Question about the angle between two targets as distance increases....

[Odal]

Distance and Optical Illusions
Here is a problem which has probably a very simple solution even though it somehow eludes me.
Imagine two targets, just like those used for shooting practice or darts, on a horizontal line, but a few meters apart.
Let us say that you can hold your rifle aimed straight at the bull's eye, at a perpendicular angle relative to one target at a time, from two different positions.
You now go stand much farther away from the targets, in a straight line from your initial position. My question is, will you still have to hold your rifle at the same angle, or will you need to adjust it, making it smaller the farther you go?
If that is the case, it would mean that the objects are indeed getting smaller with the distance... Or that you only think that you are changing the angle of your aim.
Which is it?

 

[A.T.]

You now go stand much farther away from the targets, in a straight line from your initial position. My question is, will you still have to hold your rifle at the same angle, or will you need to adjust it, making it smaller the farther you go?

The farther you are away, the less you have to rotate the gun between the targets. Is that what you trying to ask?

 

[Odal]

The farther you are away, the less you have to rotate the gun between the targets. Is that what you trying to ask?

Not exactly. Let me try the analogy of a photograph depicting train rails disappearing in the distance. On the foreground, there will be a distance between the rails, on the photograph, of say 2 cm. At the upper side of the photograph, the distance will be much less. The points have somehow shifted. If instead of rails you use targets, and you are standing just outside the photograph, the angle of aim will be different when you aim at the close targets than when you aim at the targets farther away, even if they are on the same line (two by two).
It that is the case, then it would seem that the shift is real since we have to change the angle of shot or aim.

edit:
If the the change of position as witnessed on the photograph is not real, we could keep the same angle whatever the distance with the target. I don't know if that is indeed the case.

 

[A.T.]

the angle of aim

What angle is that exactly?

 

[Odal]

What angle is that exactly?

Consider the (vertical) legs of the triangle and the rectangle as representing that angle.

 

[Bandersnatch]

I think I can speak for everybody by remarking that your description is confusing (e.g. a triangle does not have 'vertical legs')

Perhaps it would be best if you updated your picture by marking on it:
- the physical position(s) of the shooter(s) initially, and after moving further away (say, S1, S2, and S1', S2', or however you mean it)
- position of the targets (T1 and T2)
- the angle(s) you're concerned with (α1, α2)

(btw, in what sense is this a physics graduate-level question?)

 

[Odal]

I think I can speak for everybody by remarking that your description is confusing (e.g. a triangle does not have 'vertical legs')

Perhaps it would be best if you updated your picture by marking on it:
- the physical position(s) of the shooter(s) initially, and after moving further away (say, S1, S2, and S1', S2', or however you mean it)
- position of the targets (T1 and T2)
- the angle(s) you're concerned with (α1, α2)

(btw, in what sense is this a physics graduate-level question?)

Maybe the use of two targets is indeed confusing. In my mind it represented better the figures of a triangle and a rectangle. And no, triangles have no vertical legs, but I think the meaning was clear enough.
Imagine aiming at a target in a straight perpendicular line relative to the target, and then getting farther and farther away from the target, without deviating from the straight line. Will you need to adjust your aim or not?
Concerning the question whether it is a graduate level matter, I will not mind if a moderator changes the prefix to better comply with the forum criteria.

 

[jbriggs444]

Maybe the use of two targets is indeed confusing. In my mind it represented better the figures of a triangle and a rectangle. And no, triangles have no vertical legs, but I think the meaning was clear enough.
Imagine aiming at a target in a straight perpendicular line relative to the target, and then getting farther and farther away from the target, without deviating from the straight line. Will you need to adjust your aim or not?

Imagine standing in the center of the road bed of a set of railroad tracks. U.S. standard gauge, 4 feet 8.5 inches apart. You aim at the left rail, 6 feet ahead. Then you aim at the right rail. You will have shifted your aim between the two targets by some angle. You can use trigonometry to calculate this angle.

Now aim at a point 500 feet ahead on the left rail. Then the right. You will have shifted your aim by another angle. You can use trigonometry to calculate this angle.

Edit: Now, imagine a pinhole camera taking a photograph of the rails extending off into the distance. Imagine that the image plane is vertical and at right angles to the tracks. What shape is traced out on the image plane by the tracks?

 

[Bandersnatch]

Imagine aiming at a target in a straight perpendicular line relative to the target, and then getting farther and farther away from the target, without deviating from the straight line. Will you need to adjust your aim or not?

No.

I think you are confused by the fact that parallel lines tend to converge in the distance (i.e.perspective). Note that when you see that effect, e.g. with railway tracks, you're not standing on either track, but somewhere away from both (e.g. in-between). If you do stand on one of the tracks, or - as per your example - aim along a straight line perpendicular to a target - then that track or line keeps on going on straight, regardless of how far on it you're located.
A track or line next to you will appear to converge in the distance, but then again, you're not on that line - you're not aiming along it towards the target. A shooter on the line next to you does not see his target line as converging (only yours), so he doesn't have to adjust his aim.

