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Imagining the size of a second of arc
Posted by Mark Mayer on March 26, 2016 at 6:41 pmImagine you are standing in front of a standard classroom chalkboard (or dry erase board, as the case may be, according to your generation). Draw a circle on it about as big as will fit. Let’s say it’s 48 inches in diameter.
Now divide that up into 360 degrees. The circumference of your circle will be about 150 inches, each degree will subtend about 0.4inches, or about the width of your chalk (or dry erase marker).
Now divide each degree up into 60 minutes. Each minute will subtend about 1/2 of a 64th of an inch. You will need a very sharp, very hard pencil, a steady hand, and some serious optical enhancement to even hope to achieve that.
Now divide each of those marks into 60 seconds. Or even twenty 3 second segments. Hardly seems possible. The marks would be microscopic. And remember, this is a 48″ diameter circle, far, far larger than any total station.
For a 6inch diameter circle (still, I think, somewhat large) the degrees are well less than 1/16th of an inch, each minute is 1/20th of 1/64th of an inch. Dividing each of those marks up 12, 20, or 60 times is hard to visualize.
larryscott replied 8 years, 4 months ago 19 Members · 31 Replies 
31 Replies

Mark,
Now walk us through the steps to visualize sizing up that 48″ diameter circle to the diameter of the Earth, and project out the size of a second of arc on the ellipsoid…

Pierre Vernier invented the vernier scale early 1600s
I ran Dietzen transit 8yrs and every time I cleaned and replaced the black polish that filled the grooves in the scales and verniers it amazed how in the world did they make it so accurately.
,B)

This is helping me to visualize & frame my current existential dilemma. Dasein.

Mark Mayer, post: 364284, member: 424 wrote: Imagine you are standing in front of a standard classroom chalkboard (or dry erase board, as the case may be, according to your generation). Draw a circle on it about as big as will fit. Let’s say it’s 48 inches in diameter.
Now divide that up into 360 degrees. The circumference of your circle will be about 150 inches, each degree will subtend about 0.4inches, or about the width of your chalk (or dry erase marker).
Now divide each degree up into 60 minutes. Each minute will subtend about 1/2 of a 64th of an inch. You will need a very sharp, very hard pencil, a steady hand, and some serious optical enhancement to even hope to achieve that.
Now divide each of those marks into 60 seconds. Or even twenty 3 second segments. Hardly seems possible. The marks would be microscopic. And remember, this is a 48″ diameter circle, far, far larger than any total station.
For a 6inch diameter circle (still, I think, somewhat large) the degrees are well less than 1/16th of an inch, each minute is 1/20th of 1/64th of an inch. Dividing each of those marks up 12, 20, or 60 times is hard to visualize.
Another method of visualization. The chord distance subtended by 1″ of arc at:
1 foot = 0.0000048′
10 feet = 0.0000485′
100 feet = 0.0004848′
1,000 feet = 0.0048481′
10,000 feet = 0.0484814′
100,000 feet = 0.4848137′This is why most experienced people can do very tight layout work with a 5″ instrument. The techniques, prisms, and method of control matter far more than whether or not you have a 1″, 2″, 3″, or 5″ total station.
Leica has been giving accuracy spec for their ATR for only about the past six years, Trimble just started with the new S5/S7/S9 series. Robotics are just now getting to where they are as good as the human eye, and I don’t know if you’ve noticed, but consistently aiming to the same prism with a 1″ instrument and getting within 5″ agreement is pretty dang hard.

Plumb Bill, post: 364290, member: 226 wrote: …. aiming to the same prism with a 1″ instrument and getting within 5″ agreement is pretty dang hard
If you want to read angles to 1″ you are going to need more than just a 1″ gun. You are going to need higher quality tripods (or concrete pier), tribrachs, and targets as well. I believe that 3″ is about the best you can hope for when mounting on a tripod, and if you are using chinese tribrachs and glass then 5″ may be tough to see.

Warren Smith, post: 364286, member: 9900 wrote: Mark,
Now walk us through the steps to visualize sizing up that 48″ diameter circle to the diameter of the Earth, and project out the size of a second of arc on the ellipsoid…
25,000miles (earth cir)ÃŒÃ‘5280ÃŒÃ¡100=number of seconds in 360degs,,,,,,,,,,or 1 second is very close to 100′

Or you could say that 1 second = pi/648000 radians. Then use the radius of the earth as 3956.5 miles based on the GRS80 ellipsoid and get 101.3 feet.

What I use as a rule of thumb is that at 2000 feet, 1″ of arc = 0.01′. This is not exact, but it’s pretty close. Really brings it into perspective for me.

I have been checking out distances by subtense. T2, 1016 sets, SDEV usually 0.8″, std err around 0.15″.
Daytoday repeatability generally: 0.20.3 seconds. Standard wood tripod.
A 1″ angle generally requires 10 sets and statistics.
36 sets, 2 m subtense bar at 74.730 m (1 deg 32 min.), returned less than 0.003 m error, which is about 0.25 arcseconds of subtended angle, compared to multiple chainings. Used 2 different chains, Lufkin Super Highway, K&E 30 meter, cold and hot days, spring balance, 2 thermometers. Chaining agreement less than 0.002 m.
1″ is a the width of quarter at a mile.

