Relativity or something [Split from "Pressures"]

[eran_rathan]

Okay, I know see that I can avoid asking that initial question and get more directly to proof.

The naming of points and associating that name with their system.

transformation takes the selected points in one system and places them into the opposite system.


Sooo, given this depiction of a point P and Q...what would be the a proper coordinate transformation between systems....

P in Red...transforms to what in what ?
Q in Blue...transforms to what in what ?

[ my error, you can forget that i asked the "motion" question...so no answer requested ].
So, just this one diagram...just this set of two questions, please.

Taking solely that diagram (and not the previous one):

Q_Blue (2,0,0) => Q_Red (5,0,0)
P_Red (2,0,0) => P_Blue (-1,0,0)

The transformation between the systems can be expressed as
Blue = Red +(3,0,0)

[J Thomas]

this is a simple math question

This is not about physics, nor relativity, transformations, time, space, observers, dilation, etc, etc

given coincident red and blue Cartesian coordinate system with a selected point at (2,0,0) in each

If we move the Blue system,

does selected point Q move with the Blue system?

or does selected point Q in Blue stay right there, opposite of point P in Red
as the Blue system is "relocated" underneath point Q in Blue?

I would hugely appreciate any opinion...without talking about non-math stuff, would be ever more so desired.

Q and P should be coincident, so even if you redefine Blue to another origin and rotation (assuming it is scale-invariant), Q and P will remain coincident.

I don't see that in the math. You could use the same expression to describe a new coordinate system, or to describe something that moved. x' = x+3 .

I'm not clear there's anything in the math to say whether two things are coincident or not. In physics that's a real distinction. Two things which are together, are together whether or not you use two different coordinate systems to describe them, while two things that are 3 units apart will be 3 units apart in both coordinate systems.

But the only reason I see to say that his P and Q are together is that he drew two graphs and the graphs overlapped to make it look like they were together.

[eran_rathan]

given coincident red and blue Cartesian coordinate system with a selected point at (2,0,0) in each

I don't see that in the math. You could use the same expression to describe a new coordinate system, or to describe something that moved. x' = x+3 .

I'm not clear there's anything in the math to say whether two things are coincident or not. In physics that's a real distinction. Two things which are together, are together whether or not you use two different coordinate systems to describe them, while two things that are 3 units apart will be 3 units apart in both coordinate systems.

But the only reason I see to say that his P and Q are together is that he drew two graphs and the graphs overlapped to make it look like they were together.

Emphasis mine.

[steve waterman]

Okay, I know see that I can avoid asking that initial question and get more directly to proof.

The naming of points and associating that name with their system.

transformation takes the selected points in one system and places them into the opposite system.


Sooo, given this depiction of a point P and Q...what would be the a proper coordinate transformation between systems....

P in Red...transforms to what in what ?
Q in Blue...transforms to what in what ?

[ my error, you can forget that i asked the "motion" question...so no answer requested ].
So, just this one diagram...just this set of two questions, please.

Taking solely that diagram (and not the previous one):

Q_Blue (2,0,0) => Q_Red (5,0,0)
Q_Blue (2,0,0) => P_Blue (-1,0,0)

SW - This is real close, and wonderful that you have used the system names...this is real real good...
...do you agree that ...instead, this would be the transformation of our given depiction ?
Q_Blue (2,0,0) => Q_Red (5,0,0)
P_Red (2,0,0) => P_Blue (-1,0,0)

[J Thomas]

given coincident red and blue Cartesian coordinate system with a selected point at (2,0,0) in each

I don't see that in the math. You could use the same expression to describe a new coordinate system, or to describe something that moved. x' = x+3 .

I'm not clear there's anything in the math to say whether two things are coincident or not. In physics that's a real distinction. Two things which are together, are together whether or not you use two different coordinate systems to describe them, while two things that are 3 units apart will be 3 units apart in both coordinate systems.

But the only reason I see to say that his P and Q are together is that he drew two graphs and the graphs overlapped to make it look like they were together.

Emphasis mine.

OK! I missed that. I was wrong.

[eran_rathan]

Taking solely that diagram (and not the previous one):

Q_Blue (2,0,0) => Q_Red (5,0,0)
Q_Blue (2,0,0) => P_Blue (-1,0,0)

SW - This is real close, and wonderful that you have used the system names...this is real real good...
...do you agree that ...instead, this would be the transformation of our given depiction ?
Q_Blue (2,0,0) => Q_Red (5,0,0)
P_Red (2,0,0) => P_Blue (-1,0,0)

/facepalm - that's what i get for copy-pasting quickly.

[steve waterman]

Taking solely that diagram (and not the previous one):

Q_Blue (2,0,0) => Q_Red (5,0,0)
Q_Blue (2,0,0) => P_Blue (-1,0,0)

SW - This is real close, and wonderful that you have used the system names...this is real real good...
...do you agree that ...instead, this would be the transformation of our given depiction ?
Q_Blue (2,0,0) => Q_Red (5,0,0)
P_Red (2,0,0) => P_Blue (-1,0,0)

/facepalm - that's what i get for copy-pasting quickly.

