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Matching Armstrong Easements with AnyRail

Started by rlcross, December 12, 2021, 12:34:59 AM

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The attached anyrail file creates an Armstrong Broad Curve easement and then uses AnyRail easement to approximate the Armstrong easement.

Armstrong uses Length of easement and Offset to calculate the easement.  His suggested standards for an HO broad curve (30" radius) is a length of 18" with a 1/2" offset.  He uses half the easement length (9") to get to the point of tangency for the circular curve, and half the length beyond that point of tangency.  This results in an intersection of the easement with the circular curve at approximately 17.5 degrees into the curve.

In order to duplicate this Armstrong easement/curve, Using an Anyrail easement of 30 degrees and 30 radius followed by the circular curve of 30 radius yields the an almost identical Armstrong track layout.  You can select the Anyrail Easement/Circular Curve and position it over the Armstrong to see how close they are.


Looks good!

We decided to use the angle and the radius to define a flex easement, because it's a lot easier to connect to a sectional curve of a given angle that way.

But the end result is indeed very similar.

There's a lot of theory on easements, also combined with 'canting'. However, for model railroading it's probably easier to take a practical approach.
David Hoogvorst. Founder and Owner of DRail Software. Creator of AnyRail.


David, I like the way our easements work here on AnyRail.  You're right, it is much easier to match two angular easements, with a circular curve, to get a total curve.  The Armstrong method is easy to work with when you have a bendy stick and you're physically bending an easement into a curve on your train table.

That's why I have been looking into what AnyRail easement inputs equate with typical Armstrong easements so I can easily plan here, and have the other method when I am physically laying it out on plywood.  The option of printing out a 1:1 track plan from AnyRail will be easiest, but I haven't looking into getting it printed here yet.

I want to say how much I appreciate the work you and others have put into AnyRail.  It's a great product.


Fascinating! I'm comforted, although not surprised, that AnyRail and Armstrong are so close. It is also what I have found in my tests. And David is right, Angle and Radius is an easier input way to create a new easement to a given sectional curve where Radius is known.

But I still wonder what Angle to input in AnyRail for a new easement. Your heuristic from the earlier thread (30 deg etc.) is a good rule of thumb. But is there a simple way to come up with an approximation of the angle more precisely?

Attached are the NMRA and NEM discussions on the topic. Note that NMRA is publishing all new data sheets as the old ones (D3b.3 etc.) are now "legacy data sheets". The attached is the new one from 2019 (DS5.7). Both NMRA and NEM are similar to Armstrong, even if they differ in the details. (Seems like the Upload directory is full, so I couldn't upload the NEM file).

Using the NMRA method:
QuoteThe minimum practical length of a spiral easement is the coupler-to-coupler length of the longest car or locomotive expected to traverse it. Exclude the tender for steam locos.
One and a half times the minimum length is better, and twice the minimum length is best as a practical maximum.

My longest car is L = 300mm or ~12"
My sharpest curve is R = 483mm or ~19" (Fleischmann Profi R3)
QuoteNMRA Aspect Ratio: p = L/R = 300mm/483mm = 0.62
NMRA says:
QuoteBest appearance: aspect ratio less than 0.5
Good appearance: aspect ratio between 0.5 and 1.0.
Borderline appearance: aspect ratio between 1.0 and 1.5.
Grossly un-prototypical appearance: aspect ratio greater than 1.5.
(I wonder if the guideline is formulated a little strange as using 2*L instead gives a higher p for a given R, which is worse, not better. But they probably mean that R should be higher, so that our 2*L should have a Radius of 1200mm to get p = 0.5 (p = 2*300/1200 = 0.5)
QuoteNMRA Offset: D = p*L/24 = 0.62*300/24 = 8mm (5/16")
Which is very close to Armstrong; he has 9.5mm (3/8") for an 18" Radius
QuoteNMRA Minimum Direction Angle (the angle between the two tangents of the tracks):
Bmin = 57.3 * p = 57.3 * 0.62 = 36°
This would be the minimum direction angle for the two tangents (if I read this right) so each easement should be 18°; if the tangent direction angle is more than 36°, the circular curve would be added between the two easements, and if the tangent direction angle is less than 36° a parabolic easement method should be used instead.

