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!

@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

Paul was right, it's just plain 'ol Gargraves :-)

I am experimenting with using a wire and standoffs in place of a center rail. Something less-obtrusive looking. For fun, mostly, as the trains I am running on this are hardly paragons of scale accuracy. If anything comes of it, I'll post the results.

Worst-case scenario, the experiment fails and I buy myself some rail and spike it down to get back to 3-rail. If nothing else, I've learned that 2 rail is a lot easier to bend!

I don't see new Gargraves 3-rail as particularly hard to bend.  I haven't done anything with a tight radius, but O72 3-rail curves are pretty easy.  When the track gets old and the ties dry out, then it gets harder.

A very informative thread. Impressive math knowledge too! The long and graceful curves attained by easements are pleasing to the eye.

I am wondering about switches. I have mostly Ross and looking at the Premiere switches, I think that easements are engineered into the switch curves. Is that so? What about the Superline e.g. Ross regular? It would seem a shame to go to the effort of easing all the curves when laying track on a layout only to have switches that are not eased.

Bob

@Bob "O" posted:

A very informative thread. Impressive math knowledge too! The long and graceful curves attained by easements are pleasing to the eye.

I am wondering about switches. I have mostly Ross and looking at the Premiere switches, I think that easements are engineered into the switch curves. Is that so? What about the Superline e.g. Ross regular? It would seem a shame to go to the effort of easing all the curves when laying track on a layout only to have switches that are not eased.

Bob

Switch built-in easements?  Yeah, you might say so, if on a numbered switch (i.e., #4, #6, #10, etc.).

But a switch with a radius'd diverging track, then no.  It's no different than a regular piece of curved track (i.e., no easement).  They're just a lot easier to work into track plans that are already using 'like' radii curved track sections.

In either case though, always consider using the absolute biggest switch that will fit the space, if possible.  The trains will look better and operate better.

The diverging arm of a switch, even a #5 Ross or Gargraves, makes a fair to middling easement when it can be used at the end of a curve.  I've done it with curve attached to either the tail, or the diverging arm, of the switch.  Try it by eye, and you will probably agree.  Longer ones, like #8, are even better when combined with O72 and similar wide curves.

Just had a look at Ross Custom's technical pages, switch templates. If one focusses on the curved main rail in the template of the #6, #8 and #10 switches, there is a different color for the middle part of the curved rail - this appears to represent where all the curvature is. On each side of this, the rail is in a different color and looks dead straight to my eye, but well eased into the curved portion. (Not so for the tighter radius switches 031, 042, etc. Paul, above , is right.) I bet this is due to good design, not happenstance.

If Steve at RCS is listening, maybe he can comment.

Bob

I’m posting three more examples of calculated easement curves that I think may be of interest to model railroaders. After that, I shall try to avoid any further comment on this subject unless people have questions and are interested.

The first photo below shows a curve with a specified change of direction and easements at each end calculated from a spreadsheet. The user specifies the angular change of direction in the curve and the distance along the curve. The curve shown below has a 45-degree change of direction and a length of 48 inches. The coordinates were computed at 101 points and extend from a horizontal tangent at the entrance point (x=0, y=0) to an exit point at (x=42.70, y=17.69) with a slope (tangent angle) of 45 degrees. The calculated output includes the curvature and radius at each point and the minimum radius (40.74 inches) at the midpoint of the curve. For a given angular change of direction, the user can specify any curve length and adjust it until the minimum radius is as large as necessary for the intended locomotives or rolling stock. The curve has zero curvature (is straight) at each end, which means that the ends align exactly with the adjacent tangent tracks and there is no lateral acceleration as a train enters or leaves the curve. The calculated coordinates of the endpoints show the space required for the curve. By comparison, O-81 sectional track (radius 40.5 inches) with a 45-degree change of direction would require 28.64 inches in the horizontal direction and 11.86 inches in the vertical direction at the track centerline compared to 42.70 inches and 17.69 inches for the easement curve.

The plot below shows the smooth variation of curvature (blue) and slope (tangent-angle, dark red) along the length of the easement curve.