Been known to make a typo or two myself....so, do you agree that...?
Q_Blue (2,0,0) => Q_Red (5,0,0)
P_Red (2,0,0) => P_Blue (-1,0,0)

[gmalivuk]

Yes, that's fine, as several people have agreed before.

How does it lead to mistakes in the math for relativity?

[steve waterman]

Yes, that's fine, as several people have agreed before.

SW - kinda a tad unclear...what is fine, please...to what do you refer ?

How does it lead to mistakes in the math for relativity?

SW - Almost there dude, just a little more patience, and I will have finished my short proof...

do you agree that ...given the one depiction...the transformation should be ?
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)

[eran_rathan]

Yes, that's fine, as several people have agreed before.

SW - kinda a tad unclear...what is fine, please...to what do you refer ?

How does it lead to mistakes in the math for relativity?

SW - Almost there dude, just a little more patience, and I will have finished my short proof...

do you agree that ...given the one depiction...the transformation should be ?
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)

To reiterate, yes, the transformation from Blue to Red is given as

Blue = Red + (3,0,0)

[steve waterman]

Yes, that's fine, as several people have agreed before.

SW - kinda a tad unclear...what is fine, please...to what do you refer ?

How does it lead to mistakes in the math for relativity?

SW - Almost there dude, just a little more patience, and I will have finished my short proof...

SW - do you agree that ...given the one depiction...the transformation should be ?
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)

To reiterate, yes,

SW - so therefore you agree, according to the above, that QBlue (2,0,0) = PRed (2,0,0) ?

[eran_rathan]

SW - do you agree that ...given the one depiction...the transformation should be ?
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)

To reiterate, yes,

SW - so therefore you agree, according to the above, that QBlue (2,0,0) = PRed (2,0,0) ?

Absolutely not.

You are comparing apples and kumquats. The only valid description is between points on the same coordinate system. Otherwise the notation is meaningless.

Q_red (5,0,0) != P_red (2,0,0)

If you think so, you do not understand coordinate systems.

[steve waterman]

SW - do you agree that ...given the one depiction...the transformation should be ?
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)

To reiterate, yes,

SW - so therefore you agree, according to the above, that QBlue (2,0,0) = PRed (2,0,0) ?

Absolutely not.

You are comparing apples and kumquats. The only valid description is between points on the same coordinate system. Otherwise the notation is meaningless.

Q_red (5,0,0) != P_red (2,0,0)

SW - I have never said this statement, I am unclear even if you are saying so,
.
What you are missing is that ..IN THEIR OWN SYSTEM, Red = Blue...that is, (2,3,4) in Red has the same relative distance (2,3,4) in Blue,
FOREVER...this NEVER changes...no matter what gets moved or not.

let me get your official stance, please ...is this your position correctly stated?
you agree...
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)
yet say, that, the following is absolutely false....
P_Red (2,0,0) = Q_Blue (2,0,0)

Hmmm, if we have two empty systems, we agree that the coordination goes along for the ride...
and it would appear apparent that Red(x,y,z) = Blue (x',y',z')... agreed ?

[Radical_Initiator]

I have never said the above.

let me get your stance, please ...is this your position?
you agree...
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)
yet say, that, the following is absolutely false....
Q_red (5,0,0) != P_red (2,0,0)

Hmmm, if we have two empty systems, we agree that the coordination goes along for the ride...
and it would appear apparent that Red(x,y,z) = Blue (x',y',z')... agreed ?

I think I know what eran_rathan will say to the last point, but perhaps this is a good time for a question: what does it mean for two coordinate systems to be equal (equivalent)?

[eran_rathan]

let me get your stance, please ...is this your position?
you agree...
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)
yet say, that, the following is absolutely false....
Q_red (5,0,0) != P_red (2,0,0)

Hmmm, if we have two empty systems, we agree that the coordination goes along for the ride...
and it would appear apparent that Red(x,y,z) = Blue (x',y',z')... agreed ?

EDIT: if we are assuming the continuing use of Red & blue as previously defined, than sure,

Red(x,y,z) = Blue(x',y',z'), with the transformation of Blue = Red + (3,0,0) (and no rotation or scale factors).

[steve waterman]

let me get your stance, please ...is this your position?
you agree...
QBlue (2,0,0) => QRed (5,0,0)
PRed (2,0,0) => PBlue (-1,0,0)
yet say, that, the following is absolutely false....
Q_red (5,0,0) != P_red (2,0,0)

Hmmm, if we have two empty systems, we agree that the coordination goes along for the ride...
and it would appear apparent that Red(x,y,z) = Blue (x',y',z')... agreed ?

