Easement: Curves from tangent?

@Rail Dawg posted:

Have any of you used transitions from tangent to curves as described in Armstrong's book?

On my last layout I tried but wasn't too successful. Are they worth the extra work?

Thanks!

Chuck

I seem to remember that the term is referred to as "easement". I remember doing that in HO, but didn't bother with the extra work when build the 3-Rail layout, since we were using Atlas solid nickel silver track.

Not to hijack, but maybe this question is germane...Does anyone manufacture easement forms for O gauge flexible track? Or patterns that can be printed, pasted on wood and cut out?

In most cases probably not. My entire layout was built with Gargraves flex. I did read up on using easements. Didn’t really understand all the math and such involved. I made a jig to bend the track to a 072curve. Then basically bent it back to somewhat of an easement just by simply eyeing it.

I now run body mounted Kadee’s on even my 21” passenger cars. With an easement they negotiate my 180 curves just fine. One area that went from a straight portion to a fixed 072 curve. I had an instant derailment. It has since been changed. These cars can handle 072 curves. But they do seem to need a gradual turn before they see a true 072 curve.

You can avoid all the math involved and make beautiful easements by using a 1/4" thick wood lath strip like these:

You don't need a bundle, in fact all you need is one piece!

• Anchor one end of the lath where you want the curve to start.
• Then bend the lath into your curve.
• Anchor the lath somewhere near the middle of the curve.
  The lath will form a natural easement into the curve, with zero math.
• Trace the line on the benchwork.
• Remove the lath and lay the track.

Of course, this only works if you're using flex track.

With an easement, as used on full scale railroads, the transition from straight track to a constant radius curve is gradual. This causes a gradual build-up of lateral acceleration, side loads on the cars, passengers and track, and is more comfortable for the passengers, as well as being less damaging to the track. The exact shape for an easement curve can be derived mathematically.

With O gauge sectional track, the transition from straight track to a constant radius curve (say O-36) occurs suddenly, at the junction of the two track pieces, and causes sudden lateral acceleration and side loads on cars that passengers would notice at full scale. It also looks unrealistic but is not readily discernible when viewed at model scale.

For example, with Atlas O sectional track, an (approximate) easement into an O-36 curve over a 90 degree change of direction could be built by starting with an O-72 section (22-1/2 degrees) followed by an O-54 section (22-1/2 degrees) followed by an O-36 full section (30 degrees) and two O-36 quarter sections (15 degrees). While this would not be a perfectly smooth easement, it would look more realistic than transitioning immediately from a straight section (tangent track) directly into an O-36 curve.

Adding easements into curves on O gauge model railroads increases the space required to execute a specified change in direction (say 90 degrees, as in my example above) and, in my opinion, is why most O gauge model railroads don't have them.

MELGAR

Mel it's been quite a while since I've heard any refer to a change in direction as lateral acceleration.  All to often we think of acceleration only as an increase in velocity not a change in speed or direction.

The method Rich Melvin described has always been my favorite if you are using flex track. But, there is actually no way to achieve a true easement without using some sort of flexible track. I used a wood yardstick standing on edge.

This way avoids the need for any math or templates and you can control the length of the easement section by varying the amount of flex you put into the wood.

Using different radii of sectional track is better than nothing, but is not a true easement, as Melgar said.

Jim

@Hot Water posted:

I seem to remember that the term is referred to as "easement". I remember doing that in HO, but didn't bother with the extra work when build the 3-Rail layout, since we were using Atlas solid nickel silver track.

You are right! I fixed the title. Appreciate that.

Chuck

Rich, With all due respect, if I am correctly understanding what  is written,  that will not work without an X set back.  I use easements almost everywhere mostly with sets of spiral jigs which were all laid out one at a time.

Jigs:

Two styles, the flat ones for a drawn center line the braced ones for a router;

These are the kind of curves that easements produce;

When you have to lay out an easement from scratch simply establish the point where a fixed curve would normally terminate as shown by the short dash line below.  Then check the X offsets for different radii, in this case I used 1".  Then set the trammel back the 1" offset and swing your curve.  The dash line will be the center of your easement.  Take 1/2 of your radius and use that for your easement length.   Clamp your flexible lath stick over the tangent line up to the easement start then cause the flex strip to conform to the fixed radius at the curved end of the easement then make sure the mid section is at the center point of the X factor then draw a line.  Now you have your right of way C/L.