The second example shows an S-curve constructed from two easement curves of the type described above. In this case, the inputs are the arc length (for each section of the S-curve) and the minimum radius at the midpoint of each section. For this S-curve, the curve length of each section is 72 inches (total length 144 inches) and the minimum radius of each section (41.86 inches) was adjusted such that vertical offset between the entrance and exit of the S-curve is 72 inches. The coordinates of each section were computed at 101 points and extend from a horizontal tangent at the entrance point (x=0, y=0) to a horizontal tangent at the exit point (x=111.52, y=72) with a slope (tangent angle) of 65.7 degrees at the midpoint of the S-curve. The S-curve has zero curvature at each end and the midpoint, so there is no lateral acceleration at those points.

The third example shows another S-curve in which the tangents at each end are offset by 5 inches and the total distance along the S-curve is 32 inches. The slope (tangent angle) is 18.09 degrees at the midpoint of the S-curve and there is zero curvature at each end and the midpoint. The minimum radius along this curve is 33.79 inches.

The purpose of my posts on this subject is to show how easement curves calculated in a spreadsheet are useful for track planning prior to construction of a layout. The spreadsheet to do these calculations gives immediate results following input of the data (two numbers in each case) for either type of curve discussed here – with no math having to be done by the user. And, since the (x,y) coordinates of these curves are calculated as program output, marking their location on a layout does not require locating or having access to the center points – which vary along the length of the curves.

MELGAR

Melgar that is an outstanding presentation on easements!

Chuck

Agree with Rail Dawg - professorial.

Bob

Great ideas you guys. I wrote to Fast Tracks because I like their SweepSticks and other tools for laying track. Asked them about their making an easement SweepStick. They said they looked hard at that but there were too many variations to be able to produce. Makes sense.

They recommended starting the curve from the tangent using a SweepStick radius larger than the final radius and that would make for a smooth transition into the curve.

That sounded like a pretty good idea to me. Would like to hear your thoughts.

Chuck

@Rail Dawg posted:

Great ideas you guys. I wrote to Fast Tracks because I like their SweepSticks and other tools for laying track. Asked them about their making an easement SweepStick. They said they looked hard at that but there were too many variations to be able to produce. Makes sense.

They recommended starting the curve from the tangent using a SweepStick radius larger than the final radius and that would make for a smooth transition into the curve.

That sounded like a pretty good idea to me. Would like to hear your thoughts.

Chuck

Fast Track's recommendation pretty much sounds to me like simply taking curved sectional track pieces and stepping down the radius as you advance into the curve (and stepping it back up as you exit out the curve, of course).

But, for flex track, Rich Melvin's methods are the quickest and easiest way to go about it.  That's exactly the method I use too, for that matter.  Even MELGAR agrees with this approach on MODEL railroad layouts.  As a matter of fact, since you are basically drawing out the easement anyway on the layout, why not draw up a few (or several) different easement templates yourself if you plan on building a layout.  There's your own home-made easement sweepsticks right there!

As for calculating easements, MELGAR is merely showing how to mathematically accomplish this for those that might be interested in this method, which of course is very impressive.

@Mixed Freight posted:

... since you are basically drawing out the easement anyway on the layout, why not draw up a few (or several) different easement templates yourself if you plan on building a layout.

As for calculating easements, MELGAR is merely showing how to mathematically accomplish this for those that might be interested in this method...

I hadn't considered providing easement coordinates for under-construction layouts. However, it is something that could be done. I have been calculating easement coordinates with about 100 points along the curve, which is much more than necessary for laying track on a layout. A typical easement might require plotting only about ten points along the curve and could be done quickly with a yardstick placed on the layout along a tangent track that transitions into the easement. Just a thought.

MELGAR

@Rail Dawg posted:

Great ideas you guys. I wrote to Fast Tracks because I like their SweepSticks and other tools for laying track. Asked them about their making an easement SweepStick. They said they looked hard at that but there were too many variations to be able to produce. Makes sense.

