Concentric circles are parallel?

  • Thread starter Thread starter phya
  • Start date Start date
  • Tags Tags
    Circles Parallel
AI Thread Summary
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.
  • #101
When I was learning math definitions were very strict. At least in my country. Using "parallel" instead of "equidistant" would mean I will not pass the exam even I answer correct to every other question.
 
Mathematics news on Phys.org
  • #102
Upisoft said:
When I was learning math definitions were very strict. At least in my country. Using "parallel" instead of "equidistant" would mean I will not pass the exam even I answer correct to every other question.

well, mathematics tries to be precise, but that sort of nit-picking is not the best way to test your understanding of concepts
 
  • #103
Well, I don't say it was perfect testing system. I just wanted to say if the statement was that concentric circles are equidistant, there would be not 7 pages of discussions over that. Sometimes nit-picking is required. :P
 
  • #104
part of the reason that this thread is so long is that phya has failed to make a fundamental statement that applies to parallelism, i.e. what constitutes parallelism as opposed to non-parallelism
 
  • #105
G037H3 said:
err, what?...the top and bottom edges of the line? that's a pretty loose definition of parallelism, and even if they were parallel, how would that help you?

if you flip a curve (a reflex transform), it is no longer parallel. a line is.

Are you acknowledge a straight line own and own parallel? If you acknowledged that then curve own and oneself not parallel?
 
  • #106
The actual existence's line has the width. If this width is invariable, then on the one hand on the other hand its one and another is parallel. I said own and own parallel this aim at do not have the width curve to say.
 
  • #107
Upisoft said:
Concentric circles are not parallel, they are equidistant. "Parallel" is defined only for straight lines.
This was the old idea, mathematics was also progresses unceasingly.
 
  • #108
Upisoft said:
When I was learning math definitions were very strict. At least in my country. Using "parallel" instead of "equidistant" would mean I will not pass the exam even I answer correct to every other question.

G037H3 said:
well, mathematics tries to be precise, but that sort of nit-picking is not the best way to test your understanding of concepts
That's not nit-picking. In any geometry except Euclidean the set of points equidistant from a line (and on one side of it) is NOT a line. For example in spherical geometry, the equator is a line (a great circle) but the set of points at a fixed distance north of equator is NOT a great circle and so not a "line".

In hyperbolic geometry we have the concept of "equidistant curve" which is not a line.
 
  • #109
phya said:
Are you acknowledge a straight line own and own parallel? If you acknowledged that then curve own and oneself not parallel?
Yes, by any definition of "parallel" I know, a line and itself are NOT parallel. The most fundamental definition of "parallel" is "they do not cross". And that is certainly not true for a line and itself.
 
  • #110
Upisoft said:
Concentric circles are not parallel, they are equidistant. "Parallel" is defined only for straight lines.

phya said:
This was the old idea, mathematics was also progresses unceasingly.
No, that is the new idea. Your notion of "parallel" being the same as "equidistant" is the old idea.
 
  • #111
G037H3 said:
parallel curves exist, it's just that phya is using too loose of a definition for parallel :/

http://mathworld.wolfram.com/ParallelCurves.html
Although in daily massive use parallel curve, but the present scholars did not acknowledge that the curve parallel, they only think the existence equidistant curve.
 
  • #112
phya said:
Are you acknowledge a straight line own and own parallel? If you acknowledged that then curve own and oneself not parallel?

...
 

Attachments

  • parallelism.jpg
    parallelism.jpg
    23.7 KB · Views: 367
  • #113
Upisoft said:
When I was learning math definitions were very strict. At least in my country. Using "parallel" instead of "equidistant" would mean I will not pass the exam even I answer correct to every other question.
Perhaps your son will study the curve parallel theory.
 
  • #114
Were we may also say that the parallel straight line was the equal-space straight line? Actually the equal-space straight line and the equidistant curve all are parallel lines, why can the same thing use the different terminology?
 
  • #115
HallsofIvy said:
That's not nit-picking. In any geometry except Euclidean the set of points equidistant from a line (and on one side of it) is NOT a line. For example in spherical geometry, the equator is a line (a great circle) but the set of points at a fixed distance north of equator is NOT a great circle and so not a "line".

