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## Concentric circles are parallel?

 Quote by CRGreathouse 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.

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 Quote by 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.
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!

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

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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.
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 Quote by 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.
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.

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 Quote by 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.
The attention, is must simultaneously be vertical to two curves.
 To add concentric circles to our definition of "parallel lines," you would have to prove all the rules for parallel lines apply. One of those rules has to do with parallel lines intersecting other parallel lines (corresponding angles are equal). Can you show me even one that works? (ALL should work if what you say should be accepted.) Attached Thumbnails

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 Quote by AC130Nav To add concentric circles to our definition of "parallel lines," you would have to prove all the rules for parallel lines apply. One of those rules has to do with parallel lines intersecting other parallel lines (corresponding angles are equal). Can you show me even one that works? (ALL should work if what you say should be accepted.)
The straight line parallel has the phase angle to be equal, generally, the curve parallel does not have the corresponding angle to be equal.

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