Concentric circles are parallel?

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Concentric circles can be considered parallel under certain definitions, particularly when the concept of parallelism is generalized beyond straight lines. The discussion highlights the distinction between Euclidean geometry, which defines parallelism strictly for lines, and other interpretations that may include curves. Participants debate the nature of parallelism, questioning whether curves can be parallel and how definitions may vary across different geometrical contexts. The conversation emphasizes the importance of clearly defining "parallel" to avoid confusion, particularly when discussing non-Euclidean geometries. Ultimately, the definition of parallel can extend to curves if one adopts a broader perspective, but this is not universally accepted.
  • #61
"Concentric lines"...circles are concentric because of their uniform curve...two parallel lines cannot have any 'center', they can have a mid-point between them, wherein the group of all points that are halfway between the two lines create a new line that is parallel to both of the original parallel lines. Obviously.

I'm pretty sure that the strict mathematical definition of "line" always means a perfectly straight line, so I don't know what you could mean by 'concentric lines', as if this is true, only curves can be concentric from a particular point, in if the distance from the center is changed for one uniform curve to the distance (radii) of another curve, that both are exactly the same, in 2D space, with relation to the origin/center.
 
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  • #62
So then you agree that "concentric lines" is absurd, since there are many flaws and counter-examples to it. The same deal with parallel circles applies here.
 
  • #63
Mentallic said:
I beg to differ, notice that I wrote


Your diagram doesn't express what I mentioned. So I took the liberty of explaining it to you in pretty pictures.

http://img691.imageshack.us/img691/9694/parallel.th.png
This diagram is supposed to be 2 dimensional.
In the first row of examples, the red and orange lines are of equal length. The circles are of the same size.

So when you say curves are parallel, we could be confused into thinking it's two circles above. How can we know you mean concentric circles? If you do mean concentric circles, then just say so. They already have a name for those so there's no need to try and label them as being parallel.
There are inconsistencies in trying to extend the meaning of parallel since it needs to hold for all cases. Clearly, from my diagram, if you consider it parallel to be the observer from a fixed point (as in the case of concentric circles) then why aren't two lines parallel if you observe them in the same way?

And to touch up on another point. When two parallel lines are transposed (moved) they are still parallel to each other. In your diagram, this rule breaks down. You move one of those concentric circles and suddenly they aren't "parallel" to each other anymore.

You indeed have not understood my meaning. Must pay attention, we enter are not the familiar domain, therefore we are very easy to make a mistake. Must pay attention, we in exploration curve parallel, but is not straight line being parallel. In straight line parallel, after the straight line was parallel the migration, is still parallel, but in the curve - - circle was not that simple. We are conceivable, the straight line is the radius infinitely great circle, then what meaning the parallel motion straight line is? The parallel motion straight line is increasing or is reduced the circle (is certainly diameter infinitely great circle) the diameter. The concentric circle parallel migration is also so, is also changes the circle the diameter, but is not under the invariable diameter migration.
 
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  • #64
I can honestly say I have no idea what you just said.
 
  • #65
Mentallic said:
So then you agree that "concentric lines" is absurd, since there are many flaws and counter-examples to it. The same deal with parallel circles applies here.

Yes, concentric lines do not make sense, at least in Euclidean space, I agree.

I say that the argument with parallel circles is different than the obviously standard definition of parallelism (Playfair's axiom), because with parallel lines, the argument is meant for plane geometry, but can obviously be made to hold in 3D geometry. The same can be said for concentric circles -> concentric spheres, except that you must specify a 1 to 1 matching between points on each sphere, with the minimum distance value relationship that I mentioned above. With parallel lines, it is more of a conceptual consistency, with parallel curves it is always in relation to something else, if you ignore the superposition proof/argument.
 
  • #66
Well there's no problem defining concentric circles to be parallel, but the only use I see this having is for it to be a shortcut way of describing the property that the tangents on each circle are parallel.
 
  • #67
G037H3 said:
I'm not sure that last point is exactly fair, because since a line is the same in homogeneous space, and distances between circles depend on their center and radius, it is much easier to claim that circles can't be parallel.

