Concentric circles are parallel?

[phya]

In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. http://mathworld.wolfram.com/ParallelLines.html" , Therefore, The parallel essence is not does not intersect, but is away from constantly invariable.

 

[CRGreathouse]

In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. http://mathworld.wolfram.com/ParallelLines.html" , Therefore, The parallel essence is not does not intersect, but is away from constantly invariable.

So would the helix x = sin t, y = cos t, z = t be phya-parallel to the line x = 0, y = 0, z = t?

 

[phya]

Curve parallel axiom is:

If the plane B parallel to the plane D, the distance between them is E, A straight line is a straight line on the plane B, obviously, A parallel to B, A to B on the distance of any point is E.

B in the plane, we bend A, so A a circle C, so, C to B at any point to the distance is still E, then, C is parallel to the B it? Obviously, the answer is yes.

Because, B to D on the distance of any point is E, therefore, B is parallel to D, or, at any point, B parallel to the D, so, B parallel to the D. The arbitrary point A parallel to B, so C is also parallel to the B.

Therefore, the curve can be parallel to the plane.

Therefore, the curve and the curve can be parallel to each other, as long as the distance between them remained unchanged.

Curve parallel axiom is:

At most one curve can be drawn through any point not on a given curve parallel to the given curve in a plane.

Attached is an animated map shows the curve parallel to the truth.

 

[phya]

So would the helix x = sin t, y = cos t, z = t be phya-parallel to the line x = 0, y = 0, z = t?

I may tell you in the appendix animation the spiral line parallel truth. In this animation, red line and blue color line parallel, if their distance is invariable, on the column surface, they becomes two spiral lines, that they were still parallel.

 

[phya]

So would the helix x = sin t, y = cos t, z = t be phya-parallel to the line x = 0, y = 0, z = t?

The circle is also a parallel line actually. The circumference and the center of circle are parallel.

 

[Mentallic]

A point is phya-parallel too?

 

[sachinism]

A point is phya-parallel too?

whats meant by phya-parallel ?

 

[Mentallic]

I'm not exactly sure. It's Phya's definition of parallel so you should ask him

 

[HallsofIvy]

Phya has been asked repeatedly, through 189 posts on this thread and one or two other threads on basically the same thing, to explicitely give his definition of "parallel". He has not yet done so. I rather suspect that he has no idea what a mathematical definition is.

 

[phya]

whats meant by phya-parallel ?

Actually I have said many times, parallel is the constant distance. But some people just don't listen.

equivalent statements to the parallel postulate :

There exists a pair of straight lines that are at constant distance from each other.

 

[HallsofIvy]

Actually I have said many times, parallel is the constant distance. But some people just don't listen.

equivalent statements to the parallel postulate :

There exists a pair of straight lines that are at constant distance from each other.

That is true in Euclidean geometry but you have also said that you are not talking about "parallel lines".

 

[HallsofIvy]

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?

Actually, you have given a number of different definitions of "parallel" which are equivalent in Euclidean geometry.

But you titled this thread "Concentric circles are parallel?" which implies that you are NOT using Euclid's definition of "parallel" which requires lines, not curves. The statement that "two curves are parallel to each other if they are always the same distance apart" is NOT equivalent to Euclid's definition.

You have also asserted over and over again that "a curve is parallel to itself" despite repeated attempts to tell you that that violates the definition given, even for parallel curves, in textbooks. You seem to be asserting that all textbooks are wrong just because you do not agree with them.

You also are skipping over the question of how you measure the 'distance between lines'. The standard definition of the distance between a point on one curve and a second curve is the distance measured along a line perpendicular to the second line. But are you aware that two curves with a constant "distance" between them, in that sense, may intersect?

 

[phya]

That is true in Euclidean geometry but you have also said that you are not talking about "parallel lines".

The attention, my definition contains the concentric circle parallel.

 

[CRGreathouse]

I take from posts #183 and #184 that curves A and B are phya-parallel if they can be parameterized as functions a(t) and b(t) such that for all t, the distance from a(t) to b(t) is constant.