Without using an offset factor you will be fudging the curve tighter than what you think your radius is.

NMRA has recommended practices on how to layout an easement.  There are three different ways using math to project an easement.  This Armstrong method is the easiest.   When laying a good easement you will not be able to see exactly where the tangent stops and the fixed radius curve starts.

In the view below I am using a template primarily to get the easement.  Because the actual degrees of curvature is so small I will be flipping the template over and come out of the easement with only about 12" of fixed radius curvature between the two full easements.

This is now mostly occluded by a removable mountain hiding the cinder block outside corner.

The below shot is a pair of easements on a split grade.

<-The easements stop at the "T" mark

You can use easement templates anywhere.  In the photo below I used two different easements when laying out curved whisker tracks that other wise would crowd into an aisle.

My most recent project involved laying out a 13' long easement on a new peninsula.  For this I used two 14' X 1/4" X 1 3/8" lath strips which are available at real lumber yards, not big box stores.

By starting with alignment of the diverging tracks in a 3-way switch and flexing them around to see what kind of accommodations they can provide I got an idea of what looked best.  Then laying some track in it's path and loosely placing some buildings I was convinced this was what I wanted.  Most of my track planning is done with a wet thumb raised to the wind.  Having a fair understanding of the requirements of grade clearances and curve suitability I just make it up as I go along.

Using flex track and 84" / 128" curved switches gave me the final layout

@Tom Tee posted:

Rich, With all due respect, if I am correctly understanding what  is written,  that will not work without an X set back.

Correct, Tom.

The center of the curved track must be offset away from the approaching straight track. The larger the offset, the smoother the easement, but the more space required.  

Great track work in your photos.

Jim

Lots of good information posted here.  Another rule of thumb for smooth-appearing passenger train running, is that the easement be at least as long as your longest passenger cars (15, 18, or 21 inches as the case may be).  That’s the rule I’ve used, and at scale speeds the transition from tangent to curve appears seamless to my eye anyway.  

I’ve used about 1/8 inch of superelevation from tie end to tie end. If you use super-elevation for curves, it should be done over the length of the easement at the minimum, watching for wheels lifting off the rail if trucks don’t have enough equalization.  In this case I mean from front truck to rear truck, ability to tilt at the bolster, not axle to axle within a truck.  If wheels are lifting enough that a flange can climb the outside rail, that would require a longer transition to super-elevation then.  The curve easement length wouldn’t have to change.  Only one maverick Lionel GP9 of recent manufacture has had that problem on my layout.  Its identical mate except for road number has no problem.

I made a template using the ones printed in the Oct. 1969 Model Railroader article about easements, and eventually turned it into a sturdy thin plywood template could use to lay out curves.  The offset with this is not extreme, less than the 1” mentioned above, so a 3/4 to 1 inch offset using the Armstrong lath method with such a rule of thumb for the length of the easement should have excellent results.

I could photograph my plywood template and post that.  Probably these days you could copy and paste that and turn it into a scaled-up template, with the help of a commercial printer or laborious arithmetic and drawing it out.  Or find a copy of the Oct 69 Model Railroader on eBay.

Fun stuff. Watching a favorite passenger train glide into an eased, superelevated curve is a great natural releaser of endorphins!

I’m aware that this post may be of no interest to anyone but me, but it is very much on topic when it comes to easements. I decided to spend some time on this subject to occupy myself during the pandemic, instead of just running trains on my layout – which uses only sectional track and has no easements.

What I did here was to write a mathematical equation that describes an arbitrary curve and develop a process to solve the equation subject to the conditions that are required for an easement. The equation for the easement curve states mathematically that the curvature increases in direct proportion to the distance along the curve. The characteristics of the easement are that it has zero curvature at its starting point and the specified constant radius of a circle at its endpoint.