They recommended starting the curve from the tangent using a SweepStick radius larger than the final radius and that would make for a smooth transition into the curve.

That sounded like a pretty good idea to me. Would like to hear your thoughts.

Chuck

If you want to use the simple method of progressive sectional radii, using the rule of thumb to make your whole easement at least as long as your longest passenger cars will help get the appearance you want for those trains.  Whatever radii you use, this would mean the total length of the easement including all pieces of larger radii together being equal to 15, 18 or 21 inches to match your passenger fleet.

In planning my layout, I ran down the easement rabbit hole in some depth.  The following discussion is based on the semi-empirical method of easement curve layout described in John Armstrong's "Track Planning for Realistic Operation".  The process is set forth in Figure 8-8 of the original 1963 edition.   It is apparent that the table in Armstrong's figure was derived using the equations set forth in James Glover's "Transition Curves for Railways", published in the Proceedings of the Institution of Civil Engineers (UK) in 1900 (and downloadable online).  The Glover paper appears to have been the basis for prototype curve easement calculations for a hundred years.

What Armstrong's table does not state is the angular sector assumed in calculating the easement parameters.  Using Glover's equations, Armstrong's table can be completed as follows (with some 1/16" difference because I'm not using a slide rule):

"Total d" is the total effective diameter of a semicircle, including easements at both ends.  An O-60 semicircle with easements as defined by Armstrong would occupy 60-7/8" of space, plus tie width.  From these data, it is apparent that Armstrong assumed a sector of approximately 18°.  Assuming that sector and applying it to a wider range of diameters/radii, the following table results:

Use of sectional track fixes the sectors at different values.  For example, small diameters use 8 sections per circle and 2 per quadrant, so it's impractical to use more than one fixed track section per quadrant.  The transition sectors for the quadrant become half of the normal section sector, 45°.  The same holds for 12 sections per circle situations, where each section defines a 30° sector, but in a quadrant one would use two fixed section and two half-sector transitions, 15° each.  It should also be noted, as Glover stated in his paper, that the underlying mathematics apply for "small" sectors (though Glover doesn't define them).  I assumed that "small" means sectors below 25°.  So, based on sectional track, the table can be re-stated this way:

At 16 sections per circle, it is also possible to use half section sectors with three fixed sections in between.  However, this reduces the easement "shift" number significantly and thereby doesn't provide as gradual a transition.  At the widest diameters, that may not matter much, aesthetically.  It's really a matter of how much of a circle you prefer to implement with fixed sections vs. flex track.

Glover's paper also recommends easing superelevation over the transition length L noted in the tables above.  Using the math laid out in Glover's paper, more general situations can be analyzed, x-y coordinates of points along a transition curve defined, and so on, for the mathematically curious or compulsive.  There are other formulations that can be useful in certain situations.  For example, I have two horseshoe curves which are obscured in a lower level.  These are implemented using continuously varying radii that go from O-72 at the horseshoe apex to O-104 nearing the (visible) edges, on the inner track.  The curve was defined mathematically using a non-cubic approach.

--Karl

Too much math, I play with my track planning software's sectional track until it looks right for the available space.  There is never enough available space for graceful easements.

I find this entire discussion fascinating on a purely theoretical level, although the higher math is way beyond my long ago and now forgotten high school math.

Rich's simple method will create a natural easement with flex track, but don't the variables have to be fixed before it is of any use? That is you need a tangent section, a curved section, an offset, and then some way to determine where to start the easement and at what angle (slope?) to meet at a tangent with the curve. I know this is what Melgar is getting at, but I'll be ****ed if I understand it.

Karl has given us useful graphs for fixed radius track to show how little extra width you need to do a 180deg curve with easements. What would be useful are diagrams using sectional track that show which pieces are used and the overall width and length for both 90deg and 180deg turns. For example, if I want to do a layout with sectional track and a minimum radius of O-54. What are my options for easements of various lengths using wider radius? There would be several ways to do this depending on what radii were available and how gradual I wanted to make the easement or maybe how much real estate I had available without crowding my aisles.