EDIT: if we are assuming the continuing use of Red & blue as previously defined, than sure,

thank you again, for the nice direct clear answer.

Red(x,y,z) = Blue(x',y',z'), with the transformation of Blue = Red + (3,0,0) (and no rotation or scale factors).

i am saying that the galilean would be saying,
Red(x,y,z) = Blue(x',y',z') - vt
I am saying that Red(x,y,z) = Blue(x',y',z')
(you just ageeed to that directly above)

obviously, both cannot be true...can they.

[gmalivuk]

Red(x,y,z) = Blue(x',y',z'), with the transformation of Blue = Red + (3,0,0) (and no rotation or scale factors).

i am saying that the galilean would be saying,
Red(x,y,z) = Blue(x',y',z') - vt
I am saying that Red(x,y,z) = Blue(x',y',z')
(you just ageeed to that directly above)

obviously, both cannot be true...can they.

Yes, they can, because as eran_rathan points out Red(x,y,z) = Blue(x',y',z') with the transformation Blue = Red + (3,0,0). And if instead of static we're moving, that (3,0,0) changes to (v_xt, v_yt, v_zt) = vt

(And whether you're adding or subtracting vt depends only on which direction you're taking the transformation.)

[eran_rathan]

Steve -

you apparently edited some things after I responded, let me respond now to those changes:

What you are missing is that ..IN THEIR OWN SYSTEM, Red = Blue...that is, (2,3,4) in Red has the same relative distance (2,3,4) in Blue,
FOREVER...this NEVER changes...no matter what gets moved or not.

Yes, the distance between the same coordinate pairs in the two systems as previously defined will always be (|3|,0,0) (+ or - being which direction you are going, from Red to Blue or Blue to Red).

vt is essentially a scale factor between the coordinate systems, because if we are in relative motion, the transformation changes from

Blue (x,y,z) = Red + (3,0,0) to
Blue (x_t,y_t,z_t) = Red + (v_xt,v_yt,v_zt)

dammit, ninja'd

[steve waterman]

Steve -

you apparently edited some things after I responded, let me respond now to those changes:

What you are missing is that ..IN THEIR OWN SYSTEM, Red = Blue...that is, (2,3,4) in Red has the same relative distance (2,3,4) in Blue,
FOREVER...this NEVER changes...no matter what gets moved or not.

Yes, the distance between the same coordinate pairs in the two systems as previously defined will always be (|3|,0,0) (+ or - being which direction you are going, from Red to Blue or Blue to Red).

agreed

vt is essentially a scale factor between the coordinate systems, because if we are in relative motion, the transformation changes from

agreed

Blue (x,y,z) = Red + (3,0,0) to
Blue (x_t,y_t,z_t) = Red + (v_xt,v_yt,v_zt)

dammit, ninja'd

Again...you are the point...yes yes...there is relationship of +/- 3 between P in red to the transformed P in blue
yes the is a relationship of there is relationship of +/- 3 between Q in red to the transformed Q in blue

i get that...i am ONLY talking about, the coordinate values in relationship to their own system...not between the two systems, you are focused there,
that has zero to do with MY point....if P in Red (2,0,0) = Q in Blue (2,0,0), then it is not rocket science to accept that
Red (2,0,0) = Q in Blue (2,0,0) +/- whatever x value you like...therefore, logically, cannot be true.

[gmalivuk]

But the vt term means one of those coordinate systems continues to move. It can be plus or minus whatever value you pick, because the value you pick replaces the 3-unit shift you have in your pictures.

[eran_rathan]

Again...you are the point...yes yes...there is relationship of +/- 3 between P in red to the transformed P in blue
yes the is a relationship of there is relationship of +/- 3 between Q in red to the transformed Q in blue

i get that...i am ONLY talking about, the coordinate values in relationship to their own system...not between the two systems, you are focused there,
that has zero to do with MY point....if P in Red (2,0,0) = Q in Blue (2,0,0), then it is not rocket science to accept that
Red (2,0,0) = Q in Blue (2,0,0) +/- whatever x value you like...therefore, logically, cannot be true.

I am focused on the difference between the two systems because it is where your ENTIRE POINT breaks down.

Axiom 1: Blue = Red + (C_x,0,0) (for a static solution)
Axiom 2: Q_Blue(x,y,z) = P_Red (x,y,z)
conclusion: Q_Blue=P_Red is false
conclusion2: Q_Blue = Q_Red = P_Red (x+C_x,y,z) true

Alternate scenario, for moving coordinate systems:
Axiom 1: Blue = Red + (V_xt,V_yt,V_zt)
Axiom 2: Q_Blue(x_t,y_t,z_t) = P_Red (x,y,z)*vt
conclusion: Q_Blue=P_Red is false except at t=0

[steve waterman]

But the vt term means one of those coordinate systems continues to move. It can be plus or minus whatever value you pick, because the value you pick replaces the 3-unit shift you have in your pictures.

