track to avoid kinks.