An easement is actually a piece of a spiral, isn't that correct? And an easement using decreasing radii or arcs that meet is a sort of Fibonacci spiral.

Will,

To answer your question about the spiral:

An easement curve is straight (not changing direction) at the point where it begins and then gradually changes direction and becomes more tightly curved. The radius is infinite at the starting point and gradually becomes smaller. At some point, the radius may remain constant (circular - like curved sectional track) or may begin to increase until the curve becomes straight again. The spiral is curved (changing direction) at every point and not straight at any point, so no part of it contains an easement. The spiral can serve as a transition curve from one radius to another but it is not a true easement because the initial curvature is not zero.

MELGAR

MELGAR, when you say the radius is infinite at the starting point, is that because the curve has not started yet.....a radius goes from a center to a point, so no curve, no center.....Am I understanding correctly?

@Catdaddy posted:

MELGAR, when you say the radius is infinite at the starting point, is that because the curve has not started yet.....a radius goes from a center to a point, so no curve, no center.....Am I understanding correctly?

You could also say the curvature is 0.  Just another perspective.

@Will - I actually saw that spiral diagram in a paper desribing rail easements.

Regarding easements taking too much room on a layout, my thought on the matter was to determine the curve based on the maximum allowed curvature of the easement instead of using an arbitrary ending point.

Anthony

@Catdaddy posted:

MELGAR, when you say the radius is infinite at the starting point, is that because the curve has not started yet.....a radius goes from a center to a point, so no curve, no center.....Am I understanding correctly?

Catdaddy,

You are understanding correctly.

Imagine drawing a series of circular arcs starting with a small radius and progressively increasing the radius. As the radius gets larger, the arc gets closer to a straight line. As the radius becomes infinitely large (choose your number to represent infinity - 10,000 - 100,000 - 1,000,000 - or something larger) the arc gets closer to being a straight line. When the radius becomes infinite, the arc becomes a straight line. How big is infinity? That is the concept.

MELGAR

Responding to this part of Will’s post above: “What are my options for easements of various lengths using wider radius? There would be several ways to do this depending on what radii were available and how gradual I wanted to make the easement or maybe how much real estate I had available without crowding my aisles.”

As long as the constant radius you wind up with is not too tight for any of the equipment you will be likely to operate, you can sacrifice a bit on that by tightening the radius to compensate for the “real estate” loss caused by the easement offset.  That will allow you to keep your aisle width constant, and the appearance of your trains operating on the resulting curves will be better than with the slightly larger constant radius, but no easements.

Knowing that railroads use these procedures to lay real track makes this discussion all the more fascinating.

Lots of respect to all of you.

Chuck

@MELGAR posted:

Will,

To answer your question about the spiral:

An easement curve is straight (not changing direction) at the point where it begins and then gradually changes direction and becomes more tightly curved. The radius is infinite at the starting point and gradually becomes smaller. At some point, the radius may remain constant (circular - like curved sectional track) or may begin to increase until the curve becomes straight again. The spiral is curved (changing direction) at every point and not straight at any point, so no part of it contains an easement. The spiral can serve as a transition curve from one radius to another but it is not a true easement because the initial curvature is not zero.

MELGAR

Even though I defer to you on matters mathematical, as soon as the track diverges from the tangent it begins the curve and the easement at the same time, so in fact it the easement would have no straight segment or any part with fixed radius, no ? Wouldn't that make it at least some sort of spiral?

If a spiral is:

I realize an easement is not tightening around a central point, but what is it mathematically? Here is the definition I found:

": a curve (as on a highway) whose degree of curvature is varied either uniformly or according to a definite pattern to give a gradual transition between a tangent and a simple curve which it connects or between two simple curves."

So I guess we call it an easement and leave it at that. You realize that my curiosity is strictly hypothetical at this point, because I have a 4' x 6' prewar layout!