My take is actually x' = x-d ... d being the distance moved along x. so, my MATH example has no time nor velocity component...merely a distance along x.

[J Thomas]

i am saying that the galilean would be saying,
Red(x,y,z) = Blue(x',y',z') - vt
I am saying that Red(x,y,z) = Blue(x',y',z')
(you just ageeed to that directly above)

obviously, both cannot be true...can they.

Chiming in with everybody else, when v=0 or t=0 then x'=x-vt gives us x'=x and you're right.

When v and t are not zero then the two coordinate systems don't put things in the same place after all.

[steve waterman]

Again...you are the point...yes yes...there is relationship of +/- 3 between P in red to the transformed P in blue
yes the is a relationship of there is relationship of +/- 3 between Q in red to the transformed Q in blue

i get that...i am ONLY talking about, the coordinate values in relationship to their own system...not between the two systems, you are focused there,
that has zero to do with MY point....if P in Red (2,0,0) = Q in Blue (2,0,0), then it is not rocket science to accept that
Red (2,0,0) = Q in Blue (2,0,0) +/- whatever x value you like...therefore, logically, cannot be true.

I am focused on the difference between the two systems because it is where your ENTIRE POINT breaks down.

SW - the difference between the two systems is...with d (distance along x) replacing vt... so no velocity nor time be involved.
P in Red (2,0,0) transforms to [P in Blue (2.0.0) -d (3,0,0)] ...P in Red (2,0,0) transforms to P in Blue as (1,0,0) agree ?
Q in Blue (2,0,0) transforms to [Q in Red (2.0.0) +d (3,0,0)] ...Q in Blue (2,0,0) transforms to Q in Red as (5,0,0) agree ?
R in Red (2,0,0) = Q in Blue (2,0,0) agree ?
Also, when d replaces vt...we are now in the exclusive world of math.
I will not comment upon your following characterization, as I obviously disagree, with some items below, by virtue of what is stated directly above

Axiom 1: Blue = Red + (C_x,0,0) (for a static solution)
Axiom 2: Q_Blue(x,y,z) = P_Red (x,y,z)
conclusion: Q_Blue=P_Red is false
conclusion2: Q_Blue = Q_Red = P_Red (x+C_x,y,z) true

Alternate scenario, for moving coordinate systems:
Axiom 1: Blue = Red + (V_xt,V_yt,V_zt)
Axiom 2: Q_Blue(x_t,y_t,z_t) = P_Red (x,y,z)*vt
conclusion: Q_Blue=P_Red is false except at t=0

[JudeMorrigan]

P in Red (2,0,0) transforms to [P in Blue (2.0.0) -d (3,0,0)] ...P in Red (2,0,0) transforms to P in Blue as (1,0,0) agree ?
Q in Blue (2,0,0) transforms to [Q in Red (2.0.0) +d (3,0,0)] ...Q in Blue (2,0,0) transforms to Q in Red as (5,0,0) agree ?
R in Red (2,0,0) = Q in Blue (2,0,0) agree ?
Also, when d replaces vt...we are now in the exclusive world of math.

Three points:

Firstly, if you can't manage your quote tags properly, I wish you wouldn't try to do inline quoting. It's making it far more difficult to parse out who's saying what in your posts than is necessary. Bottom posting would make everyone's life easier, imo.

Secondly, I'm having a hard time seeing how your third statement follows from the first two. Could you please elucidate on the logic you're using to derive the third equation?

Lastly, I don't understand how using vt removes us from the realm of math.

[steve waterman]

P in Red (2,0,0) transforms to [P in Blue (2.0.0) -d (3,0,0)] ...P in Red (2,0,0) transforms to P in Blue as (-1,0,0) agree ?
Q in Blue (2,0,0) transforms to [Q in Red (2.0.0) +d (3,0,0)] ...Q in Blue (2,0,0) transforms to Q in Red as (5,0,0) agree ?
P in Red (2,0,0) = Q in Blue (2,0,0) agree ?
Also, when d replaces vt...we are now in the exclusive world of math.

Three points:

Firstly, if you can't manage your quote tags properly, I wish you wouldn't try to do inline quoting. It's making it far more difficult to parse out who's saying what in your posts than is necessary. Bottom posting would make everyone's life easier, imo.

Secondly, I'm having a hard time seeing how your third statement follows from the first two. Could you please elucidate on the logic you're using to derive the third equation?

Lastly, I don't understand how using vt removes us from the realm of math.

1 oops...i will bottom post, as you suggest.

2 The third statement, well, is not derived from the first two transformation statements. We are given coincident systems Red and Blue...relocate one
and is still remains the same....P in Red (2,0,0) = Q in Blue (2,0,0) agree ?