I stand by my previous post, concentric circles are parallel curves, but not parallel lines, because in mathematics lines are generally perfectly straight.
You in the past and in future border, therefore, on the one hand you acknowledge the curve parallel, on the other hand, you thought that the line is only a straight line. In fact, the line may be straight, may also be curving, the straight line is straight, but the line is not only straight.
 
  • #68
Mentallic said:
Well there's no problem defining concentric circles to be parallel, but the only use I see this having is for it to be a shortcut way of describing the property that the tangents on each circle are parallel.

Ah, wait a sec. Are you approaching this with a specific use in mind? You should just think about the objects and their relationship(s).

Also, I also have little idea what the hell phya is saying either. :/ idk if he's using translation software or just has a problem with vocabulary or grammar
 
  • #69
phya said:
You in the past and in future border, therefore, on the one hand you acknowledge the curve parallel, on the other hand, you thought that the line is only a straight line. In fact, the line may be straight, may also be curving, the straight line is straight, but the line is not only straight.

Well, I personally use Archimedes' definition that a line is the shortest distance between two points, but here is evidence that the mathematical term 'line' refers to a straight one: http://mathworld.wolfram.com/Line.html
 
  • #70
G037H3 said:
Ah, wait a sec. Are you approaching this with a specific use in mind?
Oh no not at all. Phya wanted to extend the definition of parallel to describe curves too, and I've given examples that have shown that for circles (which can be for curves too) you can think of parallel in different ways, depending on how you view it. This leaves room for confusion which is why one would need to define what exactly parallel for curves is and the properties that accompany it.

You should just think about the objects and their relationship(s).
But I have thought about them. Their relationship is that they are concentric and one of their properties is that given a line cutting through the centre of the circles, at the points where it intersects the circumferences, tangential lines at those points will be parallel.

This doesn't mean I'm ready to accept a flawed definition of "concentric circles are parallel curves".
 
  • #71
Mentallic said:
The point is, using the term "parallel curve" creates an ambiguity as I've already shown, unless it's specifically defined to be one or the other - which it isn't.

What is wrong with sticking to the term "concentric circles" anyway? I could just as easily argue that parallel lines are also concentric lines. It's just silly.

The concentric circle is only an example, this is only to explain the curve the parallel phenomenon. Certainly, curve parallel is not the limitation in the concentric circle. For example, the parabola may also have the parallel line. The curve parallel is the curve to the curve distance constant invariable. This is the curve parallel essence. Certainly is also the surface parallel essence. This is an axiom.
 
  • #72
Mentallic said:
Oh no not at all. Phya wanted to extend the definition of parallel to describe curves too, and I've given examples that have shown that for circles (which can be for curves too) you can think of parallel in different ways, depending on how you view it. This leaves room for confusion which is why one would need to define what exactly parallel for curves is and the properties that accompany it.


But I have thought about them. Their relationship is that they are concentric and one of their properties is that given a line cutting through the centre of the circles, at the points where it intersects the circumferences, tangential lines at those points will be parallel.

This doesn't mean I'm ready to accept a flawed definition of "concentric circles are parallel curves".

Well a line is a straight curve. Hum.

A transposition of two parallel lines would do nothing to change the values of any additional figures or forms created by any other lines. You are undoubtedly correct in your assertion about tangent lines to two points on the circumference of both circles being tangent, but what I was trying to communicate is that the straight line connecting the two is essential because of the 1 to 1 mapping, that is, the larger circle simple has a larger radius. Both circles have the same equation. Now, how is this really different than saying that two lines with y intercept 1 and y intercept 2 have the same equation if they are parallel? The shortest possible distance between any two points on the parallel lines is going to be 1. Obviously. Now, curves are not really any different, except for two things. The length of the curve, if it were flattened out to a straight line, and the uniformity of the curve. A line can be perfectly superimposed on any other line, but a curve can only be if they share a common value for the radii, so my contention is that concentric circles are an example of parallel lines, but if you want to transform the curves in any way, it changes things and they aren't always parallel if you transform/flip a curve.
 