My guess is that "if" above can be replaced by "iff": that is, if you cannot parameterize the curves A and B with equidistant functions, then A and B are not phya-parallel.

phya, is this right?

 

[phya]

A point is phya-parallel too?

Circumference is parallel to the center of the circle, so the circle became a circle, because the ellipse is not parallel, so the ellipse as the ellipse.

 

[phya]

Actually, you have given a number of different definitions of "parallel" which are equivalent in Euclidean geometry.

But you titled this thread "Concentric circles are parallel?" which implies that you are NOT using Euclid's definition of "parallel" which requires lines, not curves. The statement that "two curves are parallel to each other if they are always the same distance apart" is NOT equivalent to Euclid's definition.

You have also asserted over and over again that "a curve is parallel to itself" despite repeated attempts to tell you that that violates the definition given, even for parallel curves, in textbooks. You seem to be asserting that all textbooks are wrong just because you do not agree with them.

You also are skipping over the question of how you measure the 'distance between lines'. The standard definition of the distance between a point on one curve and a second curve is the distance measured along a line perpendicular to the second line. But are you aware that two curves with a constant "distance" between them, in that sense, may intersect?

If our original definition crow is the black, afterward we discovered that the crow also has the white, at this time, whether we should revise the original definition?

Newton's time's definition is different with Einstein's time's definition, after the theory of relativity appears, we should revise Newton's definition?

 

[phya]

I take from posts #183 and #184 that curves A and B are phya-parallel if they can be parameterized as functions a(t) and b(t) such that for all t, the distance from a(t) to b(t) is constant.

My guess is that "if" above can be replaced by "iff": that is, if you cannot parameterize the curves A and B with equidistant functions, then A and B are not phya-parallel.

phya, is this right?

If in the attached figure animation shows: If B plane and D plane parallel, the distance is E. A is a straight line, and A on B, therefore, A to the B distance is also E, when A becomes circumference C, C to the B distance is also E obviously, therefore, the distance is constant, therefore, C is also parallel to B.

 

[phya]

If a line (straight line either curve) to another line (straight line or curve) the distance maintains invariable, then these two lines (straight line or curve) are parallel.

 

[Dr Lots-o'watts]

Straight lines have the distinction that if I measure the distance at say 45 deg, the distance separating them is indeed invariable.

Not so for any other other type of (curved) line. Plus at some point, you have to decide if phya-parallel includes intersections, touching, phase shifts, is it limited to co-planar curved lines or not? Is a circle phya-parallel to the dot at its center? If so, then can I say the center-dot is parallel to the circumference? Can a dot then also be parallel to an ellipse? If so, then will a circle be parallel to an ellipse? If the circumferences are dotted, then two concentric (such a nice word - concentric! such a shame it would be to delete it from our vocabulary) circles don't have the same number of dots so there is no correspondence, are they still phya-parallel? If two concentric circles are phya-parallel, then are there other lines, that are of different lengths that can be parallel? What about two spirals, occupying 3 dimensions? Does phya-parallel include spirals of different wavelengths, of different radius? Are two squares, one within the other, phya-parallel? Then why not two triangles? Why not two houses? Are Russian dolls phya-parallel to each other? Where do I end?

Ah the simplicity of Euclid's definition!

(I do hope this debate is not an attack on Euclid himself because he's Greek or something, because I've always seen math as a nice place where everybody was civilized and able to get along and not waste time on such issues. Yay for Euclid and al-Khwārizmī! The ultimate tag team! http://www.fansonline.net/images/wrestling/TheMegapowers2.jpg)

 

[CRGreathouse]

If in the attached figure animation shows: If B plane and D plane parallel, the distance is E. A is a straight line, and A on B, therefore, A to the B distance is also E, when A becomes circumference C, C to the B distance is also E obviously, therefore, the distance is constant, therefore, C is also parallel to B.

That didn't answer the question. Even this many pages into the thread, you still haven't told us what you mean by parallel (what the rest of us call phya-parallel).

I'm done with this thread. If someone is able to formalize a definition for phya-parallel, feel free to message me and I'll check back to see what it is. Until then, have fun!