After developing the method, I programmed it in Microsoft Excel™ and generated the two graphs below as an example. By entering three numbers (a track radius, an angle, and a parameter α that controls the length of the easement curve), the program calculates the coordinates of the curve and plots them. With this information and a ruler, the easement curve can be plotted on a model railroad track plan or directly onto a layout. The track centerline can then be laid on top of the plotted curve. The method can also be applied to a full-scale railroad.

In the example below, an easement was calculated between a horizontal section of straight (tangent) track and an O-31 curve at a point where the curve has a slope of 45 degrees. As shown in the first photograph, the easement begins at the left end of the blue curve (x=0, y=0) where it meets a horizontal straight (tangent) track (not shown). It joins the O-31 curved track an angle of 45 degrees (to the horizontal) at the right end of the blue curve at the point (x=22.89 inches, y=6.10 inches). The easement has zero curvature (is straight) at its left end and therefore produces no lateral acceleration as a train enters the easement. At the right end, the radius of curvature is R = 31/2 = 15.5 inches to match the specified constant-radius O-31 curve at that point. The coordinates of the easement curve were calculated at 101 points. In this case, the length of the easement curve was specified to be twice as long (α=2) as the 45-degree arc of the O-31 curve that it replaces. The length of the easement can be adjusted by the parameter α. As α increases, the easement becomes longer. As α decreases, the easement becomes shorter.

The second photograph shows an initial straight (tangent) track (magenta), the easement curve (blue), and the O-31 track circle (red) into which the easement leads. Note that the diameter of the circle is 31 inches.

As a model railroader, it’s probably not necessary to design an easement mathematically, although doing it this way is useful in creating a track plan and determining space requirements. What’s more important is realizing how a railroad may go about laying out its track and curves.

MELGAR

@MELGAR:You guys are closing in on some of the fundamentals of what I do in my job...looks like you're plotting a roller coaster in that image!

Roller coasters are essentially one continuous "easement".  The formulas you derived represent a basic plan transition.  Although there's more than one way to calculate an easement (all with their own little nuggets of difference in the end result) your example is certainly an elegant solution for model railroads.

FWIW, I didn't use any easements at all on the layout we just did.  But dad's plan was for a "large toy train" not a really cool scale model like many of you guys are using!

And yes, center offsets are smaller than you would think to create a fairly long easement.  @Tom Tee and @Jim Policastro noted that offsets as little as one inch create a fairly decent transition curve.

You guys do some fantastic track work!  These things really are works of art in some sense, aren't they?

That's interesting. Clearly, roller coasters are another situation where easements need to be designed.

MELGAR

Melgar, I think your engineering and mathematical analysis is awesome. Although I struggle to fully understand it, what I do understand makes perfect sense. Arnold

@MELGAR posted:

For example, with Atlas O sectional track, an (approximate) easement into an O-36 curve over a 90 degree change of direction could be built by starting with an O-72 section (22-1/2 degrees) followed by an O-54 section (22-1/2 degrees) followed by an O-36 full section (30 degrees) and two O-36 quarter sections (15 degrees). While this would not be a perfectly smooth easement, it would look more realistic than transitioning immediately from a straight section (tangent track) directly into an O-36 curve.

Hi MELGAR,

Don't forget to let folks know, that along with lateral acceleration, one would probably like to have some lateral deceleration also.  In other words, for any eased curve, you would probably want to come out of the curve the same way you went into it.

In your above example, several combinations for a 90° curve would work.  You could go O72-O54-O54-O72.  Or O54-O36Q-O36F-O36Q-O54.  Or even O72-O36Q-O36F-O36Q-O72.

For 180° curve (1/2 circle), you could go O72-O54-O36F-O36F-O36F-O54-O72.  Or even a few other combinations too, as long as they all add up to 180°, of course.

@MELGAR posted:

Adding easements into curves on O gauge model railroads increases the space required to execute a specified change in direction (say 90 degrees, as in my example above) and, in my opinion, is why most O gauge model railroads don't have them.