3 "using vt removes us from the realm of math"...No, I am saying that NOT using vt, as in x' = x- d...removes/replaces their Physics equation, x' = x-vt, and now x] = x-d, only has math components.

.do you agree with ...as is true in this depiction...


P in Red (2,0,0) transforms to [P in Blue (2.0.0) -d (3,0,0)] ...P in Red (2,0,0) transforms to P in Blue as (-1,0,0)
Q in Blue (2,0,0) transforms to [Q in Red (2.0.0) +d (3,0,0)] Q in Blue (2,0,0) transforms to Q in Red as (5,0,0)
P in Red (2,0,0) = Q in Blue (2,0,0)

[JudeMorrigan]

Ok, that diagram actually helped me understand what you were saying with your third equality. It seems like a bit of an odd way to define the coordinate transformation (which is essentially what it does, as far as I can tell), but it doesn't seem to be fundamentally wrong. (Although I have a sneaking suspicion that the reason you're setting it up that way is because of your odd belief that a point in coordinate system moves when you translate the coordinate system. And that is wrong.)

[Pfhorrest]

I think a way of clarifying this for Steve may be thus:

There is only one point P, and only one point Q. P and Q are not different points in different coordinate systems; they have different coordinates in different coordinate systems. Furthermore, if this is the initial state of affairs:



P = Q, so there is only one point under discussion here at all.

In that diagram, P (aka Q) is at the same position relative to Red Origin and Blue Origin. For the sake of natural language, let us call +x "to the right of", +y "above", and +z "behind", since that's how they are oriented relative to the camera this photos was taken from and in the usual way of drawing coordinate systems. In those terms, P (aka Q) is two units to the right of both Red Origin and Blue Origin, and neither above of below, in front of or behind, either. Relative to Red Origin and Blue Origin, it thus has the same coordinates, (2,0,0).

If we move Blue Origin three units to its (and Red Origin's) right, this does not happen:



P = Q, and P (aka Q) has not moved just because Blue Origin moved. Q (aka P) is still right where it was before, two units to the right of Red Origin; which is now one unit to the left of Blue Origin. Q is not "in Blue", nor is P "in Red"; they are they same point, and it is wherever it is, regardless of where the origins of some coordinate systems are, and how we move those coordinate systems.

This is not to say that there is an absolute reference frame independent of coordinate systems either, mind you. Absent some external frame of reference, Blue Origin moving three units to its right is equivalent to P (aka Q) and Red Origin both moving three units to the left. It just means that points are equally "in" every coordinate system, since being "in" a coordinate system is just having some spatial relation to the origin of that coordinate system, and if these coordinate systems are all laid over the same space then every point in that space will have an (equally valid) spatial relation to the origin of all these coordinate systems. That is essentially what relativity says: there is no "true" coordinate system, they are all just laid over whatever is actually there, picking something arbitrary in the space (the origin) and referring to everything else by its relation to that thing. Anything else can be picked and things referred to by their relation to it, and in the end the whole network of relations will flesh out to be the same either way regardless of which node you choose to start from.

What does happen when you move Blue Origin three units to its right is that the point two units to the right of blue origin is now a different point that it was before; it is no longer P (aka Q). That point (P, aka Q) is now one unit to the left of Blue Origin; some other point, call it Q' if you like, is now at (2,0,0) in Blue, and is (and always has been) at (5,0,0) in Red; and used to be at (5,0,0) in Blue, too, before we moved Blue Origin.

[steve waterman]

Ok, that diagram actually helped me understand what you were saying with your third equality. It seems like a bit of an odd way to define the coordinate transformation (which is essentially what it does, as far as I can tell), but it doesn't seem to be fundamentally wrong.
(Although I have a sneaking suspicion that the reason you're setting it up that way is because of your odd belief that a point in coordinate system moves when you translate the coordinate system. And that is wrong.)

Those 3 equations, which included P in Red = Q in Blue...are what I suggest the Gailean transformation math SHOULD be...so, it is not a surprise to hear you say it looks okay.

Now. if you could please take a look at what the Galilean ACTUALLY does...given this identical scenario;
Red and blue are coincident, Red has a selected point P at (2,0,0) and Blue has a selected point Q at (2,0,0)...

If Blue right 3, then P in Red transforms to Q in Blue...so, P in Red =(2,0,0) and Q in Blue ( now transformed) = (-1,0,0)

whereas, if Red left 3, then Q in Blue transforms to P in Red...so, Q in Blue =(2,0,0) and P in Red ( now transformed)= (5,0,0)

That is, the Galilean transformation results of either going Red left 3...do not equal their results of Blue right 3.

In contrast, my suggested transformation process. ..yields identical results, just as it should/must/is mandated.

Here is a relevant page that shows two depictions for this...the final two on that page...
http://www.watermanpolyhedron.com/gg2012.html

Thanks for giving/sharing your math opinion.

