# Concentric circles are parallel?

1. Sep 23, 2010

### phya

The straight line parallel to each other is parallel. Concentric circles are parallel,too.

As shown in figure, There is a big circle,Oa，Another one is small, Oc.They are concentric circles. AB is a straight line. AB and Oa are intersections D, AB and Oc are intersections C. EF is a straight line. EF through point D. EF tangent and Oa. GH is a straight line. GH through point C. GH tangent and Oc. R is for Oa radius. r is for Oc radius.

set
β=∠BCH, α=∠BDF, G=CD

is
∠BCO=90+β

According to the cosine theorem:

G^2=r^2+R^2-2rRcos(180-(90-α+90+β)
After finishing to
G^2=r^2+R^2-2rRcos（α-β） （2）

After finishing (2)
cos（α-β）=（r^2+R^2-G^2）/2rR
If R →∞，r→∞, then
cos（α-β）→1
α→β

When R → ∞， Oa is a straight line， r → ∞，Oc is a straight line，too. This is straight line parallel!Therefore, straight line parallel to the curve of the parallel is special.

After the above discussion, I still have some conclusions are as follows:

Can mutually perpendicular lines, Curve can also mutually vertical.

Flat surface can be parallel, curved surface can also be parallel to each other.

......

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2. Sep 23, 2010

### HallsofIvy

Staff Emeritus
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.

3. Sep 23, 2010

### phya

Lines divided into curve and linear.All is not straight line.Why can't curve parallel?

4. Sep 23, 2010

### Ynaught?

I understand the definition of parallel lines to be, when two lines in a plane equidistant part at every point and never intersecting they are parallel. Nothing in that states the lines need to be straight. Which must be the true part you refer to. But I don't see the not true part... Unless maybe that lines of latitude when viewed from the pole appear to be concentric circles but they are not parallel because they do not lie in the same plane?

5. Sep 24, 2010

### phya

Three-dimensional space straight line can be parallel.Three-dimensional space curved line can also be parallel.

Last edited: Sep 24, 2010
6. Sep 24, 2010

### phya

Does not intersect was not equal to that is parallel. For example, two curves do not intersect, but actually not necessarily is parallel. Two straight lines do not intersect only then possibly are parallel.

Why said that possibly is parallel? Please read in the appendix the chart. In chart two straight lines not parallel, also does not intersect. These two lines are infinite long, but they do not intersect, they are not also parallel.

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7. Sep 24, 2010

### Mentallic

If you have some curve y=f(x) then you can use your own definition to say that another curve y=f(x)+c where c is some non-zero constant is parallel to the other. This is generally not considered the true definition of parallel. Parallel is only used to describe lines and planes, not curves.

How are the two lines in that diagram infinitely long? You can see where they end!

8. Sep 24, 2010

### HallsofIvy

Staff Emeritus
Read in what appendix? Again, please state your definition of "parallel"! These statements are true for some definitions of parallel and not for others. If you do not definie "parallel" people will be forced to assume you mean "parallel" as defined in Euclidean geometry where these statements you make are NOT true.

9. Sep 24, 2010

### Mentallic

He means the diagram at the end of his post.

10. Sep 24, 2010

### phya

The definition is reflects the nature, is to the nature induction and the summary. A parallel definition kind of natural phenomenon. The curve parallel is also one kind of natural phenomenon. Should also contain this kind of phenomenon in humanity's parallel concept. A parallel key character is the distance maintains invariable. Regardless of being the straight line, the curve is so. This is the parallel essence. It is the straight line or the curve, this is unimportant.Humanity's understanding is parallel starts from the straight line, therefore the humanity is limited easily. Therefore the humanity knows the non-European geometry with difficulty.

11. Sep 24, 2010

### phya

The definition is reflects the nature, is to the nature induction and the summary. A parallel definition kind of natural phenomenon. The curve parallel is also one kind of natural phenomenon. Should also contain this kind of phenomenon in humanity's parallel concept. A parallel key character is the distance maintains invariable. Regardless of being the straight line, the curve is so. This is the parallel essence. It is the straight line or the curve, this is unimportant.Humanity's understanding is parallel starts from the straight line, therefore the humanity is limited easily. Therefore the humanity knows the non-European geometry with difficulty.

12. Sep 24, 2010

### Mentallic

You obviously don't know what the word "define" means, since we've mentioned it to you dozens of times already.

lol non-Euclidean?

13. Sep 25, 2010

### phya

Non-Euclidean Geometry？

Last edited: Sep 25, 2010
14. Sep 25, 2010

### phya

We may the narrow definition parallel, is also parallel is only about straight line between being parallel. But it in fact, parallel may also be generalized. Before we think the geometry only then the Euclid geometry, but does not have other geometries, now we knew that also has the non-Euclid geometry. After having discovered non-Euclid geometry, we may say that the Euclid geometry is only the narrow geometry.Not right?

15. Sep 25, 2010

### Mentallic

If you don't know what euclidean geometry is, please explain what you meant by european geometry :tongue2:

In non-euclidean geometries such as spherical and hyperbolic geometries, we use the term "geodesics" rather than straight lines so as not to confuse the two. Since parallel lines can intersect in spherical geometry which disobeys our definition of parallel (it is assumed in these definitions that we are using euclidean geometry anyway).

The definition of parallel is not generally extended to curves, but you can make it that way if you like. This doesn't mean you are going to convince us all that the definition of parallel needs to be extended because you think it does.

16. Sep 25, 2010

### phya

The slip of the pen is the very normal matter. Has the understanding to the geometry the human, actually does not know the non-Euclid geometry, this possible? But ridicules others because of others' slip of the pen, should not.

17. Sep 25, 2010

### HallsofIvy

Staff Emeritus
No one has ridiculed you. We have, however, repeatedly asked you to define what you mean by "parallel" any you have not done so.

18. Sep 25, 2010

### phya

Ask when the Euclid geometry how to define parallel?

19. Sep 25, 2010

### HallsofIvy

Staff Emeritus
I assume that English is not your first language. Euclidean geometry does NOT "define parallel". It does define "parallel line" and, as I said before, in Euclidean geometry, parallel only applies to straight lines. So apparently, in your question you are NOT talking about Euclidean geometry. I ask, for the third time, how do you define "parallel" in this question?

20. Sep 26, 2010

### phya

Indeed, in the Euclid geometry, parallel is between straight line being parallel. I have a question, ask, the line segment is may also be parallel? Sorry, please use is or is not replied.