Angles close to zero

i think a sine is a sine whether it is close to zero or 90 or 180.
Maybe this was true in older times when you had to refer to logarithmic tables with limited decimal point values?
The total station will give results up to nearest seconds?
Today you have spreadsheets that show results up to 10 decimal places.
You are limited by present tools & technology. Just give the results as best gotten from these instruments. No one can blame you in 10 years time that you were off by 1-2 mm~:-/

Thanks - Netherlands? You should be in bed!

I would worry more about the stability of the fire hydrant than having angles near zero. We do structural behavior measurements all the time, and stable control is the most important thing. One of our Lock Walls has a 1 st Order bench mark on the end of it, and we check the elevation difference between it and a near by PBM that is not in the NGS Database.

I don't think there is a problem in using the sine of a small angle unless you are dividing by it.

The best way to check your application is to do a computation with a change of whatever you think your angle accuracy is, and see how much everything moves because of it.

I hope you have backup control. What happens if the water department decides to replace the hydrant?

Strength of Figure still applies, even to computers, not just tables of logarithms.

I dug out my Wolf and Ghilani Elementary Surveying and found this: (page 295 10th edition)

"Because of the nature of trigonometric functions, computations in some coordinate geometry problems will become numerically unstable when the angles involved approach 0 degrees or 90 degrees. Thus if coordinate geometry is intended to be used to determine location of points, it is generally prudent to design the survey so that triangles used in the solution have angles between 30 degrees and 60 degrees."

I guess this is what I'm getting at...(and that say's 90 degrees too....)

Algorithms. Approximations. This is what you are getting at. Those lovely numbers so quickly generated electronically are based on algorithms that approximate the ideal circumstance. Approximations are a lot like assumptions, something best to avoid when possible.

Back in my university years, while the Earth was still cooling and the first creatures were leaving the sea and sprouting leg-like appendages, we had a computing laboratory that filled the equivalent of three classrooms and required massive air-conditioning systems. You assembled large quantities of rectangular pieces of paper that were slightly truncated on one corner that had weird little holes whittled in them. You bowed before the keepers of "the computer" and offered up your stacks of holey paper bound together with rubber bands. A day or two later you would discover that they had finally fed your sacrifical offering to "the computer" and you were presented with a lengthy green and white huge strip of something resembling toilet paper, which upon educated deciphering told you something you hoped to learn or, more commonly, told you you had a hole in the wrong place or a card out of order.

I had a large project that involved determining the amount of solar energy that could be captured by a specific type of solar collector panel set at various angles relative to the sun and at various angles relative to terra firma. I had it go through an extensive iterative process gradually altering each of these angles to find the ideal orientation for the collector. An odd thing happened. Whenever one of the angles approached a number somewhere close to 57 degrees (roughly one radian), the computer would go nuts. The magic equations (algorithms) included hyperbolic sines and cosines and other bizarre trignometric functions. Somehow, when the angle became too close to exactly one radian, the approximations resulted in some resultant number being a teeny, tiny, wee bit more than 1.0000000. When the inverse function was required for this number greater than the absolute maximum possible value, "the computer" had a spaz attack.

Ummmm, NO.

>in some coordinate geometry

You will find such difficulties when locating a point by two lines that intersect at a small angle. Finding an adjacent side from the value of a small angle is bad. Finding the side opposite the angle is ok. The classical strength of figure applies to propagating measurements by triangulation angles, and is not so important if you also use EDM.

A particular software program might have a problem with some configurations, but there could be a different set of formulas/sequence of solution that works just fine with the same data.

For measuring the perpendicular distance from a point to a base line, the accuracy of the result depends mostly on the distance from the instrument and the angle tolerance, and will not be improved - indeed may be hurt - by having a larger angle unless you reduce the distance from the instrument.

You need to try your particular geometry to see if there is an unacceptable sensitivity to an angle, and not use an over-generalized rule of thumb.

Ummmm, that's what happened. If I had it skip over an iteration near the magic number, the program would continue working great. The geniuses (genii?) in the computing center searched for the solution and finally blamed it on the algorithms and the eventual creation of a value slightly greater than one. The next step would always explode.

If your angle is say 3 seconds, and the shot is 200 feet, the offset perpendicular is 0.003 ft., the computational accuracy of the sine of a 3-second angle on a modern computer is plenty fine for that. 200 x sin 3 seconds = 0.003. That is probably a lot better than you can physically measure. I agree with previous poster that when you try to do intersect or two point resect with this you should avoid it, but for figuring a single right triangle, I see no problem. Strength of figure for perpendicular pier movement based upon angle measurement is tighter near zero than it would be at 90.

I'm assuming you're looking for motion perpendicular to your baseline. So, if you can see down the baseline, and see each bollard to measure angle & distance, why not just sight down the baseline, place a 6' rule against each bollard, and read the offset directly?

"Because of the nature of trigonometric functions, computations in some coordinate geometry problems will become numerically unstable when the angles involved approach 0 degrees or 90 degrees. Thus if coordinate geometry is intended to be used to determine location of points, it is generally prudent to design the survey so that triangles used in the solution have angles between 30 degrees and 60 degrees."

i think this refers to intersection & resection methods of point positioning. i don't think it applies to EDM methods.

if this were the case, you would surely have read this as a strong selling point in most total stations.

-our total stations can be used in determining correct positions even in angles close to zero - 🙂

You can see from the graph of the Sine function that as it approaches 90 degrees the rate of change slows down and flattens out. This means that smaller incremental measurements of the angle result in less change in the resulting Sine value. For the Cosine Function the rate of change flattens out as it approaches 0 degrees. So from this, I would think that when evaluating the Sine function at angles near 90 or the Cosine function at values near 0, calculations would be affected differently then when the rate of change is greater along the graph.

This survey work is ideal for Least Squares. The beauty of the LS is that you can pre-design your survey and test your expected accuracies prior to any fieldwork.

> I'm assuming you're looking for motion perpendicular to your baseline. So, if you can see down the baseline, and see each bollard to measure angle & distance, why not just sight down the baseline, place a 6' rule against each bollard, and read the offset directly?

I did exactly that one year and it is my preferred method - overuled by boss though - wanted coordinates and movement north/south as well as east/west, but I agree with you.

Am I missing something here?

The accuracy of your field work lies in the precision of the angles turned and the distances measured. The "bearing" of the base line has no significance at all.

Vern

I wondered the same thing. From the original post:

"It got me thinking - we are turning very small angles from the baseline to the punchmark on the bollards - it doesn't seem to make sense, but would arbitrarily changing the baseline azimuth to 90 degrees or 45 degrees improve the integrity??"

It seems to me all that would accomplish would be to change the numbers recorded. The angle would still be the same. The calculations would still be based on the angles and distances. If the bearings were rotated some, the angle would still be bearing1-bearing2.

But I may be missing something here.

Extreme acute angles are BAD for triangulations. That is, they are weak geometrically, for computing DISTANCE.

This concept is a carry over, from the NON edm days.

Think Bilby Towers, Winding up angles 10 times. Working at evening, and morning. Turning angles to a lighted target. Acute Angles, Triangulation. NGS. Back when surveyors were still wearing leather lace up boots, and looked like woodsmen!

Nate

Bilby Towers

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