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Old Apr 26th 2017, 06:07 AM   #1
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Need a link to proof that distance is invariant (for a discussion)

Hello!
I need a reliable link to proof someone that distance doesn't depend on a reference frame in Euclidean space.

For example: if there are two points on the Earth the distance is the same no matter if we use parallels and meridians as coordinates or Cartesian coordinates, associated with the centre of the Earth.

I have such links, but in other language. I need them in English, but can't find. Please help me.
Thanks in advance.

Last edited by Fox333; Apr 26th 2017 at 06:14 AM.
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Old Apr 26th 2017, 08:13 AM   #2
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This sounds like a math question, not a physics question. Since you asked this in a physics forum I will point out that under Special Relativity distance is NOT invariant - it depends on the relative velocities of the reference frames. But I suspect that is not what you're interested in, since you mentioned Euclidean space. I suggest you post this question on our sister site: Math Help Forum - Free Math Help Forums
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Old Apr 26th 2017, 08:47 AM   #3
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Originally Posted by Fox333 View Post
Hello!
I need a reliable link to proof someone that distance doesn't depend on a reference frame in Euclidean space.

For example: if there are two points on the Earth the distance is the same no matter if we use parallels and meridians as coordinates or Cartesian coordinates, associated with the centre of the Earth.

I have such links, but in other language. I need them in English, but can't find. Please help me.
Thanks in advance.
I suppose I'm being a bit picky here, but your statement is not correct. If we restrict your comment about "reference frames" to "inertial reference frames" then it holds. I don't have the time now to give you the full proof, but the main concept is to is to have you take your velocity of a trial mass to be $\displaystyle v_1$ ie. $\displaystyle x_1 = v_1 t$ and write it out for the other reference frame to be $\displaystyle x_1' = (v_1 - v_{rel} ) t'$. Now, t' = t since we are talking about a Galilean coordinate transformation. Now you have $\displaystyle x' = x - v_{rel} t'$. You can use this to calculate the position of the endpoints of a length in each coordinate system and you get $\displaystyle \Delta x' = \Delta x$.

-Dan
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Old Apr 26th 2017, 10:50 AM   #4
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Thanks!
I'll post it on the math forum.
My opponent is rather stubborn, so I need trustworthy links from reliable sources to convince him.
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Old Apr 26th 2017, 12:05 PM   #5
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Is this what you want?

I posted it on the maths forum

Hello, is distance invariant in Euclidean space?

Note the above stupid piece of text is a clickable link.

@Dan when will we get the links working properly please?
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Last edited by studiot; Apr 26th 2017 at 12:07 PM.
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Old Apr 26th 2017, 12:11 PM   #6
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@Studiot, yes, that is.
Thanks.
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Old Apr 26th 2017, 03:38 PM   #7
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Originally Posted by studiot View Post
@Dan when will we get the links working properly please?
As soon I can convince mash to come back and do the work. Your guess is as good as mine. However, are you aware that we can color the links ourselves?

-Dan
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