A physical representation, which is what matter as far as aiming is concerned, has two parallel lines stay parallel forever (on a flat plane). The shooters on either line do not have to adjust their aim. As such, the picture you drew is not a correct representation of what you wanted to depict. It would be, it there was a third shooter at the apex of the triangle, in-between the two other shooters. The third shooter is the one who sees the lines of aiming of his neighbours converge, but he himself does not aim perpendicular to either target.It's too bad you did not draw the picture as suggested, with marking each shooter and each angle - just doing that would likely dispell the confusion.

 

[Odal]

Thank you both for your feedback. I would like to add the following question:
When we are looking at a distant object, say the moon, and we aim our telescope at a point , say on the left side of it, on the outer curve as it were. Are we aiming at a point in space as it appears to us, or at a point in space, with the real dimensions of the moon?
I understand your explanations, they are very familiar to me, but they do not solve my puzzlement concerning the relationship between perspective and space.

 

[Bandersnatch]

Are we aiming at a point in space as it appears to us, or at a point in space, with the real dimensions of the moon?

There's no difference between those two - disregarding atmospheric diffraction, etc.

 

[Odal]

There's no difference between those two - disregarding atmospheric diffraction, etc.

That is what I find disconcerting.
Imagine two astronomers on a space station close to the moon, each one aiming at respectively the left and right extremity of the diameter. The space station moves in a straight line away from the moon. Do the astronomers have to adjust the position of their telescopes? If they do, then somehow the moon has shrunk.

 

[Bandersnatch]

That is what I find disconcerting.
Imagine two astronomers on a space station close to the moon, each one aiming at respectively the left and right extremity of the diameter. The space station moves in a straight line away from the moon. Do the astronomers have to adjust the position of their telescopes? If they do, then somehow the moon has shrunk.

O.k., that's a different situation than before.
In particular, when you have one astronomer looking at one point on the Moon, you can then imagine the astronaut receding backwards along his line of sight, without having to adjust his telescope's aim.
But, with two astronomers 'attached' to a rigid station, you can not make both recede along their individual line of sights. If you did that, they would have to physically move apart from each other. By making them both sit on the space station, you're introducing a new condition, which results in them having to change their lines of sight to keep pointing at the original points on the Moon.
Again, just to hammer this point down - these two astronomers can not keep looking along their original lines of sight, because they're stuck on the station. They must look at the Moon from different directions than before.

 

[Odal]

O.k., that's a different situation than before.
In particular, when you have one astronomer looking at one point on the Moon, you can then imagine the astronaut receding backwards along his line of sight, without having to adjust his telescope's aim.
But, with two astronomers 'attached' to a rigid station, you can not make both recede along their individual line of sights. If you did that, they would have to physically move apart from each other. By making them both sit on the space station, you're introducing a new condition, which results in them having to change their lines of sight to keep pointing at the original points on the Moon.
Again, just to hammer this point down - these two astronomers can not keep looking along their original lines of sight, because they're stuck on the station. They must look at the Moon from different directions than before.

Would it make any difference if there were only one astronomer, aiming at the same point at any time?
Note that we are trying not only to describe perspective, but also to understand it. How does space and our perception thereof relate?

 

[Bandersnatch]

The whole idea is that if you move the vantage point away from the original line of sight, you have to adjust your aim. If you don't do that, then you don't change the aim.

How does space and our vision thereof relate?

Best I can answer the way the question is stated is: by trigonometry.

 

[sophiecentaur]

I think you are confused by the fact that parallel lines appear to tend to converge in the distance (i.e.perspective).

You have to insert the words "appear to" in that sentence. Parallel lines are - well- parallel.
It seems to me that the OP is all about perspective and not to do with Physics. I started thinking in terms of bullets dropping on their path to a target but this is not relevant to the OP - I think.

Perspective is a funny thing. For instance, the perspective of a photograph of a scene, taken with a wide angle lens is exactly the same as the perspective of the same scene taken from the same place with a telephoto lens. I am assuming that neither lens actually introduces any distortion - which is valid - and a cropped and enlarged part of the wide angle picture will look the same as the telephoto picture. This is counter intuitive. There is a well known trope, used in cinema where the camera moves backward as the zoom length is increased, keeping the main subject of the shot the same size. It is powerful and disturbing to watch the perspective changing - not something we tend to see in real life. Perspective only changes when the observer moves relative to a scene.

You don't need to consider the Moon for this - it only makes it harder to set up the experiment. Just consider two lasers, lined up parallel so that their beams are a meter apart at ten metres and at 50 metres. Then you move a screen further and further away. The spots will still be one metre apart, even when the spots have faded and gone fuzzy due to the atmosphere. From the viewpoint of the guy with the laser, the spots will appear to get closer and closer (angle subtended at the eye) although their actual separation hasn't changed.

 

[Odal]

I have to agree with that but the rules of the game were not specified in the OP; I read it as a practical situation. If you want to take things beyond Euclid then fair enough.

I do have to concede my own confusion which I am trying to clarify with this discussion. I find indeed space and perspective very confusing and I think that there is much more to say about the subject that what geometry and trigonometry can tell us. I am afraid that all I can do is express my confusion and hope that clarity will be attained with the help of feedback from others.

 

[berkeman]

<< Edited by Mark44 >>
Since the OP apparently wanted to head down a philosophical path (in posts now deleted), this thread will remain closed. Thank you to everybody who has contributed to try to help the OP's understanding of the science and math involved.