Parallax, angle to observer etc.

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Discussion Overview

The discussion revolves around the geometric relationships between an observer and a long object with evenly spaced marks, exploring whether the angles observed can determine the object's orientation relative to the observer. The conversation also touches on a related scenario involving sonar observations from a submarine, examining the implications of angle measurements in determining a ship's course.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that it should be possible to determine the angle of the long object based on the observed angles, noting specific cases for extreme angles (90 degrees and parallel to the line of sight).
  • Another participant agrees that there is an intrinsic connection between the angles but expresses uncertainty about how to mathematically express this relationship, mentioning the law of cosines as potentially relevant if additional parameters were known.
  • Some participants challenge the initial premise by asserting that it is false, providing a counterexample where the observer is perpendicular to the measuring rod and only three marks are present, suggesting that this configuration allows for a definitive conclusion about the rod's orientation.
  • A later post introduces a submarine sonar scenario, questioning whether angle observations can determine a ship's course, while acknowledging that more data could be available in practice.

Areas of Agreement / Disagreement

Participants express differing views on whether the angles alone can determine the object's orientation, with some asserting it is possible while others provide counterexamples that suggest limitations. The discussion remains unresolved regarding the mathematical formulation and implications of the scenarios presented.

Contextual Notes

Participants note that the problem's resolution may depend on the number of marks on the object and the specific angles observed, indicating potential limitations in the assumptions made.

MathYew
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Hi,

I'm a mathematics analphabet, but still, sometimes I come across a problem, that I just can't keep away from. I hope someone finds this one worth a thought.
Here it is:
An observer is looking at a long object with marks spaced equally along its length (like a measuring rod or a check-board). The distance between the marks (or sizes of colored fields) is not given. So is not the distance from the observer. What is known, are all the angles between the marks from the observers standing point. My question is, is it possible to determine the angle of the "long object" in respect to the observers location given only the mentioned angles?
 
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It should be possible, yes.

A simple test is that you can easily identify the limits
- 90 degrees to line-of-sight. Angles at near and far end will be symmetrical.
- parallel to LoS: all angles are equivalent and equal to 0
so it should be possible to derive a unique answer for any angle in between.
 
I think so, too. It seems obvious, there must be some intrinsic connection between these angles, but how to express it mathematicaly...? Some collegues of mine told me, it could be easily solved (with the law of cosinuses, I think) if one more parameter was given. Like the distance of the line of sight for example. But that's not the answer I'm looking for...
 
As stated, it's false. Place the observer perpendicular to the measuring rod, and put only three hashes on it.
 
zhentil said:
As stated, it's false. Place the observer perpendicular to the measuring rod, and put only three hashes on it.

What do you mean?

If the observer sees three hashes, and the angles formed by the outermost two hashes are equivalent, the observer knows conclusively that the measuring stick is perpendicular to his line of site.

QED.


BTW, who said there is a limit on the number of hash marks? Seems to me, if not stated, we have our choice. (Not that it matters. Three is all you'll need.)
 
Last edited:
Ok, here's the original form of this problem. Maybe it will make it more interesting. =)

A submarine sonar man hears a ship propeller noise at a certain angle from the still submarine (AOB - angle on bow). He makes timed observations of the AOB. Considering that the RPM of the propeller is constant, he can assume the distance traveled by the ship in equal time to be equal, but he can not tell the exact distance from the submarine, nor can he determine it's speed. Can he determine it's course?
In reality, I'm certain, he can get much more data than this, but still... :) Maybe his instruments have been depth-charged. :D
 

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