True, it does take a little more room.  But in the case of sectional track, unless you're sticking with a simple oval with 180° curves, or a simple square with 90° curves, building a more complex layout incorporating easements and using the manufacturer's standard available pieces might be an exercise in futility trying to get everything to line up close enough to fit together.

Oh, and nice work on the easement formula!

When you get Tom talking about layouts and laying track, you really need a comfortable chair and an adult beverage, he is full of ideas and techniques for creating a truly one of a kind layout.   I do have to get down to his place and see the basement full of trains he's built one of these days, the pictures are really something!

@Tom Tee posted:

Tom Tee, your photos make a good argument against scenery. 

Good morning!

@MELGAR - I'm curious if you attempted to solve your easement equation using Calculus?  It has Ordinary Differential Equations with Boundary Values written all over it.  I'm also wondering if you would be willing to share your Excel spreadsheet or, at the very least, the formula you used.  I'm assuming you used increments to generate the points.

@Mixed Freight - I came to the same realization last night that an incoming easement would need an outgoing easement for what would normally be a semicircle.  Doing so does increase the effective width of the semicircle and I can see easements being difficult to implement for those who are space constrained.  (P.S. Rock Island is one of my main lines.)

Anthony

@A. Wells posted:

Good morning!

@MELGAR - I'm curious if you attempted to solve your easement equation using Calculus?  It has Ordinary Differential Equations with Boundary Values written all over it.  I'm also wondering if you would be willing to share your Excel spreadsheet or, at the very least, the formula you used.  I'm assuming you used increments to generate the points.

Anthony

The expression for curvature is a result from elementary calculus. The equation is listed in the first line of the text box in the figures and equates the curvature to the product of an arbitrary constant and the arc length. So, it is an ordinary differential equation for the curvature that can be integrated by inspection. The boundary conditions at the beginning and end of the easement are imposed to determine the curvature and the slope of the easement curve as a function of distance along the curve. Once the slope of the easement curve is established, it is integrated along the arc of the curve (using the equation on the second line of the text box) to determine the (x,y) coordinates of the easement. In the spreadsheet, both integrations are done numerically rather than in closed form. Send me a Private Message to discuss further.

MELGAR

Easements are magical, and not only for large layouts with scale equipment and fixed couplers. I use easements wherever I can on my bedroom-sized traditional Lionel layout. (Never mind the missing rail - that will be added later.)  Both these curves are O-36 at the apex.

I follow Armstrong's method, and it works well for me. In his book, he provides a chart with values for horizontal offset and length of easement based on the radius of the curve being used - whether it is a "sharp curve" or a "broad curve" and in what scale, is really irrelevant. I know the radius of the curve I will use, I find it on the chart (listed under HO scale, no doubt) and use the values provided for curves of that radius. It is easy enough to extrapolate the values for curves not on the chart.

With small radii, the easement adds very little extra real estate. The offset for an 18" radius curve (i.e., O-36) is only 3/8". And tighter curves with easements generally look better than broader ones without them. As long as the equipment will physically navigate it, cheating an extra inch by turning your O-36 semicircle into an O-35 is worth it.

I think the real reason more 3-railers don't use easements is inertia. We grew up with sectional track, and, though I have not tried it, the layout-planning software appears to favor the use of sectional track. That, and "they" say that flex track is only good for super broad curves. But I have bent it down to O-36 quite successfully, and some have gone smaller. Making smooth joints on curves can be difficult, but I will happily take a little kink in the flex track over a huge and instantaneous change from tangent to curve.

Quoting Nickaix: "As long as the equipment will physically navigate it, cheating an extra inch by turning your O-36 semicircle into an O-35 is worth it."  Yes.  If  you want your longer passenger cars and locos to look better on their minimum radius or close to it, as long as the minimum is not violated (or not too much), they look better in an eased curve whose constant radius is tightened to keep the overall dimension of a 180 or 90 degree curve within space limits, than on the non-eased curve with slightly larger radius, because the offset between car ends entering and leaving is reduced and happens less noticeably.  Try it,  you'll probably like it.  Actually, the tighter your minimum radius curves are, the more good the easements do to improve how your passenger trains look running.  Using progressive sectional track radii is better than nothing.  You could get fancy by cutting the shortest convenient partial sections of each radius to make it closer to a "real" easement.  Sectional track has some elasticity, and can be gently bent to a smaller or larger radius to make the degrees of the arc come out right, correcting for minor problems with total curvature or letting  you make more different radii for  your progressive sectional "easement."