This discussion of tiny distances on small circles made me think of much bigger “circles” that I stumbled on recently:
[MEDIA=vimeo]139407849[/MEDIA]

Never thought about that before. Still trying to figure out how to make “12:00” stop blinking on the VCR. ðŸ˜‰

For: Larry Scott
I just read your post and would like to comment on some things.
1. Wild says that their “Precision invar subtense bar” accuracy is 25mm at 100 m.
2. At a angle of 1d 32m (exact) your distance should be 74.729 m not 74.730 m.
3. The DIN 18723 specification for a Wild T2 theodolite is 0.8 seconds (The Standard deviation of a mean direction measured in
face left and face right (FL+FR)/2 = 0.8 seconds
4. 1 second in a mile is 7.8 mm (a quarter is 24.2 mm in diameter, nickel is 21.24 mm in diameter, penny is 19.01 mm in diameter
and a dime is 17.75 mm in diameter.
5. In your taping of the distance( you say chaining but I know you DO NOT used a Gunter chain) you have close agreement but nothing was said that
the 2 tapes had been calibrated at NIST nor was the tension handle or thermometers calibrated.With #3 above I would think that you could turn less sets and get very good results.
How long have you been turning angles?
What type of T2 (old style or new).
What procedure do you use when you turn your angles?JOHN NOLTON

The typical gun is something like four inches from the pivot point to the front. So using 1 foot = 0.0000048′ suggests that the radial movement of the front is 0.0000016′ for one arc second.

Well, I didn’t post the full precision of the resulting angles. And I indicated using a Lufkin Super highway ‘chain’ which is a tape. Along with a K&E metric tape. And I’ve never used the verb taping when pulling a tape. Old style T2.
The chains are not calibrated. And they agree with each other at 20 degrees C, 10 pounds, fully supported. They also return 1:20,000 agreement to GPS determined distances. So, the chains, 2 different manufactured, very different ages, are “probably” okay for this exercise. NIST charges $500/ea. I didn’t go with that at this time – since they measure approximately the same, fit distances determined by other means, they are only assumed to be suitable. The thermometers are high quality industrial mercury. A pair, returning temperatures so similar I can’t tell them apart. And I tested them in an ice bath. The spring balance was tested against a static load of 12 liters of water, easily satisfactory for this application.
In the wild subtense documentation they state the precision can be improved by increased observations. The 25 mm per 100 m is only 1:4000. Not stated in my Wild documentation. In the wild manual that I have, it states 1:10,000 is readily achievable at reasonable effort, and the length and stability of the bar can support 1:20,000 with additional effort. Doubling and quadrupling the number of sets. In the manual they refer to a standard procedure of 10 sets at or below 1″ std dev. for the expected 1:10,000. And longer distances simply require more.
Also in my manual it states the obvious: that a 1″ second instrument can readily far surpass 0.5″ accuracy. Also per the manual: “the inverse square law applicable to errors offers the possibility of holding to a predetermined accuracy by the judicious combination of angle and distance measurement …. … the figures given in this manual for the mean errors to be expected can be substantially reduced where needed”.
And what about (FL+FR)/2 = 0.8″? Is that per set?
It’s a matter of statistics. The t2 is quoted as 1″, attainable accuracies can easily be better than 1″. With larger, consistent data sets, precision improves. Per Wild, and other academic sources, 30, 60, or even more, outliers edited, present better that 0.2″ precision. And to reduce that amount of data I use a spreadsheet.
I was trying to see how well I could measure distances with a subtense bar, without doubt. What is the upper limit of precision? So, I set up a parking lot network (10 pts), and turned every possible angle to over determine the network. Using Starnet, a priori angles: 2″. Centering was done by transiting 2 instruments (T1, T16) at close range to center the targets better than the optical plumb of the tribrachs. Centering error assumed approx 1 mm. Minimally constrained, angle/direction residuals well distributed (+/) and 05″ with exception on the really short sides. 48 sets typically, many angle/direction sets repeated.
All distances chained (or taped) multiple times. Tension temperature, with wide swings in ambient temperature. A priori distance error set 50 ppm (1:20,000). Residuals (+/) 00.006 m, with exception on only the longest line, 12 mm.
30 distances. Median distance 85 m, max distance 168.787 m.
Additionally, I incorporated 6 points held to 5 mm std err, determined by GPS, and the residuals to both distance and angles were essentially unaffected. i.e. the chain distances fit the GPS well, very small scalar component, very low noise data. (I then removed the GPS positions so that only chain/subtense/angle data are adjusted.)
Then, I measured all distances by subtense. The long distances, 168 m, several times, with approximately 0.25″ daytoday repeatability. I accumulated repeat measures, and several of the longer lines have 80100 sets collected from multiple days. Calculated Standard err typically 0.12″ and better, std dev <1". And shorter lines 816 sets, std dev <1", std err 0.2"  approximately. All in all about 1300 sets.
In the adjustment, the subtense distance residuals were indistinguishable to chain distance residuals. Including the 168 m, (550 foot) legs.
95% confidence regions, semi major axis, 0.0000.003.
The effort was well beyond a paying job. But Starnet can reveal error, blunders and accuracy so well, that a 2 day goof around became a couple of month investigation.
What I found is that subtense distances, well beyond 75 m, up to the 168 m, repeat easily and reliably at 1:20,000, and several of the distances, those with the most accumulated data 1:30,000.
I’ve seen a lot a questions about “can a 1″ instrument measure a 1″ angle?”. The answer is yes, but it’s a matter of statistics.
All I can say is that, I have repeated measure of subtense angles with 1020 sets on different days and the angle always repeated less than 1/2 second, often repeated 0.15″. A real world observation, not a quote from a text. And that returns about 3 mm in 90 m using a subtense bar. 1:30,000 best, 1:20,000 normal.
I’ve seen many publications that cite subtense as marginal, and the upper limit as 75 m. And the question “is subtense as good as chaining (taping)?” The answer is: is chaining is as good as subtense? Yes, if fully supported, elevation profile, temperature, tension applied. In catenary, 1:20,000 chaining is difficult. 1:20,000 subtense (less than 168 m) easy. 60 sets, about an hour.
I even checked the length of the bar in blaze sunshine in summer, and again in winter. A temp diff of 50 deg C. I was not able to detect a change in length. I was verifying its temp stability since it was an eBay purchase, and it may have been broken.
So, in a 1965 setting, if you had to cross a 400 ravine 12 hrs of subtense wouldn’t be the weak link in traverse. Try pulling a 400 ft tape at 1:20,000 in one pull.
If I started on it today, I would use fewer angles. But, I was trying determine: systematic error, scale error, realistic provable accuracy, procedures, and all completely analog. An EDM would not be better, just faster.
I yield on the quarter/mile. A quarter is 3″ at a mile.

