Inv tan on a calculator is what is called the arctan function.
When you hit tan the number the calculator gives is the ratio of the appropriate sides of a triangle with the angle you plugged in.
When you hit inv tan you are plugging in the ratio and the calculator is telling you what angle produces that ratio.
Prior to scientific calculators trig tables were used. Here is an online one
You mean you don’t have a book of trig and logarithm tables sitting on your most used books shelf? [:)] Seriously, at one time someone had to sit down and manually do the long division so-to-speak for each entry put into the tables.
Being older than dirt, in a British middle school equivalent, I was taught that by memorizing just 4 entries in the log tables (log 2, 3, 5, 7), you could do most any operation. So we memorized them to 4 figures. And memorizing a few critical square roots (2, 3) and pi gave you the trig tables for 30, 45, and 60 degree angles. Finally, knowing that tangent and sine functions for less than 10 degrees are pretty close to each other, and directly proportional to the angle took care of the real small angles.
In my junior year of high school I learned slide rules, both circular and straight (I prefer circular).
Then I went to college to gain more knowledge. I bought my own copy of the CRC tables, which has all the formulae, tables, and stuff I would ever need plus some. Incoming freshman were required to take a short course in how to use a slide rule. In my junior year, HP came out with their scientific calculators (HP-35 and HP-45) discounted to the amazing prices of $220 and $300. Overnight, slide rules became obsolete, and the use of round numbers for homework and tests ended. Some profs really objected to calculators and computers because folks were putting answers down with 8 digit precision. One of my graduate profs started marking wrong any problem where your answer had more precision than the entering data did.
Now, when I tutor senior high school and junior college math students, I usually take my CRC tables with me. I’m still often faster than they are with their graphing calculators. And I do prefer my HP-45 over all other calculators I’ve ever used, and still use it regularly. The Reverse Polish Notation, 4 register stack, 9 memory registers, and easy polar to rectangular and back conversions make it a
Thanks for the trig and algebraic formulas on this thread. The 62’ chain trick is really neat!
These were the missing pieces in my ability to seek precision and foresee compromises in my layout planning.
I am pleased to learn that other model railroaders find this both interesting and useful. It took me since January, working by myself, to get to learn and understand enough of trigonometry to reach this point, and while I loved the process- not all will.
This leads me to want to provide a compilation of these and the other calculations I have discovered to be useful to the ldsig site.
Thanks, also, to Harry for starting this thread just as I was wrestling with this last bit.
Crews
Hotspur, my boy, you are mixing apples and oranges here.
Let me give you an example; if you have a circle and it traverses 100 feet in every degree then your circle is going to be divided into 360 segments and the circumference will be 36,000 feet (360 ÷ 1 X 100). Divide by pi (3.141592654) and you get a diameter of 11,459.16 feet and a radius of 5,729.58 feet which equates to an HO-Scale radius of 789.51 inches.
Lets change the figures a little bit and create a circle that traverses 100 feet in every 15 degrees; your circle is divided into 24 segments and the circumference will be 2,400 feet (360 ÷ 15 X 100). Dividing by pi gives us a diameter of 763.94 feet and a radius of 381.97 feet which equates to an HO-Scale radius of 52.63 inches.
Here’s where apples and oranges get mixed up: both circles are 360 degrees around - that’s what a circle is - but I think that you will agree that that 15 degree circle is going to produce a far tighter/sharper curve than the 1 degree one. If you were in the parking lot at your local Wal-M at 2 in the morning you could spin a U-turn at a heck of a lot faster speed than you can cut a U-ey in the street in front of your house.
You can do the same math on the dynamics of a 4-8-8-4 Big Boy which was engineered for 20 degree curves: 360 ÷ 20 X 100 ÷ pi ÷ 2 X .137795 = 39.48 inches which is a very broad curve in HO-Scale yet we engineer our model locomotives to negotiate 32.90 degree - that’s 24 inches - curves, nearly 64.5% tighter than the prototype’s capability.
All this math has my head spinning… and I’m majoring in computers (guess that is a failing of using calculators for everything…)
Let me see if I understand the math right R.T. Using a NYC K-11 Pacific as the locomotive of choice, it was engineered for an 18° 30’ curve - which is 18.5° if I remember the conversion from minutes to degrees (I think it is 60’ per 1°).
That means the K-11 would be able to negotiate a 286.5 foot (radius, rounded up to nearest tenth) curve (1:1 scale) or that same 39.48" curve (rounded up to nearest hundredth) in HO scale, correct?
