I had to as-built some rather odd steel today.
One of the requirements was to determine the center point of two pieces which protrude from the top of the steel, to the left of the blue tower in the center. They appear to be flat in the pic but are actually tubes.
I know the diameter of the tubes, but I took the opportunity for a "teaching moment" in the field with one of the crews.
1. In order to determine the diameter of an unknown round object, we can use the approximate formula;
D = theta * d / 206265.
Where D = Diameter of the object,
theta = the angular width of the object in arcseconds, and
d = the distance to the center of the object.
The first difficulty, of course, was determining the angular width of the object. If it was horizontal, we would average angles to each side, likewise vertical. From the pic it's obvious we have neither ideal case.
2. We can determine the angular width of a tilted object by using the Pythagorean theorem.
Merely consider that any unit entered into the formula (feet, meters, smoots, arcseconds) will be returned in the answer.
So, we split the horizontal and vertical angles, considered each a component of a right triangle, and determined the hypotenuse in arcseconds. This also allowed us to get the distance shot on center.
Our next problem was that we had measured the distance to the face of the round object and not the center.
But using the formula anyway we generated a preliminary diameter.
3. By adding half the diameter to the horizontal distance we were able to fine tune the results through an iterative process, repeatedly adding the new radius generated from the formula result to the original horizontal distance until the answer remained stable.
Now that works reasonably well for small objects at a distance. However, imagine a large storage tank at a short distance. Our angle would not be tangent to the circumference at the extreme edges of the object. We would be measuring, in effect, a sector of the circle rather than the total width of the object.
So, is there a better method?
P.S. Lets not get distracted. The pic does not represent the instrument station and this wasn't the method of as-builting the object.
Three point triangulation and the same method used in subtense bars would seem to be more definitive and less iterative. However, it may require longer time regarding multiple occupations and with reflectorless, it may be moot, but should return the same answer if done properly.
This is a very interesting project. I believe your approach is correct, the only problem is that it is subject to an array of random errors, and systematic errors that could be difficult to predict. I like your reasoning, however, what instrument will you use to prove it? I would use in this particular case a laser scanner. You can model and slice the point clouds representing the structural members using different reference planes to prove your measurements against a plumb line set on site for control. Now, that's assuming the structure is stable enough. I would still measure & calculate in the field a small portion (say 10%) to backup the accuracy of measurements you get with the scan data. Am I going off on a tangent? Sorry if I did!
I can't add anything to the method for determining the angle. But the formula I've used is:
R = (sin A * d) / (1 - sin A)
R radius
A 1/2 angle
D measured horizontal distance to face
Based on:
Sin A = r/(r+d)
This looks like a perfect application of a laser scanner. Plus your QC of checking the measurements with a plumb line is very interesting. Primitive tech meets high tech.
The important thin is that there must be some type of QC on the taking of the ma=measurements.
You could also convert the two shots to coordinates and average the x,y, & z's to get the center point, but that would not teach them about Pythagoras.
I don't know if this is a better way or not.
Shoot 3 points. Perpendicular bisectors between them will pass through the radius point. A three point curve solution basically. Requires use of right triangles via inverse and lat. dep., and use of law of sines. But I think you would need distance to points shot rather than only angles between them. Might not be an option without reflectorless edm.
I'm not exactly sure what I'm looking at, but if this is a new structure couldn't you get the fabrication plans for it?
> But I think you would need distance to points shot rather than only angles between them. .
I was thinking the same thing. i.e. the angle between the left edge of the cylinder and the center of the cylinder, would not be the same as the angle between the right edge of the cylinder and the center. (Unless you assumed that the diameter of the circle is much smaller than the distance between the transit and the cylinder.)
as-builts
> I'm not exactly sure what I'm looking at, but if this is a new structure couldn't you get the fabrication plans for it?
Just because the plans say something, don't make it so. Plus, it's for edumacation.
as-builts
I'd be very surprised if something shaped like that wasn't constructed with some pretty close tolerences...
Of course, I am surprised by almost everything I encounter these days!
Is it true theoretically that a circle viewed at any angle other than perpendicular to its center will become an ellipse, and that the long axis of the ellipse is equal to the diameter of the circle?
I don't understand the application of the Pythagorean theorem in step 2. Can someone help me out?
By "splitting the angles" in H and V, do you mean that you take the difference between the angle to the upper edge and the angle to the lower edge and this is the V leg of the right triangle, and similarly the difference between left and right edges for the H leg of the right riangle?
Keep thinking about this one. A very interesting theoretical exercise. If I understand your method in step 2, it would tend to over-estimate the angular width of the object. If the tubing is circular in cross section, I believe that it will appear as an ellipse at any angle other than perpendicular to its center. If the ellipse is tilted, the vertically "highest" point of the ellipse does not occur at the end of its major axis, also similarly the horizontal. Whereas, you are looking for the angular width of the object, which would occur along the major axis, no?