In hyperbolic geometry we have the concept of "equidistant curve" which is not a line.
Ask that the great-circle own and oneself is parallel?
 
  • #116
phya said:
Were we may also say that the parallel straight line was the equal-space straight line? Actually the equal-space straight line and the equidistant curve all are parallel lines, why can the same thing use the different terminology?

But they're not the same; this depends on the underlying geometry. It is a theorem in Euclidean geometry that parallel lines are equidistant. This is false in hyperbolic geometry. In elliptic geometry, of course, parallel lines are not only equidistant but also equal. (In the usual formulation, parallel lines in elliptic geometry are also unicorns.)
 
Last edited:
  • #117
HallsofIvy said:
Yes, by any definition of "parallel" I know, a line and itself are NOT parallel. The most fundamental definition of "parallel" is "they do not cross". And that is certainly not true for a line and itself.

Is a straight line own and own parallel?
 
  • #118
HallsofIvy said:
No, that is the new idea. Your notion of "parallel" being the same as "equidistant" is the old idea.
But the old concept did not acknowledge that the equidistant curve is a parallel line.
 
  • #119
Actually, variations of Euclidean geometry did acknowledge that- I believe Euclid himself mentions it.

But now we know that there exist geometries in which "the equidistant curve is a parallel line" is NOT true- it is not even a line. I have repeatedly given examples to show that, yet you keep asserting it is true!
 
  • #120
G037H3 said:
...
What your chart mainly wants to explain is what?
 
  • #121
that if you have two parallel lines, and flip one over a line (a reflex, reflection, transform), that the line that is flipped remains parallel with the other line

the same is not true with curves, it becomes an inverse curve to the one that remains in position
 
  • #122
CRGreathouse said:
But they're not the same; this depends on the underlying geometry. It is a theorem in Euclidean geometry that parallel lines are equidistant. This is false in hyperbolic geometry. In elliptic geometry, of course, parallel lines are not only equidistant but also equal. (In the usual formulation, parallel lines in elliptic geometry are also unicorns.)

Hyperbolic geometry's parallel concept is wrong. Does not intersect was not equal to that is parallel.
 
  • #123
HallsofIvy said:
Actually, variations of Euclidean geometry did acknowledge that- I believe Euclid himself mentions it.

But now we know that there exist geometries in which "the equidistant curve is a parallel line" is NOT true- it is not even a line. I have repeatedly given examples to show that, yet you keep asserting it is true!
Sorry, why isn't the curve a line?
 
  • #124
G037H3 said:
that if you have two parallel lines, and flip one over a line (a reflex, reflection, transform), that the line that is flipped remains parallel with the other line

the same is not true with curves, it becomes an inverse curve to the one that remains in position

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.
 

Attachments

  • 关于同心与不同心圆的说明.gif
    关于同心与不同心圆的说明.gif
    12.3 KB · Views: 485
  • #125
phya said:
Hyperbolic geometry's parallel concept is wrong. Does not intersect was not equal to that is parallel.

Mountain, meet Mahomet.
 
  • #126
phya said:
Perhaps your son will study the curve parallel theory.

I certainly hope not. I'll try to find the best school for him.

phya said:
Is a straight line own and own parallel?
What do you ask? If a straight line is parallel to itself? If so, the answer is "no". Parallel lines are defined as lines that do not share common points. And a straight line has every point common with itself.

I understand what you are trying to do. It is not wrong. What is wrong is "how" you are trying to do it. You certainly may have a geometry in which your concentric circles are parallel. What you must do is to define consistent set of axioms that define such geometry. Good luck!
 
  • #127
CRGreathouse said:
Mountain, meet Mahomet.
:smile:
 
  • #128
Upisoft said:
I certainly hope not. I'll try to find the best school for him.


What do you ask? If a straight line is parallel to itself? If so, the answer is "no". Parallel lines are defined as lines that do not share common points. And a straight line has every point common with itself.

I understand what you are trying to do. It is not wrong. What is wrong is "how" you are trying to do it. You certainly may have a geometry in which your concentric circles are parallel. What you must do is to define consistent set of axioms that define such geometry. Good luck!

