Concentric circles are parallel?

[phya]

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.

 

[phya]

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.

 

[HallsofIvy]

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

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.

 

[HallsofIvy]

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.

 

[HallsofIvy]

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.

No, that is the new idea. Your notion of "parallel" being the same as "equidistant" is the old idea.

 

[phya]

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.

 

[G037H3]

Are you acknowledge a straight line own and own parallel? If you acknowledged that then curve own and oneself not parallel?

...

 

[phya]

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.

 

[phya]

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?

 

[phya]

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?

 

[CRGreathouse]

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.)

 

[phya]

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?

 

[phya]

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.

 

[HallsofIvy]

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!

 

[phya]

...

What your chart mainly wants to explain is what?

 

[G037H3]

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

 

[phya]

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.

 

[phya]

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?

 

[phya]

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.

 

[CRGreathouse]

Hyperbolic geometry's parallel concept is wrong. Does not intersect was not equal to that is parallel.

Mountain, meet Mahomet.

 

[Upisoft]

Perhaps your son will study the curve parallel theory.

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

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!

 

[phya]

Mountain, meet Mahomet.

:rofl:

 

[phya]

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.

 

[phya]

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.

 

[HallsofIvy]

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.

 

[phya]

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.

 

[HallsofIvy]

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.

 

[G037H3]

phya, i would like to reiterate:

lines are perfectly straight curves

all lines are curves, but not all curves are lines

 

[phya]

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?

 

[phya]

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.

 

[G037H3]

but a line is perfectly straight

so a 'curving line' is just a curve that isn't a line

 

[phya]

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.

 

[Mentallic]

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.

 

[phya]

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?

 

[G037H3]

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