[steve waterman]

I think a way of clarifying this for Steve may be thus:

There is only one point P, and only one point Q. P and Q are not different points in different coordinate systems; they have different coordinates in different coordinate systems. Furthermore, if this is the initial state of affairs:



P = Q, so there is only one point under discussion here at all.

In that diagram, P (aka Q) is at the same position relative to Red Origin and Blue Origin. For the sake of natural language, let us call +x "to the right of", +y "above", and +z "behind", since that's how they are oriented relative to the camera this photos was taken from and in the usual way of drawing coordinate systems. In those terms, P (aka Q) is two units to the right of both Red Origin and Blue Origin, and neither above of below, in front of or behind, either. Relative to Red Origin and Blue Origin, it thus has the same coordinates, (2,0,0).

If we move Blue Origin three units to its (and Red Origin's) right, this does not happen:



P = Q, and P (aka Q) has not moved just because Blue Origin moved. Q (aka P) is still right where it was before, two units to the right of Red Origin; which is now one unit to the left of Blue Origin. Q is not "in Blue", nor is P "in Red"; they are they same point, and it is wherever it is, regardless of where the origins of some coordinate systems are, and how we move those coordinate systems.

This is not to say that there is an absolute reference frame independent of coordinate systems either, mind you. Absent some external frame of reference, Blue Origin moving three units to its right is equivalent to P (aka Q) and Red Origin both moving three units to the left. It just means that points are equally "in" every coordinate system, since being "in" a coordinate system is just having some spatial relation to the origin of that coordinate system, and if these coordinate systems are all laid over the same space then every point in that space will have an (equally valid) spatial relation to the origin of all these coordinate systems. That is essentially what relativity says: there is no "true" coordinate system, they are all just laid over whatever is actually there, picking something arbitrary in the space (the origin) and referring to everything else by its relation to that thing. Anything else can be picked and things referred to by their relation to it, and in the end the whole network of relations will flesh out to be the same either way regardless of which node you choose to start from.

What does happen when you move Blue Origin three units to its right is that the point two units to the right of blue origin is now a different point that it was before; it is no longer P (aka Q). That point (P, aka Q) is now one unit to the left of Blue Origin; some other point, call it Q' if you like, is now at (2,0,0) in Blue, and is (and always has been) at (5,0,0) in Red; and used to be at (5,0,0) in Blue, too, before we moved Blue Origin.

P = Q, so there is only one point under discussion here at all.

This is part of the trouble...P in Red = Q in Blue....P in Blue does not equal Q in Blue...so your P = Q statement, is at best, ambiguous.
NOTE - 'P transformed to Blue"...the galilean have no such corresponding thing...
they only have one P...always in Red and only ever one Q...always in Blue...
without a P in Blue, then P in Red was not transformed properly
without a Q in Red, then Q in Blue was not transformed properly
they have neither...

P in A transforms to Q in Red [Galilean] vs P in A transforms to P in Blue [me]

If we move Blue Origin three units to its (and Red Origin's) right, this does not happen:

That is why I actually called this concept an axiom...
http://www.watermanpolyhedron.com/gg2y.html

From a private group of some well-known mathematicians, some while ago, i received a consensus in agreement with this axiom. Here so far...two who disagree with this axiom. I will go and look at what i said there...that perhaps I wrote/depictions used to convince a few of them there re this axiom...

[chenille]

This is part of the trouble...P in Red = Q in Blue....P in Blue does not equal Q in Blue...so your P = Q statement, is at best, ambiguous.

The problem here is you are shifting between whether P and Q represent points or coordinates. In the math that Galilean systems come from these are entirely different concepts: a coordinate system is a map, in your case an isomorphism, from some underlying space S to R³. If we want to be formal, your first image essentially shows maps RED and BLUE where P = Q and RED(P) = BLUE(P) = (2,0,0).

In the version you reject, your defining BLUE₂ three units right gives BLUE₂(P) = BLUE₂(Q) = (-1,0,0) and defining RED₂ three units left gives RED₂(P) = RED₂(Q) = (5,0,0). These transforms shouldn't have "identical outcomes". It's true you are shifting BLUE and RED the same way relative to one another, so

RED₂(BLUE^-1(x)) = RED(BLUE₂^-1(x))

But this is for coordinates x in R³ not points in S. It's not something you can apply to P = Q, and since you are shifting BLUE and RED in opposite ways relative to that point, there's no reason to expect any corresponding equality for the images of that point under the different maps.

At any rate, simply changing the maps RED and BLUE can't possibly make P and Q different within the space S. It can change their images under the maps, but the underlying space S is not actually being altered in any way. In your second set of examples you're really giving a new point Q' in S as well as a new map; it's no surprise the Galilean transformation won't work like that, because it's an entirely different scenario.

[Pfhorrest]

I made some pictures of my own to better illustrate this.