For: Larry Scott
Thanks for going to the trouble to post. I too would play with the subtense bar in the military in the 1960’s , in my spare free copious time.
There are several old articles in Survey Review dealing with the subtense bar and the accuracy you can get with them. If you would like I can get the references for you? What I found (1963) was the accuracy of somewhere between 1/2500 to 1/10,000. I did get 1/20,000 more than once but only turning 2 D&R. The DIN 18723 is what you can expect to get in the angle you turn or another way of saying it would be if you turn 1 D&R with the Wild T2 then the
standard deviation of the angle would be 0.8 seconds.JOHN NOLTON

I’m familiar with DIN standards. A standard deviation of one angle is not mathematically significant. So (D/R)/2 = 0.8″ has no real certainty.
The population of measurements to compute a significant Standard deviation, standard error, has to be large enough to determine a Gaussian distribution of the error. Minimum population of 30 has been often cited. I was trying for 2 sigma. That’s why I checked my spring balance, and thermometers.

When my wife says “Just a second” it could mean anything. In fact, my wife’s people are usually late. But, I cannot complain. I would not have a wife, if this were not so. My wife’s great grandparents had tickets on the Titanic.
You guessed it. They missed the boat.
Now, you know, as they say, the “rest of the story”!N

I remember the discussion and computations in class about subtense bar. Never have been that close to one outside a display at an equipment show and museum.
The concept stuck and I have used a few versions several times with a tape and even a level rod before when we would need to measure across a bayou.
Normally we would clip the several babbit tapes together to make something several hundred feet long to measure with.
Problem was getting it to the other side when the water was too deep to wade across.
When the bayous around here are that far up, the undertow is not something to chance.
The results were far from precise because the best instrument we had was a 20sec gun and we worked with what we could see on the target end, like a plumb bob string thrown over a mark on a tape that was close to being square to the line of site.
We would also get a stadia reading and do the same from each end and let the boss decide what results to use.
Our other option would have been to force close that distance and I’ve seen that done many times around here along the unmeasured line.
At the time, we concluded that nobody was gonna know what that distance was until something better came along, which turned out to be about 10yrs down the road. 
A T2 is an amazing instrument.
What I saw in the results, just like the documentation said, the upper limit of subtense distance accuracy is user defined. In my experiment, although up to 550′, a cm was uncommon, 15 mm an outlier.
So in a pre electronic setting, 400′ subtense distance could compete with chaining. If circumstances left no plan C. And it’s not just take an average. Reliability of the distance is how well you can home in on better than 1/4 second. And that’s not outside the ability of T2.
That’s the nostalgia of surveying. A very old, highly understood science. But, GPS and total stations have taken a lot skill out the game.
I have a 1925 book by the Royal Engineers. It’s in depth, and high science. And surveys didn’t get better, just faster cheaper.
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