Easement Curve Information

[Anonymous]

I once read an article in a Model Railroad magazine, I believe it was in the April 1980 issue, where a mathamatical formula was given which calculates the x-y coordinates for parabolic or easement curves for model railroads. Does anyone know this formula?

[AggroJones]

I don't know the formula. I free-hand those muthas. Coming off the straight, I begin the curve with 40" radius but only for a few inches (5 or 6"). Then my real curve begins.

[johncolley]

I prefer to take a piece of thin material such as masonite or a door skin and using the batten method I make templates of the easement with all points (marked on both sides so it can be used right or left), plus 6"tangent and 6" or more of curve for each radius I plan to use. that way I can put them anywhere on the layout and mark my track centerline with a fine sharpie pen. The templates can even be cut close with a saber saw and finish sanded to the line.

[camarokid]

I would look in "Track Planning for Realistic Operation" by the late great John Armstrong. Page 47.

[ericboone]

The first web page has the formulas I used in CAD to create easements.
http://www.trackplanning.com/easements.htm

These two articles are interesting.
http://www.trains.com/Content/Dynamic/Articles/000/000/001/647dsuww.asp
http://www.trains.com/Content/Dynamic/Articles/000/000/001/546jidvq.asp

[mcouvillion]

SwitchFrog,

If you decide to use easements on your layout, build a series of curve templates out of 1/4" luaun plywood first. I built a set at 3" intervals and it made laying out the curves so fast and easy. The benefit is that you only have to properly lay out the easements once. The templates can be readily adjusted to fit your space, or readily changed if it doesn't.

For those in the Houston area, the templates (for HO) are available to rent from Papa Ben's.

Mark C.

[Paul Milenkovic]

Pulling out my shopworn copy of Track Planning for Realistic Operation, the three magic number governing an easement are 1) radius of curvature, 2) length of easement into that radius of curvature, and 3) "tangent offset." On p 75, for 18" radius, Armstrong recommends a 12" easement resulting in a 3/8" tangent offset.

What you do is lay out the tangent (straight track feeding the curve) and curve without the easement. Next, you displace the tangent in the amount of the tangent offset (the 3/8" in this case), and that displacement is in the direction that takes up more space (no free lunch). Then you divide the 12" inch easement in two halves -- the first 6 inches is taken up out of what would have been straight track and the remaining 6 inches is taken up out of what would have been curve. Once you have done all of that, you can eyeball the transition, use a bent stick, use flex track and sees what makes trains run smoother, and so on.

The 12-inch transition length doesn't come from On High (sorry, it comes from Track Planning for Realistic Operation so I guess it does). You could have longer or shorter transition lengths (see ericboone's Web references). A longer transition length may help if you are trying to bend Superliner-II models around 18" radius -- it won't remove the overhang on the sharp curve, but the longer the transition the less back and forth "Chicago El/tinscale model" lurching of the car ends. A shorter transition length can save some space -- who runs scale-length Superliner-II's on 18" radius anyway? I have been using 18" radius, a 10" transition length, and a 1/4" tangent offset.

One of the properties of these easements is that the initial part of the easement looks like a very shallow curve indeed -- it looks like it isn't doing anything and there may be some temptation to cheat. The classic "cubic transition spiral" has a linearly-increasing curvature over the transition length. Note that I said curvature, not radius -- curvature is the reciprocal of radius. Straight track has a radius of infinity or a curvature of zero -- the curvature starts at zero -- it increases to half that value at the midpoint and reaches the target curvature at the end. What this means is the transition is nearly infinite radius at the start, 36" radius in the middle, and 18" radius at the end, and substituting a piece of 24" radius track or the divergent route of a #4 switch (Armstrong says Atlas #4 is really #4.5) as a "poor man's transition curve" does a poor man's job of the transition.

You know, there is such a pronounced lurch coming out of a #4 switch into 18" radius that I am planning on investigating if a proper transition can be designed to connect #4 into 18" radius -- it is not the initial turnout entrance that is the worst lurch offender.

Getting transitions right requires a bit of time-consuming fiddling, and making up some templates can help. You can eyeball, free-hand, or bent-stick transitions without fancy math, but it really helps to use the 3 numbers -- curve radius, transition length, and tangent offset -- as a general framework. And to paraphrase TV's Mr. Food, your track will by "oh, so good!" if you do. The track will look realistic and well-engineered, your equipment will look good on it, and it has benefits in reducing derailments from couplers yanking sideways on car ends.

[Paul Milenkovic]

Something I forgot to say about the transition length. What should the transition length be? Should we just accept Armstrong's formulas?

I don't have any math on this, but my intuition says it has to do with the length of the car you want to operate. A 40' HO car (about 6") occupies half of a 12" transition. An 80' HO car (about 12") occupies the full 12" transition when it crosses it. My guess is that the transition length needs to be somewhat greater than the longest car you want to operate. The reason Armstrong specs an 18" long transition when going to 24" radius is on the notion that folks running broader curves are doing it to run longer equipment. If you are not running long equipment (24" radius is still iffy for Superliner-II's), you could use less than an 18" transition with 24" radius curves, especially if you are pinched for space.

[mcouvillion]

When I built the templates for the curves, which included the transition spirals and tangents, I used the following dimensions for laying out the transitions. Some were from John Armstrong's data, the others were deduced from his information. (Sorry guys, I can't get the columns to line up properly.)

Radius \ Spiral \ Length of
(inches) \ Offset \ Spiral
(R) \ (X) \ (L)

18" \ 3/8” \ 12”
21" \ 13/32” \ 14”
24" \ 7/16” \ 16”
27" \ 15/32” \ 17”
30" \ ½” \ 18”
33" \ 17/32” \ 19”
36" \ 9/16” \ 20”
39" \ 19/32” \ 21”
42" \ 5/8" \ 22”
45" \ 21/32” \ 23”
48" \ 11/16” \ 24”

Laid out on a 4x8 sheet of 1/4" luaun plywood, it looked like a series of huge boomerangs. Once cut out and sanded, it was extremely easy to "fit" a curve in the space available, even curves that were much larger in radius than you would think were possible. The template can be moved around until it fits - it only takes a minute to determine if it will fit or not. If you lay out the radius and the spiral by hand every time you lay out a curve, you'll never get them done. You get two radii per template, and two examples of each radius, except for the smallest and largest of the group. On a 4x8 sheet, you can get from 18" to 48" radius templates. I did 21" to 45", but now that I know better, I could get the others too. It did not take long and was without a doubt worth the time invested.

I have a detailed procedure for laying out the templates, and for using them. I was hoping to get it published but haven't pursued it. If there is interest, I could pursue that, or just make it available to interested parties.

Mark C.