Say we have some points P and Q in a space S. They have no intrinsic coordinates, they just are.

points0.gif (1.55 KiB) Viewed 5441 times

We define a "Red" coordinate system over this space, taking P to be the origin, Q to be the direction of +x, and the distance between P and Q to be one unit. We didn't have to do this, we just chose to because it was convenient. P and Q now have coordinates, namely (0,0) and (1,0), because we defined Red that way.

points1.gif (3.47 KiB) Viewed 5441 times

We can also define a "Blue" coordinate system over this space, on the same basis as "Red". P and Q thus have the same coordinates in Blue as they do in Red.

points2.gif (4.12 KiB) Viewed 5441 times

But then we can decide that we'd rather redefine Blue to be centered on Q instead, translating it +1x, but lets not rotate it at all yet. The x component of both points' Blue coordinates thus becomes one less, while staying the same in Red, and now P and Q have different coordinates in the two now-different coordinate systems, but the points themselves are still the same as they ever were, and haven't moved.

points3.gif (4.51 KiB) Viewed 5441 times

We can also translate Blue +1y, centering it on nothing in particular that we have named in this space. The y components of both points in Blue thus become one less, but P and Q and still the same as they ever were, and still have the same coordinates in Red, because we haven't moved it or them relative to each other.

points4.gif (4.45 KiB) Viewed 5441 times

We can even rotate Blue and the two points' coordinates in it will change accordingly, but the points are still the same and have the same coordinates in Red.

points5.gif (4.42 KiB) Viewed 5441 times

We could also scale Blue to make its units twice as big, or skew it, punk or bloat it, twirl it, distort it any way we want, and all the points in S will have their coordinates in Blue changed accordingly, but the points themselves are still the same as they ever were.

Erm, my embedded attachments are fail. Technical help please?

[eran_rathan]

they look fine.

And you (and everyone else except Steve) is entirely correct.

THE TWO POINTS DO NOT MOVE RELATIVE TO EACH OTHER, REGARDLESS OF WHAT IS DONE WITH THE COORDINATE SYSTEMS.
(in a static solution).

its when stuff starts moving relativistically that things get weird (Lorentz contraction etc).

EDIT: Steve - its like this. You saying that because something is labelled in Latitude/Longitude that it is not the same point as if it is labelled in UTM X/Y, because there are different translation/scale/rotation factors. And you are wrong.

[JudeMorrigan]

2 The third statement, well, is not derived from the first two transformation statements. We are given coincident systems Red and Blue...relocate one
and is still remains the same....P in Red (2,0,0) = Q in Blue (2,0,0) agree ?

Other people have already covered this, and better than I have, but I went back and reread the post I'd replied to previously. While phrased oddly, I was ok with what you were saying when referencing the specific picture in the post. I want to be very clear that I do not agree that if you start with two coincident systems and translate one, you wind up with the scenario that was represented in the picture.

In a real-world scenario, if you have a satellite located at a given point in space, its ECI coordinate is NOT the same as its ECEF coordinate if you are not at the ECI epoch time. You'd have to actually move the satellite in order for that to be the case (like you had to actually move your "P" and "Q" tablets), which would clearly defeat the entire purpose of doing the coordinate transformation.

[gmalivuk]

Right. The satellite (P) is at the satellite's location (Q) no matter what you do with the coordinate system. P and Q are referring to points, not coordinates. And so if P=Q, they are the same point, and transforming coordinates doesn't affect that.

[steve waterman]

working definitions...coordinate points, selected points, relocation, transformation...

coordinate points....all (x,y,z) locations...or more generally, all ( abscissa, ordinate, applicate )
selected points...a coordinate point designated with a corresponding name...example, point P in Red at (2,0,0)

This mystery cannot be unraveled, without ALWAYS keeping track of each selected points system...that is just saying point P,
and not saying point P in Red or point P in Blue, contributes to confusion...
x' = x - vt ........................x' in what ? = x in what - vt to what ?

we all agree the coordinates move when their system moves.

we disagree, in that selected points move as their system moves.

I would like to stick with selected point P in Red at (2,0,0) and point selected point Q in Blue at (2,0,0) at coincidence, as given.

i would then like to agree upon what the galilean results are...under condition 1 ( Blue right 3 ) and condition 2 (Red left 3).

condition 1 - Blue right 3, then selected point P in Red = (2,0,0) trans to their newly named selected point Q in Blue =(-1,0,0)
or their inversion...
condition 2 - Red left 3, then selected point Q in Blue = (2,0,0) trans to their newly named selected point P in Red = (2,0,0).

do we agree upon these being the galilean transformation results using the given selected points P in Red and Q in Blue?

i ask for a differentiation to made between
a RELOCATION ( moving a system relevant to the other ) and quite specifically, NOT transferring of ANY points ( coordinates or selected points )

and a TRANSFORMATION is the adding in of corresponding selected points to their opposite system,
transforming say, selected point P in Red means...ADDING another selected point into the opposite system ( so, into Blue ) and naming it P in Blue.
and not naming it as the receiving selected POINT name..NEVER to Q in B but as the sending system's point name...so, P in Red should trans to P in Blue.