  • #73
G037H3 said:
I don't disagree, but I feel that the main point of the thread is to assert that concentric circles are parallel, which they are, because they're parallel curves. As long as the OP understands the difference between parallel lines (all the points that the line consists of can be rotated about themselves 360 degrees and the line remains parallel) and parallel curves, I don't really have a problem with the assertion.
I was not only saying that the concentric circle parallel, I was also saying the curve parallel, moreover was also saying the non-Euclid geometry exists the question, sees #43 to paste:https://www.physicsforums.com/showpost.php?p=2919121&postcount=43
 
  • #75
G037H3 said:
I'm sorry, but could you please elaborate on what you mean by transposition? :)

My knowledge in this kind of topic comes from studying Heath's version of Euclid's Elements, btw. I'm only about 10% through it, but it is a very beautiful book.

：）
His meaning is the parallel migration. I already replied him.
 
  • #76
If I were to think of two parabolas being parallel, then I would instantly think of some parabola y=ax^2+bx+c and then one that is parallel to it would be of the form y=ax^2+bx+k.

But apparently to have the parabolas follow this same rule as the concentric circles do, if say the "centre" of the parabola y=x2-1 is (0,0) then for another parabola that is a ratio of m:1 distance further from this parabola, it must be of the form y=x2/m-m

So which do you consider to be parallel parabolas? May I remind you this will be a definition that you create and which is not already widely accepted. I hope you can see why by now.
 
  • #77
G037H3 said:
Well, I personally use Archimedes' definition that a line is the shortest distance between two points, but here is evidence that the mathematical term 'line' refers to a straight one: http://mathworld.wolfram.com/Line.html

I understand your idea, but, the present mathematics is also in historical mathematics, but has not been separated from the historical mathematics. In the past, the humanity did not have the negative number concept, but afterward had. At present the humanity did not think that the curve is parallel, but the future will think not like this? Is indefinite. In the spherical surface between two spot most short distances are the straight line? On ellipsoid surface? On column surface? In paraboloid? In random surface? The line is the broad concept, but the straight line is the relative narrow concept.
 
  • #78
phya said:
I understand your idea, but, the present mathematics is also in historical mathematics, but has not been separated from the historical mathematics. In the past, the humanity did not have the negative number concept, but afterward had. At present the humanity did not think that the curve is parallel, but the future will think not like this? Is indefinite. In the spherical surface between two spot most short distances are the straight line? On ellipsoid surface? On column surface? In paraboloid? In random surface? The line is the broad concept, but the straight line is the relative narrow concept.

Negative numbers were known. The thing is that the Greeks treated mathematics in a very pure manner, so that things like the square root of 2 really bothered them because of their desire for clean numerical relationships.

After that, negative numbers were mostly ignored because mathematics was used mainly for physics and physical applications, so negative results were thrown out as invalid. Which they are, if they're describing something physical on their own, with nothing to compare them to.

Concentric circles are parallel curves, but not parallel lines, because you can't perfectly place a section of the smaller circle onto a section of the larger circle and have them fit.

The shortest distance between two points is always a straight line. Non-Euclidean geometries only change this by changing the rules of the space so that it is not homogeneous in all directions as Euclidean space is. For Riemannian (elliptic) geometry, an example of the planet Earth is given, that two points on the surface are connected by a great circle, but still the shortest distance between those two points is a straight line going through the Earth, but in Riemannian geometry this sort of thing isn't allowed...basically, its still Euclidean geometry, just with somewhat different rules. Both models can describe space adequately.

Again, I think that you are mistaking the difference between a curve and a line. A line is always a straight line, a line is a straight curve. Get it?
 
  • #79
Mentallic said:
Oh no not at all. Phya wanted to extend the definition of parallel to describe curves too, and I've given examples that have shown that for circles (which can be for curves too) you can think of parallel in different ways, depending on how you view it. This leaves room for confusion which is why one would need to define what exactly parallel for curves is and the properties that accompany it.

But I have thought about them. Their relationship is that they are concentric and one of their properties is that given a line cutting through the centre of the circles, at the points where it intersects the circumferences, tangential lines at those points will be parallel.

This doesn't mean I'm ready to accept a flawed definition of "concentric circles are parallel curves".
Certainly, if two curves are parallel, then their corresponding points of curvature circle must concentric, this is certain. The curve parallel defined me already to say. The concentric circle is only a curve parallel phenomenon, but is not all.
 
  • #80
G037H3 said:
Well a line is a straight curve. Hum.

