Can curves be parallel to each other？

In summary, the concept of parallel can be applied to both straight lines and curves. Straight lines not intersecting is a specific example of parallel, while there are many examples of curves that can be considered parallel, such as concentric circles. The defining characteristic of parallel is a constant distance between the lines or curves. Even in non-Euclidean geometry, parallelism can still exist between lines or curves, but it may not always be defined by their equidistance.
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mg1912
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they can. when we are talking about parallel, it can be concluded into 2 situations, the parallel of straight lines and the parallel of curves. the situation of straight lines is just a particular example of paralle.
there are many example of parallel of curves, for example, concentric circles, tortuous railway and so on. the essential character of parallel is a certain unchange distance, the concentric circles is a good example of this. at the same time circles are also curves. so curves can be parallel to each other.
 
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  • #2
Even straight parallel lines can intersect in non-Euclidean geometry, what's new?
 
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mg1912 said:
they can. when we are talking about parallel, it can be concluded into 2 situations, the parallel of straight lines and the parallel of curves. the situation of straight lines is just a particular example of paralle.
there are many example of parallel of curves, for example, concentric circles, tortuous railway and so on. the essential character of parallel is a certain unchange distance, the concentric circles is a good example of this. at the same time circles are also curves. so curves can be parallel to each other.
Are you asking or simply asserting?

You are going to have to tell us what you want to mean by "parallel". The usual definition of "parallel lines" is "lines that do not intersect". Of course there exist curves that do not intersect. Another is "have constant distance apart"- "equidistant lines". There exist curves of that kind: two concentric circles with different radii are, as you say, equidistant. (Of course, in non-euclidean geometry, equidistant line may not be parallel.
 

1. Can curves be parallel to each other?

Yes, curves can be parallel to each other. Parallel curves are defined as curves that maintain a constant distance between each other at all points. This means that the curves never intersect and have the same slope at every point.

2. How do you determine if two curves are parallel?

To determine if two curves are parallel, you can check if they have the same slope at every point. If the slopes are the same, the curves are parallel. Another method is to check if the distance between the curves remains constant at all points.

3. Can parallel curves have different shapes?

Yes, parallel curves can have different shapes. As long as the distance between the curves remains constant at all points and they have the same slope, they are considered parallel.

4. Are parallel curves always in the same direction?

No, parallel curves do not have to always be in the same direction. They can be parallel in opposite directions, as long as they maintain a constant distance between each other and have the same slope at every point.

5. Can curves in different planes be parallel?

Yes, curves in different planes can be parallel. Two curves are considered parallel if they maintain the same distance between each other at all points, regardless of the planes they are in. This means that two curves can be parallel even if they are not in the same plane.

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