To be 100 percent clear...mathematically, we can do a relocation and NOT do a transformation...this is critical to be aware and make this distinction, as well as that between selected points and coordinate points...(yes, all selected points are also coordinate points)
do you acknowledge these distinctions ?

AHA...since coordinates move with their system, ( I believe you agreed upon that )...then it must be true that...AFTER either or both systems have moved...that (x,0,0) in Red = (x',0,0) in Blue!!! This has now become my KEY question...if you respond to nothing else...do please, weigh in on this one...

[eran_rathan]

???

I can't really follow what you are saying here, but I'm going to try (again) to explain what I think you are trying to say.

Given:
Coordinate system, and points P and Q which are non-coincident. P represents the origin of the coordinate system, Q is a point at (x,y,z).

1. Create a new coordinate system at (x,0,0) and call it Blue, where x is the X-value of Q (making Q the origin point of Blue). P_Blue is now at (-x,y,z).

2. The distance between Q and P, regardless of coordinate system used, is constant.
3. Distances between coordinates can only be measured using the same system, otherwise you are comparing apples and oranges.

[steve waterman]

???

I can't really follow what you are saying here, but I'm going to try (again) to explain what I think you are trying to say.


Coordinate system, and points P and Q which are non-coincident. P represents the origin of the coordinate system, Q is a point at (x,y,z).



1. Create a new coordinate system at (x,0,0) and call it Blue, where x is the X-value of Q (making Q the origin point of Blue). P_Blue is now at (-x,y,z).

2. The distance between Q and P, regardless of coordinate system used, is constant.
3. Distances between coordinates can only be measured using the same system, otherwise you are comparing apples and oranges.

Coordinate system, and points P and Q which are non-coincident. P represents the origin of the coordinate system, Q is a point at (x,y,z).

NO. ...selected point P in Red is not an origin...as it is at (2,0,0) in the Red system.

No . selected point Q at (2,0,0) in Blue is not at the VARIABLE (x,y,z)...it is SELECTED/named/given a CONSTANT VALUE for (x,y,z) = (2,0,0) in the Blue system.

Does that makes any clarity for you?
btw, you could not possibly follow ANY consequential math logic here...as you were using the wrong parameter from the get go.


WOW...by virtue of trying to do "bottom of the page writing" it is so hard to follow who is saying what...way confusing. So, it is now is a presentation form that can stand-alone...even any depictions are not necessary as it is hopefully, all trimmed to some basics, itemized directly below...
....................................................................................................................................................................
MY mathematically GIVEN....
1 Red and Blue coincident system ( where coincidence implies equality in all mathematical aspects )
2 selected point P in Red at (2,0,0)
3 selected point Q in Blue at (2,0,0)

the transformations between systems results with...
using the Galilean
Red left 3
then Q in Blue at (2,0,0) trans to P in Red at (5,0,0)
OR
Blue right 3
then P in Red at (2,0,0) trans to Q in Blue at (-1,0,0)

my mandate
Red left 3. then
Q in Blue = (2,0,0) transforms to Q in Red = (5,0,0) AND P in Red = (2,0,0) transforms to P in Blue = (-1,0,0)
Blue right 3. then
Q in Blue = (2,0,0) transforms to Q in Red = (5,0,0) AND P in Red = (2,0,0) transforms to P in Blue = (-1,0,0)

observation 1...
comparing the transformation results of Red left 3 against that of Blue right 3...

the Gailean has opposing/variant/variable/inconsistent/conflicting results.
my mandate has equivalent/equal/identical/proper/mathematical valid results.

observation 2...
since coordination is affixed to its' own system, then If one of two coincident system relocate from that coincidence,
then
Red (x,y,z)= Blue (x',y',z') is true.

observation 3
the Galilean contends that
Red (x,y,z)= Blue (x',y',z') - d, is true

Conclusion: both equations cannot be mathematically correct
...................................................................................................................................................................
steve waterman

[eran_rathan]

a lot of stuff, where he confuses a coordinate with a point.

You are comparing apples and oranges. It has been sufficiently explained, multiple times. You do not understand coordinate geometry, sirrah.

[eran_rathan]

If P and Q are coincident, then you can simply use P as the point, with the subscript blue or red for the system you are writing the coordinate in.

Moving Blue +3 wrt Red:
P_Red is (2,0,0)
P_Blue = P_Red + (3,0,0) = (5,0,0) in Blue

Moving Red -3 wrt to Blue:
P_Blue is (2,0,0)
P_Red = P_Blue - (3,0,0) = (-1,0,0) in Red