Not that this really matters for the purpose of converting prototype degree of curve to model but railroads all (to the best of my knowlege) used cord definition so the radius of a one degree railroad curve would be 5729.65’
Just want to make sure someone who is following the math doesnt get confused.
??? Majoring as in college? A BS in computer science usually requires a minor in mathematics. Calc I, II, III, and either differential equations or linear algebra. If your advistor hasn’t mentioned this you might want to speak to them about it. What we are dealing with on this thread is simple geometry & trig. All very essential to computer science especially if one gets into graphical programming for image creation, recognintion, and processing.
Oops going off topic here. This is just another way that model railroading is a great source for educating young people.
Maybe I just need to stop frequenting the forums after a particularly exhausting bout of programming… makes so much more sense now. OTOH, my major is more focused on business application rather than the nitty gritty of straight programming (ie web design, database integration, general programming, etc without a real focus on the theory or the advanced maths).
I still think calculators are the bane of everything, 2 years ago (when I was still taking math courses) I don’t think I would have been running headlong into a brick wall thinking about this…
In the railroad business this IS a “business application”.
Dave H.
I didn’t really do the math to figure precisely what the arc would be for a 100 foot chord on one degree curve but it may well measure out to 5729.65 feet.
360 ÷ 18.5 X 100 ÷ 3.141592654 - 50% X .137795 = 42.68 inches.
360 ÷ 18.5 X 100 = 1,945.95 feet; divide that by 3.141592654 and subtract 50% renders a prototype radius of 309.71 inches which equates to a 42.68 inch HO-Scale radius curve.
A round number for pi has always been 3.1416; HO-Scale is 3.5mm to a foot and a millimeter is .03937 inches which renders an HO-Scale decimal measurement of .137795 inches but your measurement is not going to come out more than a couple thousandths off from my calculations.
Ah hah!
I see where my mistake came in, I was looking at the K-11 (18.5° minimum curve) but plugging in the numbers for the big boy in your example… and not double checking what I was doing.[D)]
OK, I think I have it now… just a bit slow on the pickup [:)]
I am still having a ball with this stuff!
Radius of 1 degree curve (100’ chord).
Angle from tangent point to second point of chord = .5 degree
R = 5729.65’
By the way, you only need 429.72 inches (about 36 feet) in N-Scale!
Crews
From the far, far reaches of the wild, wild west I am: rtpoteet
Question: What’s the difference between Political Correctness and Mindless? Answer: Thirteen Letters!
Only dead fish go with the flow! - - - - - Sarah Palin
============================
Political Correctness and Mindless:
Politica ~ Mindless [8 letters each]
That leaves the final l from Political, plus all the letters in Correctness.
Counting in groups:
lCo + rre + ctn + ess = 12 letters, by my count: the space between the words isn’t a letter (in English).
Of course, if you intended that as a joke indicating what’s wrong with our nation (from the proletariat and the bourgeoisie to the bureaucracy), then you did pretty well.
All.
The math to calculate curves is not particularly difficult. Spiral curves (what model railroaders incorrectly call easements) are somewhat more complicated. Just keep in mind that if any two elements of a curve are known, all other elements can be calculated.
The radius of a 1 degree chord definition curve = 5729.65’
The radius of a 1 degree arc definition curve = 5729.58’
To obtain the radius of a 5 degree curve, simply divide the above radii by 5, a 15 degree curve by 15 etc. Then all other elements can be found. (including stationing).
During my career, I have put many design curves (albeit arc definition curves) onto the ground by “walking the curve” which required the survey instrument to be moved along the center of the arc at 100’ intervals (or intervals divisible by 100), and calculations performed to set the next centerline point(s). I have also staked curves with the survey instrument located at the P.C., P.T., and P.I. of design curves, and calculations performed to determine points on the center of the arc at varius intervals. As I said, these calculations are not difficult, but a basic understanding of the math involved must be learned. Your scientific calculator, or PC is not going to teach this.
These methods can also be used with a protractor (not a high end farming impliment) and rule directly on the layout surface, without need to locate the radius point of the curve. This is in fact how I lay out curves on my layout.
As a note of clarification, the standard surveyor’s chain was 66’ long and had of 100 links 7.42" long. Many railroad engineering forces had chains 100’ in length, with 100 links 12" long. These chains were VERY heavy, and cumbersome to use. They were suseptable to expansion and contraction due to temperature variation, thus correction calculations were frequently required. They were intended to measure horizontal distances, thus had to be used “level” to acheive their highest degree of accuracy. T
http://www.1728.com/circsect.htm
circle calculator