 

[phya]

Is a circle phya-parallel to the dot at its center? If so, then can I say the center-dot is parallel to the circumference? Can a dot then also be parallel to an ellipse?

Please note, my words were such say:Circumference is parallel to the center of the circle, so the circle became a circle, because the ellipse is not parallel, so the ellipse as the ellipse.

 

[Mentallic]

I'm done with this thread.

Considering you landed smack bang on the 200^th post, I hope you truly are done with it

I've been able to argue for weeks at a time over the net with others, but those were on controversial topics such as global warming and what-not. This is just getting ridiculous though... I could never see myself reading that entire book which proves 1+1=2, and simultaneously I could never see myself pursuing this discussion any further than I already have.

If you feel like you have just theorized the relativity of mathematics, then by all means take your findings and present them to a professor or something.
Oh and then get back to us on the verdict

 

[phya]

What about two spirals, occupying 3 dimensions? Does phya-parallel include spirals of different wavelengths, of different radius? Are two squares, one within the other, phya-parallel? Then why not two triangles? Why not two houses? Are Russian dolls phya-parallel to each other? Where do I end?

You had not understood that the concentric spherical surface was certainly each other parallel.

 

[phya]

What about two spirals, occupying 3 dimensions? Does phya-parallel include spirals of different wavelengths, of different radius? Are two squares, one within the other, phya-parallel? Then why not two triangles? Why not two houses? Are Russian dolls phya-parallel to each other? Where do I end?

Starts, the red straight line and the blue color straight line is parallel, afterward they became the spiral line, if their distance were invariable, therefore they were each other parallel. This is not the very simple truth?

 

[phya]

That didn't answer the question. Even this many pages into the thread, you still haven't told us what you mean by parallel (what the rest of us call phya-parallel).

I'm done with this thread. If someone is able to formalize a definition for phya-parallel, feel free to message me and I'll check back to see what it is. Until then, have fun!

If a line (straight line either curve) to another line (straight line or curve) the distance maintains invariable, then these two lines (straight line or curve) are parallel.

 

[HallsofIvy]

If a line (straight line either curve) to another line (straight line or curve) the distance maintains invariable, then these two lines (straight line or curve) are parallel.

I suspected that you did not know what a mathematical definition was. What you are giving, over and over again, are "characterizations" or "examples', not definitions. You seem to be saying that two "lines" (which, in your definition can be curves) are parallel if and only if they "maintain" a constant distance. But for that to be a complete definition, You must tell exactly how you are defining the "distance" between two curves- and there are a number of quite different ways of doing that. And, as I said before, depending on exactly how you define that distance, you might find that there are examples of curves that are "parallel" in your definition but also intersect!

 

[phya]

you might find that there are examples of curves that are "parallel" in your definition but also intersect!

Please give an example!

 

[HallsofIvy]

A "conchoid" is a curve that loops back on itself. Another conchoid, just slightly distant from the first, where the "distance" is defined along a mutual perpendicular, will have constant distance yet intersects- the points of intersection on the curves not having a mutual perpendicular so that, even though the curves intersect, the "distance" between them is not 0. As I said, you have to be careful how you define the "distance" between two curves.

 

[phya]

A "conchoid" is a curve that loops back on itself. Another conchoid, just slightly distant from the first, where the "distance" is defined along a mutual perpendicular, will have constant distance yet intersects- the points of intersection on the curves not having a mutual perpendicular so that, even though the curves intersect, the "distance" between them is not 0. As I said, you have to be careful how you define the "distance" between two curves.

You said right, on the exploration path, we must be careful. In the plane, between two curve's distances to be vertical to two curve line segment length.

 

[phya]

A "conchoid" is a curve that loops back on itself. Another conchoid, just slightly distant from the first, where the "distance" is defined along a mutual perpendicular, will have constant distance yet intersects- the points of intersection on the curves not having a mutual perpendicular so that, even though the curves intersect, the "distance" between them is not 0. As I said, you have to be careful how you define the "distance" between two curves.

The attention, is must simultaneously be vertical to two curves.