Fellas, with all due respect, all this math is simply not necessary!

Many years ago I had an HO scale layout in the basement. I used 12-foot long strips of wood lath to lay out the track. Wood lath will make natural easements into curves, with no math.

Along the front wall of the basement I had a track with a broad curve.

A friend of mine, who was a numbers guy, was visiting the layout. I noticed him standing over near this section of track, looking at this long, broad, graceful curve for quite a long time. Finally I went over to him and asked him what he thought of the layout so far. He said, "How did you plot the radius of this curve? The center point is out in your front yard somewhere."

My answer? I said, "I didn't plot anything. I just moved the wood lath until it looked nice."  To this day I have no idea what the radius of that curve was, nor did I care! It looked nice.

And that is my point. If you use a piece of wood lath, or a yardstick as Jim Policastro mentioned, you don't need any math! Just lay out the curve until it looks nice. The easements will form all by themselves.

MATH--->  ONLY when I must use math to describe curves to my laser cutter; otherwise no!

Rich nailed it.  For model trains a long bent stick works wonders.

Here are two for cosmetic curves. No math.  Just a flexed trim molding.   I lay out a lot of curves so a template comes in handy.

One wood, one plastic.

I mounted them on a piece of plywood for a router pattern.  My factors are the amount of deflection in 8'.

Looks like how a lot of the curves on my benchwork were derived Tom, funny thing about that.

Math is essential for some things but not for easements on a model railroad - if that's the only part of the subject that someone wants to understand. But, it can be used for easements on a model railroad and it is required on a full-scale railroad. To quote my previous post:

"As a model railroader, it’s probably not necessary to design an easement mathematically, although doing it this way is useful in creating a track plan and determining space requirements. What’s more important is realizing how a railroad may go about laying out its track and curves."

MELGAR

@MELGAR posted:

Math is essential for some things but not for easements on a model railroad - if that's the only part of the subject that someone wants to understand. But, it can be used for easements on a model railroad and it is required on a full-scale railroad. To quote my previous post:

"As a model railroader, it’s probably not necessary to design an easement mathematically, although doing it this way is useful in creating a track plan and determining space requirements. What’s more important is realizing how a railroad may go about laying out its track and curves."

MELGAR

@MELGAR - The reason I'm so keen on the math aspect of easements is that I like to put together 3D models of my layout and to be able to represent an easement via a formula will allow me to add this aspect to my model.  Also, I'm not sure how to send you a PM, even though I'm very keen on doing so.

You may wish to back track in your planning to incorporate soft curves into your bench work plans too.  Cosmetic curves work visual wonders for bench work also as Gunner just noted.  Here is the before and after of one of the towns along the branch line.  Using flowing bench work helps to keep awkward boxie table surfaces to a minimum, reduces reach and efficiently follows the meandering nature of a cosmetic curved right of way:

4 staging tracks below

An add on blister provides additional real estate.

<- Meandering 3 rail add on.

This mainline section was originally laser straight and parallel to the edge of the benchwork when the layout was being built. It was laid out by a member who was a civil engineer and it always bugged a couple of us, so when we decided to change out the scenery, out came the "sacred stick of curvature". When a Big Boy rolls through the curve by the water tower, the boiler is still over the track.

AGHRHowie and I have a running joke. "There are ten things people do wrong when building a model railroad. We've done 12 of them!" So, every time new track goes in to replace old track, the design is revisited to omit errors.

As an example, this photo shows a curve with a 180-degree reversal of direction that includes two easements (blue and cyan) and a 90-degree arc (red) of O-31 sectional track. With the easements, the vertical distance between the entry and exit points is 34.12 inches compared to 31 inches for an O-31 semi-circle. The starting point of each easement has a 1.56-inch vertical offset from points A and B on the O-31 circle. In the horizontal direction, the entire reverse curve requires 27.43 inches compared to 15.5 inches for an O-31 semi-circle, and the horizontal offset from the start of the easements to points A and B is 11.93 inches, which is the extra horizontal space required to include the easements. Measurements refer to the track centerline.