A transposition of two parallel lines would do nothing to change the values of any additional figures or forms created by any other lines. You are undoubtedly correct in your assertion about tangent lines to two points on the circumference of both circles being tangent, but what I was trying to communicate is that the straight line connecting the two is essential because of the 1 to 1 mapping, that is, the larger circle simple has a larger radius. Both circles have the same equation. Now, how is this really different than saying that two lines with y intercept 1 and y intercept 2 have the same equation if they are parallel? The shortest possible distance between any two points on the parallel lines is going to be 1. Obviously. Now, curves are not really any different, except for two things. The length of the curve, if it were flattened out to a straight line, and the uniformity of the curve. A line can be perfectly superimposed on any other line, but a curve can only be if they share a common value for the radii, so my contention is that concentric circles are an example of parallel lines, but if you want to transform the curves in any way, it changes things and they aren't always parallel if you transform/flip a curve.
You know what I mean?
https://www.physicsforums.com/showpost.php?p=2921163&postcount=63
 
  • #81
Mentallic said:
If I were to think of two parabolas being parallel, then I would instantly think of some parabola y=ax^2+bx+c and then one that is parallel to it would be of the form y=ax^2+bx+k.

But apparently to have the parabolas follow this same rule as the concentric circles do, if say the "centre" of the parabola y=x2-1 is (0,0) then for another parabola that is a ratio of m:1 distance further from this parabola, it must be of the form y=x2/m-m

So which do you consider to be parallel parabolas? May I remind you this will be a definition that you create and which is not already widely accepted. I hope you can see why by now.

https://www.physicsforums.com/showpost.php?p=2921406&postcount=79
 
  • #82
So then the second column of curves in this picture aren't supposedly parallel?
http://img691.imageshack.us/i/parallel.png/"
 
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  • #83
G037H3 said:
Negative numbers were known. The thing is that the Greeks treated mathematics in a very pure manner, so that things like the square root of 2 really bothered them because of their desire for clean numerical relationships.

After that, negative numbers were mostly ignored because mathematics was used mainly for physics and physical applications, so negative results were thrown out as invalid. Which they are, if they're describing something physical on their own, with nothing to compare them to.

Concentric circles are parallel curves, but not parallel lines, because you can't perfectly place a section of the smaller circle onto a section of the larger circle and have them fit.

The shortest distance between two points is always a straight line. Non-Euclidean geometries only change this by changing the rules of the space so that it is not homogeneous in all directions as Euclidean space is. For Riemannian (elliptic) geometry, an example of the planet Earth is given, that two points on the surface are connected by a great circle, but still the shortest distance between those two points is a straight line going through the Earth, but in Riemannian geometry this sort of thing isn't allowed...basically, its still Euclidean geometry, just with somewhat different rules. Both models can describe space adequately.

Again, I think that you are mistaking the difference between a curve and a line. A line is always a straight line, a line is a straight curve. Get it?

Why wants online front to add on " Straight " ? Therefore so the rhetoric has the reason, because the line is containing the curve. Your line's concept is forms in the history, therefore has the limitation. In the early history, we speak of the human often to understand the man, actually person this concept contains the woman, also contains the man.
 
  • #84
Mentallic said:
So then the second column of curves in this picture aren't supposedly parallel?
http://img691.imageshack.us/i/parallel.png/"
You give the link is unable to open.
 
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  • #86
HallsofIvy said:
What do you mean by 'parallel'? There are some definitions of "parallel" in "concentric circles are parallel" is true and some in which it is not. The "usual" definition of parallel in Euclidean geometry specifically defines only "parallel lines" and so, with that definition, it is not true.
Euclid's parallel is narrow parallel, my parallel is generalized being parallel.
 
  • #87
In the spherical surface does not have the parallel line, this viewpoint is not correct. Should say that in the spherical surface does not have the straight line parallel line, but has the curve parallel line.
 
  • #88
phya said:
Euclid's parallel is narrow parallel, my parallel is generalized being parallel.
So, once again, you state that you are not using the "standard" definition but refuse to state what definition you are using.
 
  • #89
Well, his own definition. It's generalized, and it's flawed. There's nothing else to it really.
 
  • #90
generalized parallelism as a definition doesn't work because you can just flip one of the figures over and they aren't parallel anymore
 

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