Will,

This would be much easier to understand with mathematics rather than words. The spiral curve you presented is not an easement because, as I said before, the spiral is curved (changing direction) at every point and not straight at any point other than at infinity, so no part of it contains a practical easement. A railroad track easement is exactly straight at the point where it begins (and intersects the straight track) and gradually attains a specified curvature (reciprocal of the radius) in a finite arc length (distance along the curve). The graphs I presented show some easements derived from mathematics. I've attempted to clarify this in words as best I can.

MELGAR

Will,

I will update my tables to include total track lengths for quadrants and semicircles, among other things.  I thought of doing that about 2 hours after posting the tables - and your request is encourages doing that.  What I want to look into first is the impact on easement sector span on the "look" of the easement.  That's an evening or two in Excel with some plots to visualize the situation.

So I am going to take a shot at this. MELGAR, correct me if needed please, as I am not 100% on this.  The spiral curve is always changing, growing or shrinking, it never straightens out, even at the very end. It wraps into itself. If you were to splice into the spiral, Or follow a tangent line from the curve....that makes it an easement.  Look at the tangent line graph in MELGAR graph. Hopefully, I put out a good explanation, please tell me if I did not.  Think of the spiral like a ‘French curve” that connects two straights.

Drawing a tangent line to a point on the spiral does not make the curve an easement because the curvature changes abruptly from zero on, and at the end of, the straight section to curvature = 1/radius of the spiral at the point of tangency. Railroads use easements to transition from straight track into a curve with a smooth increase in curvature, rather than an abrupt increase. That prevents sudden sidewise (lateral) acceleration as a train enters the easement, which is its purpose.

To repeat what I have presented before, the plot below shows two tangent sections of straight track (magenta) leading into and out of two easements (blue and cyan) and transitioning into an O-31 model railroad curve. The easements have a 15.5 inch radius (curvature = 1/15.5 = 0.0645) where they meet the O-31 circle and are exactly straight (curvature = 0, radius = infinity) at the points where they meet the tangents.

I have done my best to explain this and will leave further discussion to others.

MELGAR

Yes, I see it, @MELGAR.  I guess I was thinking that every point of the easement was in fact a curve however slight, but you are saying because it starts from a tangent, it can't be a spiral. That makes sense. But it makes me curious. Is an "easement" strictly a practical term used in the physical world, or does it have a use ( or even exist) in pure mathematics? Your formula seems to imply a mathematical model.

@Catdaddy posted:

So I am going to take a shot at this. MELGAR, correct me if needed please, as I am not 100% on this.  The spiral curve is always changing, growing or shrinking, it never straightens out, even at the very end. It wraps into itself. If you were to splice into the spiral, Or follow a tangent line from the curve....that makes it an easement.  Look at the tangent line graph in MELGAR graph. Hopefully, I put out a good explanation, please tell me if I did not.  Think of the spiral like a ‘French curve” that connects two straights.

Yes, that is exactly what I was thinking, except a French curve that connects a straight to a circle or arc. Or two French curves mirrored that connect two straights.

I had forgotten that I used to use French curves back in the days of hand drafting for just this application. In CAD, I think this would be a curve with bezier points. You would snap to the straight and to a tangent on the circle and then pull your bezier handles to get your easement. I have designed news desk tops using this technique.

I have seen the word "easement" used only in connection with railroad curves and have not seen it used in mathematics to describe the behavior or shape of a curve.

MELGAR

I threw in French curve was for you Will, even though it was a bad explanation, I thought perhaps it would help you see the connecting part, spiral has no constant section that is eased into,  I focused on wrong part of  the  deal. .......MELGAR is correct the pictures say it best...... thank you all for discussion

an easement is like a right of way of , or a pathway thru a property as far as I know

I'm going to use the FastTrack "sweepsticks" when laying my Atlas flex track. Will build an easement out of them that departs the tangent with a larger radius and then a progressively smaller radius until I hit the final radius. Can glue them together and use them all over the layout for easements.

Watching you guys discuss this stuff is pretty cool. Am learning a lot.

Chuck

One other comment about easements that nobody has brought up -  When space is a concern and you simply have to make x degrees of turn in y inches, easements may still be a good idea, but you've got to pay the piper.