If has straight line parallel line A and B, in A infinite approaches B in the process, they always parallel, when they superpose in together time, they no longer were parallel? They certainly parallel, therefore the straight line own and oneself is parallel, the curve is also so.
 
  • #129
If has moving point a and moving point c, straight line B through a and c, the direction which a and c move is always vertical to B, and a and the c straight line's distance maintains invariable, then a and the c path is a parallel line, regardless of this path is a straight line, or is the curve.
 
  • #130
phya said:
If has moving point a and moving point c, straight line B through a and c, the direction which a and c move is always vertical to B, and a and the c straight line's distance maintains invariable, then a and the c path is a parallel line, regardless of this path is a straight line, or is the curve.
That completely ignores the question of where two objects moving in such a way will move along straight lines, which is the whole point.

You seem to be insisting on Euclidean geometry while refusing to use "parallel" as Euclid defined it.
 
  • #131
HallsofIvy said:
That completely ignores the question of where two objects moving in such a way will move along straight lines, which is the whole point.

You seem to be insisting on Euclidean geometry while refusing to use "parallel" as Euclid defined it.

His definition has the question, parallel should define maintains invariable for the distance.
 
  • #132
Well, that was not his definition. For one thing Euclid define "parallel" only for lines, not curves. If you would say "equidistant curves" rather than "parallel curves", I would have no trouble with what you say.
 
  • #133
phya, i would like to reiterate:

lines are perfectly straight curves

all lines are curves, but not all curves are lines
 
  • #134
HallsofIvy said:
Well, that was not his definition. For one thing Euclid define "parallel" only for lines, not curves. If you would say "equidistant curves" rather than "parallel curves", I would have no trouble with what you say.
Mathematics is develops unceasingly, might not Euclid say anything, therefore forever was anything. Newton said that the space and time is absolute, but Einstein said that the space and time is relative. Euclid said that the parallel line is only a straight line, but we said that the parallel line is also the curve, the curve may also be parallel. This is not to the geometry development?
 
  • #135
G037H3 said:
phya, i would like to reiterate:
lines are perfectly straight curves
all lines are curves, but not all curves are lines

...↗ straight line
Line
...↘ curving line

This is my concept.
 
  • #136
but a line is perfectly straight

so a 'curving line' is just a curve that isn't a line
 
  • #137
G037H3 said:
but a line is perfectly straight

so a 'curving line' is just a curve that isn't a line
Why in front of this word adds “ straight” in “the line”, is for the line of demarcation extension. The line has two kinds, one kind is a straight line, one kind is a curve.
 
  • #138
phya has a point there.

If two particles travel in the same direction and at any point in time they are tangential to each other, then the path they trace will be two equidistant curves. This doesn't necessarily mean they are parallel curves though.

Each case of curves that have these special properties are already given names to distinguish between each other, there are equidistant curves, concentric (curves?) circles etc.

You've given examples of both equidistant curves and concentric circles, and claim that both parallel. Well I'm telling you that there needs to be a distinction between the two because it creates confusion. And plus they already have been given names and their properties are well known.
 
  • #139
In the analytic geometry,
Supposition
the straight line L1 equation is y=kx,
the straight line L2 equation is y=kx+c,

then L1∥L2 is parallel,
if reduces c, then still L1∥L2.
When c=0,
L1 and L2 superposition, still L1∥L2,
if L1 and L2 not parallel, then L1 and L2 will not superpose, will intersect.
Therefore the straight line own and oneself is parallel, otherwise the straight line will not be a straight line, will intersect.
The curve is also so, the curve is also own and own parallel, therefore the curve is also may mutually parallel.
Does my this logic have what question?
 
  • #140
phya said:
Why in front of this word adds “ straight” in “the line”, is for the line of demarcation extension. The line has two kinds, one kind is a straight line, one kind is a curve.

you think that a 'line' is any 1 dimensional continuum, this is not so

all 1 dimensional continuums are curves, and the curves that are perfectly straight (parallel with at least one other straight line, etc.) are called lines
 
  • #141
G037H3 said:
you think that a 'line' is any 1 dimensional continuum, this is not so

all 1 dimensional continuums are curves, and the curves that are perfectly straight (parallel with at least one other straight line, etc.) are called lines
I not too understand.
 
