# Horizontal Alignment Curve Parameters



## chuljin (Mar 9, 2009)

It's with some interest that I have been perusing the many documents available at NTSB's docket for the Metrolink 111 incident.

Most of the documents (especially the interviews with emergency personnel) are a source of considerable sadness, and after skimming only a few, I've consciously avoided them.

However (and it's a shame that only tragedy made it public), one document that has been a source of some enjoyment (since it's not related to the incident) is the Metrolink Ventura Sub Track Charts.

One question emerging from that enjoyment, however: how does one interpret the various parameters listed for curves in the 'Horizontal Alignment' area? I've googled and googled, and found many many formulae, but none that do what I'm trying to: take the three parameters listed (Degree Curve, Length Spiral, Length Curve) and turn them into the total number of degrees by which a train's course has changed during the curve. I actually majored in math, but the track I chose (no pun intended) involved very little geometry or trigonometry. 

Two uncomplicated examples, both on page 7:

The curve between tunnel 28 and the Chatsworth station:

Curve Number (from the chart): 224

Degree Curve (from the chart): 06°00'00"

Length Spiral (from the chart): 360'

Length Curve (from the chart): 1100'

Heading north of the curve (measured by me with a GPS): 90°/270° (i.e. exactly east-west)

Heading south of the curve (measured by me with a GPS): 0°/180° (i.e. exactly north-south)

Change in heading (calculated by me from the previous two): 90° clockwise (when heading south).

The curve between the Chatsworth station and the 'straight' section all the way to Burbank Junction ('straight' because it is, to the naked eye, though the chart shows 20+ minor curves):

Curve Number (from the chart): 225

Degree Curve (from the chart): 01°30'00"

Length Spiral (from the chart): 369'

Length Curve (from the chart): 4360'

Heading north of the curve (measured by me with a GPS): 0°/180° (i.e. exactly north-south)

Heading south of the curve (measured by me with a GPS): 105°/285° (i.e. 15 clockwise off exactly east-west)

Change in heading (calculated by me from the previous two): 75° counterclockwise (when heading south).

As mentioned above, googling found me no complicated formula that I could just drop these into, and the simplistic method of (Length Spiral+Length Curve)/100*Degree Curve yields a bit less total heading change than I actually measured. (I'd expect it to be more, because the full curve degree is only in effect in the 'Length Curve' part, IIUC).

Can anyone more skilled in the science point me to a good document that is, essentially Track Charts for Dummies?

Thanks!

Chris


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## George Harris (Mar 9, 2009)

Having years ago done surveying for a railroad company, I can tell you that track charts are somewhat approximate at best. Quite frequently when you ran a traverse around a curve and attempted to develop a proper spiraled curve for the turn NOTHING would match the track chart information, not the degree of curve, not the purported length of spiral, nothing. Generally, the total central angle would match or be very close, but that was because, according to one old two page letter written by one of the people involved in the 1916 Valuation surveys, they took a compass reading at each end of the curve and established the total central angle therefrom. Degree of curve was usually done by stringline checking on a couple of random spots. Where plan information was available, and that was usually only on lines built in the late 1800's forward, the degree of curve and spiral lengths were extracted from them. However, over time adjustments by stringlining could definitely modify the degree of curve slightly and frequently the spiral lengths by quite a bit.

By using (Length of Sprial plus Length of Curve) you blundered into the right sum. Remember, there is a sprial on both ends of the curve, so you have the Length of Spiral twice. However, since a spiral is a variable radius curve, a length of spiral has one-half the central angle of an arc of the same length. That is to say, if the spirals are not of equal length, which can happen, you should be able to find the central angle by using (LC + 0.5*LS1 + 0.5*LS2)/100 * Dc.

You also have to contend with the level of accuracy of the GPS. Since I would consider it unlikely that the headings would be EXACTLY at multiples of 5 degrees, it suggests to me that your GPS will only give you directions in 5 degree increments. Therefore, when you read 90 degrees, it means that the bearing is somewhere between 87.5 degrees and 92.5 degrees. Thus, the difference when subtracting bearings has an accuracy that could be as far off as plus or minus 5 degrees.

Curve 224: (11+3.6) * 6 = 87.6 degrees

Curve 225: (43.6+3.69) * 1.5 = 70.935 degrees

These numbers are will within +/- 5 degrees of what you get from the GPS, so I would tend to beleive the track charts for these curves. Since this has been a major main line for many years, the track charts are far more likely to be right on than if it were a minor branch.


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## chuljin (Mar 9, 2009)

Thank you for that, Sir!

I hadn't considered the fact that the spiral curve is on both ends. Er, well, I had, but I'd thought that the 360' or 369' listed was the total length of both bits, not the individual length of each.

You're quite right about the GPS...that's not actually what I saw, but what I assumed based on what I saw: there's a little jitter, so the headings it reported varied from about 88 to about 92, from about 178 to about 182, and from about 103 to about 107, and I assumed this meant 90, 180, and 105, respectively. The 90 and 180, I'm almost implicitly certain about, just having seen maps. The 105 I'm not as sure about, but it just made sense to me, as a multiple of 15 (1/24th of a circle). It should be said that it's a rather cheap GPS: the one incorporated into the phone I got for free by extending my AT&T contract for 2 years. Though it is 'real' GPS, not cell-tower pseudo-GPS. 

And of course I didn't dispute the figures on the charts. I'm sure they're pretty accurate, since, as you say, it's a well-traveled main line, and the charts of very recent creation/update. I simply disputed my own ability to turn those figures into something more meaningful to me (the said total change in heading).

At any rate, it's very kind of you to clue me in. I do appreciate that.

Thanks!

Chris


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## George Harris (Mar 9, 2009)

Degrees to radius:

The relationship is closely approximated by 5,730 feet / Dc = Radius in feet.

The exact formula for railroad curves is: 50 feet / sin(Dc/2) = radius in feet. This is also called a chord definition curve.

The exact formula for highway curves is: (18000 feet / pi) / Dc = radius in feet. This is also called an arc definition curve.

If you try to do the calculation in excel, don't forget that excel does angles in radians, so if your angle is in degrees, you must multiply it by pi / 180 to convert it to radius to make the numbers come out right. This applies to the chord definition calculation, but not to the arc definition calculation.

For the two curves in question:

Radius of 6 degree curve is 955.36 feet.

Radius of 1.5 degree curve is 3819.83 feet.


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## jackal (Mar 10, 2009)

Wow. All I can say is wow.

Careful, guys--this isn't "Trigonometry Unlimited." I think you just might make some fellow posters' heads explode with this topic. I'm not a mathematician by any stretch of the imagination, but I did pass trig in high skoo', and this came very, very close.


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## George Harris (Mar 10, 2009)

jackal said:


> Wow. All I can say is wow.
> Careful, guys--this isn't "Trigonometry Unlimited." I think you just might make some fellow posters' heads explode with this topic. I'm not a mathematician by any stretch of the imagination, but I did pass trig in high skoo', and this came very, very close.


:lol: Railroad alignment for dummies in a few confusing lessons? 

I can try to make it simpler for those interested, but I need to know what is confusing people.


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## chuljin (Mar 10, 2009)

George Harris said:


> jackal said:
> 
> 
> > Wow. All I can say is wow.
> ...


For me, it was actually quite edifying and helpful, thanks again.

By way of follow-up, in looking at the NTSB-produced diagrams, the track south of curve 224 is indeed exactly north-south, but north of curve 224 it is not exactly east-west, but a few degrees counterclockwise off that.


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