So, for this example, we should enter Angle=18° in AnyRail for a new easement with 36° or greater direction tangent angle? And Radius=483mm.
(Interestingly, the standard easement in Fleischmann Profi turnouts happen to be 18°. I wonder if it's a coincident or reflect their calculations as I have used Fleischmann car and curve data in the example?)

And if p = 0.5 (Best), we would get Bmin = 28° and therefore each easement in AnyRail should be Angle = 14°?


Magnus, some great background work on trying to find the best Angle to enter into AnyRail for the best easement generation.  Finding an easy way to figure what angle to put in would be a huge step forward.

As we all know, there's a big difference between the resultant AnyRail easement if you put in 10 degrees, or 40 degrees, so finding what's best would really make planning easier.

In my example, I used the Armstrong method with his broad curve L (length of easement) and Offset rules of thumb to construct the sample on the left.  This was done with precise placement of the circular curve offset and L, the targeted amounts.  This kind of precision is tedious when planning each easement, but was useful to generate a single example.  It physically created the point where the easement would start, and end, and showed that this method ends where the circular curve is at approximately 17.5 degrees from the offset tangent point.  You can use trig to calculate this to be exactly 17.45 degrees for the L, offset and Radius used in the example, so it confirmed the physical layout validity.

I then tried an AnyRail easement using 17.45 degrees and the 30" Radius.  However, this easement did not match up with the tangent track.  It came out 3-7/8" short and only a 3/16" offset instead of the target 1/2".  It's not that AnyRail is wrong.  It just has to do with the internal math built into the AnyRail easement calculator.  Even though it had the actual Angle for the Armstrong curve and the correct Radius, it generated an easement that was not nearly as gradual as Armstrong required.

However, when I created an easement using the much higher, 30 degree angle and the correct radius then it generated an easement that closely approximated the total Armstrong easement/curve.  Why the 17.45 degrees didnt work I dont know.  It's part of the blackbox of the software.  But knowing that 30 degrees generates a result that matches is a step forward.

So it appears to me that if we want to generate easements that, when combined with the circular curves to generate a total curve that closely matches the Armstrong, NMRA, other standards that you noted, we just have to use higher numbers for Angle to get AnyRail's easement calculator to match up.

I've observed that many track plans are presented without easements, so the discussion is not relevant to many I'm sure.  For them, AnyRail is a good tool for just working out the big issues, and the details can be worked out on the plywood when laying out the actual track.  It appears you, like I, am looking for those finer details at the planning stage here in the software, that help us build a great operating layout.  Plus I enjoy getting into the weeds on these things.

I am going to lay out the physical Armstrong easements for Sharp and Conventional curves too.  To see what AnyRail angles give the similar easements.  That way I can have my personal angle standards for Sharp/Conventional/Broad curves that can be easily plugged into AnyRail easements.  I doubt they will all be the same, but we'll see.

Regards,  Richard


To get a better feel for easements I did some more tests in AnyRail:

I tried the 18°/30°/No Easement in 2 different scenarios, see attached.
One is 90° curve and the other 180° curve.

As you see in the 90°, you can get the tangent tracks to line up perfectly, BUT you have to start the easements much earlier, see note where the curves start; it is quite a difference. (I aligned the tangent tracks vertically, but the horizontal sections I staggered to make it easier to see them; you can easily align the horizontal tangents as well and you will see that the curves are within a few mm of each other.)

As you see in the 180° curve, if you start the curve (with or without easement) at the same point, the curves will require quite different space.