MELGAR

@nickaix posted:

Easements are magical, and not only for large layouts with scale equipment and fixed couplers. I use easements wherever I can on my bedroom-sized traditional Lionel layout. (Never mind the missing rail - that will be added later.)  Both these curves are O-36 at the apex.

I follow Armstrong's method, and it works well for me. In his book, he provides a chart with values for horizontal offset and length of easement based on the radius of the curve being used - whether it is a "sharp curve" or a "broad curve" and in what scale, is really irrelevant. I know the radius of the curve I will use, I find it on the chart (listed under HO scale, no doubt) and use the values provided for curves of that radius. It is easy enough to extrapolate the values for curves not on the chart.

With small radii, the easement adds very little extra real estate. The offset for an 18" radius curve (i.e., O-36) is only 3/8". And tighter curves with easements generally look better than broader ones without them. As long as the equipment will physically navigate it, cheating an extra inch by turning your O-36 semicircle into an O-35 is worth it.

I think the real reason more 3-railers don't use easements is inertia. We grew up with sectional track, and, though I have not tried it, the layout-planning software appears to favor the use of sectional track. That, and "they" say that flex track is only good for super broad curves. But I have bent it down to O-36 quite successfully, and some have gone smaller. Making smooth joints on curves can be difficult, but I will happily take a little kink in the flex track over a huge and instantaneous change from tangent to curve.

I do not want to hijack this thread but would like to know more about the track you have selected and how you plan to add the third rail.  Is that two rail that has been hand laid with a third rail in the future?  I have been toying with the idea of modifying MTH Scale Trax tie spacing or using a two rail track and adding a third rail.  Thanks 

I suspect that he runs 2-rail and doesn't plan on a 3rd rail!

I thought that could be the case. 

Thanks, 

@Former Member posted:

I do not want to hijack this thread but would like to know more about the track you have selected and how you plan to add the third rail.  Is that two rail that has been hand laid with a third rail in the future?  I have been toying with the idea of modifying MTH Scale Trax tie spacing or using a two rail track and adding a third rail.  Thanks

Looking closer at the track picture, I'd say it is GarGraves 2-rail O-gauge track.

Note: If you get on GarGraves website, you will find that they DO offer 2-rail O-gauge flex track (in both tinplate and stainless, if memory serves).

@A. Wells posted:

...Also, I'm not sure how to send you a PM, even though I'm very keen on doing so...

Private Messaging on the forum is a feature available only to Digital Subscribers.

Only $29.95 gets you access to all 316 issues that OGR has published since 1969!

I remember reading a discussion about this long ago, and I think it was Ace that pointed out you could use defined curve pieces of a larger radius and slowly work to your minimum.  This precludes the need to hand bend track

I've included a pic using GarGraves and FasTrack demoing this concept

This doesn't necessarily answer the question asked,

But it gives an answer to a different question,

Therefore it's answer by acccident

I have always liked Math, yet I have resolved that I like the KISS theory better. Being aware of the easement requirement for double and triple track curvatures, and making various curvature jigs,

I decided to experiment with "pre made" sectional curves. I found by using Ross pre made curves,  "Outside Track" 4 sections of Ross 096 to make a 90 degree curve, "2nd Track" I could use a section of 104 easement to 2 sections of 088 for the "apex" of the curve and back out with another 104, "3rd Track" for the next track in, use 112 easement to a 080, 080 "apex"  return with 112.  I even used 112 to 072, 072 "Apex" and returned with 112, Playing with these was Simple, put them together, take them apart, move them, and make many variations, some times I even used Ross 120's

Looked good enough to me, accomplished "curvature clearances"  for MTH Big Boys and 21" cars allowing to Start and Return to 4.5" on center straights and some cases 4.25" .

Made Advanced Math "Simple" for me

Dean

@RWDeano - "Simple" math is no fun!