  • #142
I propose the creation of a new word: "Phyallel!"

The definition of "phyallel"?:_____(insert definition here)______

:-)
 
  • #143
lol :approve:
 
  • #144
Dr Lots-o'watts said:
I propose the creation of a new word: "Phyallel!"

The definition of "phyallel"?:_____(insert definition here)______

:-)

In the analytic geometry,
Supposition
the straight line L1 equation is y=kx,
the straight line L2 equation is y=kx+c,
then L1∥L2 ,
if c→0, then still L1∥L2.
When c=0,
L1 and L2 superposition,
if L1 and L2 not parallel, then L1 and L2 will not superpose, will intersect.
L1 and L2 superpose, not intersect.
Therefore still L1∥L2,
Therefore the straight line own and oneself is parallel, otherwise the straight line will not be a straight line, will intersect.
The curve is also so, the curve is also own and own parallel, therefore the curve is also may mutually parallel.
Does my this logic have what question?
 
  • #145
Your statement that superposed lines do not intersect is incorrect.

If two lines do NOT intersect there is NO point that lies on both lines. That is obviously incorrect for superposed lines.
 
  • #146
So two lines phya-intersect iff they intersect but are unequal.

Has anyone yet figured out what it means to be phya-parallel ("phyallel" as Dr Lots-o'watts puts it)?
 
  • #147
HallsofIvy said:
Your statement that superposed lines do not intersect is incorrect.

If two lines do NOT intersect there is NO point that lies on both lines. That is obviously incorrect for superposed lines.
Your view is also not correct, the superposition is two straight line all corresponding points overlapping, but intersects has a spot superposition, but other spots do not superpose. Therefore the superposition is not the intersection.

“If two lines do NOT intersect there is NO point that lies on both lines” is correct.but the superposition is not the intersection.
 
  • #148
HallsofIvy said:
Your statement that superposed lines do not intersect is incorrect.

If two lines do NOT intersect there is NO point that lies on both lines. That is obviously incorrect for superposed lines.
The intersection is two straight lines has a common point, but superposes is two straight lines becomes a line straight line. Therefore the intersection and the superposition are different.
 
  • #149
I agree with HallsofIvy. If you have two straight lines that are parallel (no intersection points) and you translate one of them so that it coincides with the other line, then the two lines intersect at every point.

No doubt it's a problem with your command of English. "To intersect" does not necessarily mean that the lines have to cross at some nonzero angle.
 
  • #150
Mark44 said:
I agree with HallsofIvy. If you have two straight lines that are parallel (no intersection points) and you translate one of them so that it coincides with the other line, then the two lines intersect at every point.

No doubt it's a problem with your command of English. "To intersect" does not necessarily mean that the lines have to cross at some nonzero angle.

Actually we may regard as a straight line are two superpose in the together straight line, they look like are a straight line, under such situation, these two straight lines was not being parallel? Obviously, although they superpose in together, but they are parallel.
 

Similar threads

MHB Can You Prove that AC//BD and AF//BE are Parallel in this Geometry Problem?
Replies
1
Views
2K
MHB State the angles in terms of x , Circle theorems
Replies
6
Views
2K
B Analysis of Simple Pendulum Motion
Replies
2
Views
846
B Understanding A Middle Ordinate In Terms Of Geometry
Replies
2
Views
2K
B Can You Solve This Simple Geometry Problem with Just a Ruler and Pencil?
Replies
23
Views
3K
What is the difference between superposition and intersection?
2
Replies
50
Views
7K
Solving Rotating Ice Skaters Problem: Is their Straight Line Parallel?
Replies
7
Views
2K
What is the equation of a circle
Replies
1
Views
2K
Complex Geometry: EQN of Circle, Parabola, Ellipse & Line
Replies
1
Views
1K
I Follow-up Einstein Definition of Simultaneity for Langevin Observers
Replies
4
Views
470

Hot Threads

Recent Insights

Back
Top