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@WMDave 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!

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.

MELGAR_2021_0109_EASEMENT_FIGURE_4_CHANGE_OF_DIRECTION

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

MELGAR_2021_0109_EASEMENT_FIGURE_5_PARAMETERS_CHANGE_OF_DIRECTION

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.

MELGAR_2021_0109_EASEMENT_FIGURE_6_S_CURVE2

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.

MELGAR_2021_0109_EASEMENT_FIGURE_7_S_CURVE1

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

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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.

... 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):

Armstrong Easement Table, Sectors & Total Diameter Added
DiameterRadiusSectorLength LShift xTotal d
(in.)(in.)(deg.)(in.)(in.)(in.)
361819.112  5/1636 11/16
482419.116  7/1648 7/8
603017.218  7/1660 7/8
643217.9201/265  1/16
844217.05255/885 1/4
1085415.93011/16109 3/8



"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:

Flex Track Easement Table - 18° Easement Sectors
DiameterRadiusLength LShift xTotal d
(in.)(in.)(in.)(in.)(in.)
361811  5/16  5/1636  9/16
422113  3/163/842 6/8
482415  1/163/848 6/8
542716 15/16  7/1654 14/16
643220 1/81/265 
723622 5/8  9/1673 1/8
804025 1/811/1681 3/8
884427 5/83/489 4/8
964830  3/1613/1697 5/8
1045232 11/167/8105 6/8
1125635  3/1615/16113 7/8
1286440  3/161  1/16130 1/8



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:

Sectional Track - Half Section Easements <16 per Circle, Full > 12 per Circle
DiameterRadiusSectionsSectorLength LShift xTotal d
(in.)(in.)(deg.)(in.)(in.)(in.)
3115.5822.512  3/163/831 13/16
3618822.514 1/8  7/1636 15/16
4221822.516 1/2  9/1643  1/16
4824121512  9/16  1/448  9/16
5427121514 1/8  5/1654 5/8
72361622.528 1/415/1673 7/8
80401622.531  7/161   82  1/16
88441622.534  9/161 1/890 1/4
96481622.537 11/161 1/498  7/16
104521622.540 13/161  5/16106 11/16
112561622.544   1  7/16114 7/8
120601622.547 1/81  9/16123  1/16
128641622.550 1/41 5/8131  5/16



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

Last edited by KarlDL

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.1520129887014

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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

Last edited by MELGAR
@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

Last edited by 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.

@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:

"1. a winding in a continuous and gradually widening (or tightening) curve, either around a central point on a flat plane or about an axis so as to form a cone.
2. a spiral pattern. "
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

Last edited by 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.

Last edited by Catdaddy

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

MELGAR_2021_0105_EASEMENT_FIGURE_3

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Last edited by 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 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

Last edited by Catdaddy

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

Last edited by Rail Dawg

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.

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