The result is not surprising, but it reminds me that although easements do extend the length of the curve significantly, you can compensate for that extension by cutting down the length of the tangent track an equivalent amount.

The effect will be a longer easement without requiring the extra space.

Back to our question: What Angle to enter in AnyRail? After this experiment I think 18° or 30° might both be good options as you can compensate for the extra space required by starting the curve earlier.

The choice should be aligned with what the actual curve is - sharper curve -> longer easement. And the visual appearance you want.

@Richard, I'm curious why your 17.45° did not line up? Did you figure out why?

I was talking to some local friends here about this topic. Two of them followed the Armstrong guidelines and was happy with the result. One guy is building a new layout and had just read Lance Mindheim's new book "How to Design a Model Railroad" (Kalmbach 2021). In that book Lance apparently suggests to just ignore Easements, they are too much hassle and not really needed. But apparently he also suggests at least 32" radii for HO! That is of course one explanation, but we speculated that it also could be because today we tend to run trains slower with all the interest in operations and switching, so the need for easements for high speed trains might be less?

(Can't seem to upload the .any file @David, can you check? I get this error message when I upload:

QuoteThere was a problem during the uploading of Easements test.any.
The upload directory is full. Please contact an administrator about this problem. 31.1 KB


Can you please retry to upload the file? It looks like there was a maximum set on the upload directory indeed.
David Hoogvorst. Founder and Owner of DRail Software. Creator of AnyRail.


Magnus, I look forward to seeing what you've found when you can upload your file.

In the meantime I sharpened my pencil and looked for exact equivalents for the three (Sharp 18", Conventional 24" and Broad 30") Armstrong curve easements.  I fine-tuned the Broad to get 25.5 degrees in AnyRail.  The Conventional came out to 26.5 degrees and the Sharp came out to 27.5 degrees.

Why the calculated angle for Armstrong's Offset/Length doesnt work exactly when input into AnyRail's generated easements angle is probably just a function of the calculation algorithm built into the coding.  After all, Armstrong uses "rules of thumb" for laying out an easement on plywood.  Very useful in final building.  The program probably uses a quadratic equation of some type to generate true spiral curves.  All I am looking for is a good angle to plan curve easements that works in the computer, and can be translated using Armstrong for drawing on the plywood.

I think we've found it.


Here is the .any file of my easement tests. See the previous post for analysis.

Your Armstrong easements look precise, but how did you get the actual angle (25.5/26.5/27.5)? Did you iterate in AnyRail until you found a good match or was it a calculation?

Using the 2019 NMRA method :

QuoteBmin = 57.3 * (L/R)

on Armstrong's assumptions for HO and N scale would get the following angles for AnyRail input (see attached spreadsheet couldn't upload a .xls file...)

NMRA Easement Angles with Armstrong data.jpg

Seems like all angles are around 18°. Perhaps that's what NMRA was shooting for? I wonder in the formula where the constant 57.3 comes from?


Magnus, it was strictly an iterative process.  With smaller angles, the result left too little offset.  Larger angles overshot the tangent track creating too large an offset.  Settled on those values just by trial and error.


Thanks, that's what I guessed.

Inspecting closely my experiments, seems like the 30° has a too large offset and the 18° has a too small offset compared to Armstrong's number, but it's off by only a couple mm. Perhaps it's the same distortions you saw.

I find that close inspection of the geometry in Anyrail is helped by changing in the Show tab: Centerline, and turning on Guides. It's much easier to inspect the geometry then.


Hey Magnus.  I looked at your NMRA document about easements and your analysis of it in your Dec 14th post above.  It's amazing how detailed something as simple as turning a train can be. 

My feelings about the NMRA background confirmed that I want to use easements.  For realism/aesthetics mainly, but also for any operational benefits.