In other words, by giving yourself the luxury of including track at the entrance/exit to your curve that is broader than the average radius, you necessarily will include track in the middle of your curve that is tighter than it would have been without the easements.  You have to balance your desire for easements against the corresponding reduction in your minimum radius to create the best looking/running curve you can in the space available.  I don't think there is a formula for that situation.  Some equipment may be more sensitive to the change in lateral acceleration, while other equipment may be more sensitive to the absolute minimum radius.

I’ve always used the term “transition” for this sort of geometry, an “easement” being an access or right of way.

Working with Fastrack, or any other sort of set track, the main thing is to keep track of the total curvature of the bend, versus the required curvature. Distribute the total curvature symmetrically and ensure that the total curvature is correct, ie that a 180 degree curve adds up to 180 degrees, and there isn’t really much to go wrong. Half-length O48 curves are your friend here, because two O36 = three O48, so you need one and a half O48 to equal one O36 or two O72.

Provided the resulting curve fits into the available space (SCARM or similar is your friend here!) you just need to check overhang clearances.

One general comment I would make is that the first place to look for problems with side-throw clearances, is outside exit curves.

I was working on the math involved with easements while I was trying to fall asleep the other night.  I was hoping to find a closed form solution using the integral of dK/dO and the equation of K(O).  However, I was not able to achieve separation of variables in polar coordinates but I could see the solution of the curve in my mind as well as the separate function describing the effective radius for the angle (the osculating circle).  I was aware that I had seen that curve before.  It's not just a plain spiral and it has double quadrant mirrored symmetry.

It is a "Euler Spiral" and the solution is most likely based on the Euler functions.  Check out (https://en.wikipedia.org/wiki/Euler_spiral) and the section on Track Transition Curve.  Given boundary conditions (angle of transition (eg. 45 or 90) and the final curvature (eg. 1/36 <= 1/(O-72/2)), you can calculate the exact points for the easement.

Anthony

Working on the math for easements would put me to sleep!

@gunrunnerjohn posted:

Working on the math for easements would put me to sleep!

Luckily there is a cubic curve approximation for those who are less math inclined.

I just eye-ball it!

and I have an engineering degree,  too.

On all my trackage I use Midwest Cork O-gauge roadbed. On the outside rail of all curve trackage + 10 inches in and out straight trackage, I use 1/2 of a N gauge cork roadbed ( gives about 1/8 inch rise). Looks good and is not overkill. Works on 0-54-thru 0-72 diameter curves. No calculating and gives a prototypical 'elevated curve' look. I only use K-line Super Snap track and successor pieces.

Walter

I spent a lot of time last year exploring the curve easement topic, consulting a turn-of-the-prior-century British civil engineering paper on the matter and far more recent work by some Greek civil engineering professors, as well as the empirical procedure outlined by the late John Armstrong.  The result was a spreadsheet that calculates the x-y coordinates of points on an eased curve, as well as templates for curve entry and exit, with emphasis on situations where sectionalized track is used.  I hope to post the write-up before summer.  As for superelevation, my layout-under-construction uses N gauge cork roadbed strips at the outer edge of curves, between easement points and tapered down within such easements by careful application of a belt sander.  The area between the edge of the N half-roadbed and the O inner track is tapered using lightweight spackling compound, to provide a uniform surface for the foam O gauge roadbed atop.  The 1/8" superelevation not only looks good, as Walter notes above, but is prototypical, from what I've read.

Yep, I found that the vertical easements are probably more important for reliable running of large locomotives.  I had to do a little "tuning" to make sure that certain locomotives made a smooth transition onto and off of the grade.  The horizontal easements aren't always critical for reliability as long as curves are wide enough, they do enhance the appearance running through the curves.  However, the vertical easements will really bite you if you don't consider them.  Oddly enough, some of the significant offenders for vertical easement issues are things like the Lionel scale GG-1 and the steam with lots of fixed drivers like the UP 9000 4-12-2.  I found most of my articulated stuff much more forgiving on the grades.