The NMRA, like Armstrong and others, focuses on the practical application of laying out an easement/curve physically.  How to plan the offset from the tangent track.  How to plan the length of the easement and given these, where it meets the circular curve.  Key elements in the "Bent Stick" approach to laying out the track centerline on the roadbed.  This is exactly how I intended to lay out my track plan to lay track and cut roadbed.  As an aside, if I can arrange to print my track plan in 1:1 scale and lay it out on wood, I will definitely prefer to do that.

AnyRail is a great tool to plan all this prior to drawing lines on wood and gluing down track.  My intentions for AnyRail are to work as many kinks out of my plan as possible before starting construction.  Easements are an important element for me to incorporate as I want a plan that has graceful curves that look good and operate well.

So all this background is really good for me to understand and use in my plan.

I see the NMRA formulas for using the Bent Stick method are great for the stage where I am laying out the plan for installing track.  They give me the right Offset to use, and where to place the stick for my various curves as planned in AnyRail.

The difficulty comes in translating the physical (Offset and Length from the formulas) into the AnyRail generator.  It appears to me that AnyRail's easement calculator does not work with the angles from the formulas.  Note:  The AnyRail easement generator works great, it just doesnt use similar inputs.  Whatever calculations go into the black box, it generates the correct offsets only if you use larger angles than the Min angle the formulas suggest.  Neither is wrong.  Just they calculate differently.  That is why I overlay AnyRail against the full circular curve to find the AnyRail inputs that give the right offset.  Whatever input into AnyRail that generates the offset I want is what I am looking for.

From what I've seen, from my tests and your tests, it appears that somewhere around 25 degrees is what generates the offset that I am looking for (25.5, 26.5, 27.5 for three types of curves, rounded to 25 degrees for simplicity).  I liked your work to determine that 14 degrees is what the formula suggests for easements, however, I think the AnyRail generator requires higher angles to get the same shaped easement. Why?  It's a detail in the software and how AnyRail builds easements, but I dont need to know that detail.  I can work with the angles that generates the offset I need when I get out my stick and start drawing lines on wood.

I've had a lot of fun digging into this issue with you.  And I know my trackplan will be better for it.  I'm imagining gentle curves flowing through a forest and crossing gorges as we speak :)

Oh, one additional point.  I think putting AnyRail easements back-to-back generates a parabolic curve easement.  I am using them in this manner when I am looking for a gentle 10 degree curve around a foothill for example.  I put two 5 degree, say 60" radius easements together to turn the track 10 degrees.  Seems to work well.  I also use just connecting the two tangents together by dragging one tangent to another and using Smooth Flex.  It too give a nice gentle curve.




Yes, this has been a very interesting and educational journey on easements.

I agree, I also like the NMRA approach, even though it's not identical to Armstrong since it is more general, but it produces close results. (NEM has a slightly different approach, but I have not dug into it as deep.)

AnyRail is different from both Armstrong and NMRA as we have seen. (David explains the easement math used in AnyRail here:,2616.0.html)

All of them produce reasonable easements, although I'm not sure which is best under what circumstance.

My layout data entered into the NMRA formulas gave me an 18° angle to use in AnyRail as I showed earlier. But your analysis of Armstrong offsets gives an angle of ~25°.

So I was just about to conclude this thread by saying:
QuoteI plan to use AnyRail Easement angles of between 18° and 25°.
when I decided to search this forum for similar discussions. The topic of easements comes up once in a while, and I was surprised to see various recommendations:

Quote[Easements] works best with angles of 30 to 60 degrees.
from David here,2682.msg19873.html#msg19873
(Perhaps this is more a comment about the math used in AnyRail as you suspected.)
QuoteI find that the resulting easements are very reasonable if I keep the angle within 5 and 15 degrees.
from chaz here,2616.msg19505.html#msg19505

Wow! I feel we are back to square one of what Angle to enter in AnyRail Easements? For now, I will stick with my conclusion:
QuoteI plan to use AnyRail Easement Angles of between 18° and 25°.
until someone convinces me otherwise.