Parallelism and Concentric Circles
Parallel lines. I was playing with a lock in geometry
class on day and my teacher asked me if the outside line and the
inside line of the lock are parallel. said that it was, but he told
me that it could not be, but he never gave me a reasonable
explanation. I think that the inside and outside lines could in fact
be parallel. Could you please give me an explanation as to why they
are not considered parallel? Everything I have looked up about them
never says anything about the lines not being able to curve like
that in an arch.
By definition, parallel lines must be straight lines in the same plane.
So even though the lines do not intersect, they are not parallel.
Scott P. Smith
Lines are, according to Wikipedia, perfectly straight. This coincides with
the idea that equations of lines are of the form ax + by = c or the possibly
more familiar y = mx + b (slope - intercept form). So your teacher is
correct. Two arcs that never meet, your lock hasp, would be considered
concentric - that is they have a common center but different radii.
Concentric arcs never meet either.
It may seem like semantic games, but the field of mathematics is one for
rigid and precise definitions within a strict logical and semantic
Mathematics is often in the forefront of human thought and deals with
questions that are so "far out" they may not be obviously tied to the
"real world". The way a mathematician can be sure they are correct is
if other mathematicians can follow each and every step of their argument
and find no "holes" or logical errors. Thus they are extremely precise
and tend to break things down into a series of steps, each of which has
its own proofs.
That being said, there is very little mathematics out there that has NOT
be applied to other problems once technologists and other scientists get
hold of it. So if math seems a bit over precise, cut it some slack. We
all benefit from the work mathematicians do and have done.
The answer does in fact depend on what you mean by parallel. In the
strictest sense, a line does not curve. Parallel lines must be in the
same direction everywhere. The curves are only parallel at the closest
points. The top of one circle is not parallel to the left side of the
In some areas of science and engineering, the term parallel can be used
in a variety of ways. One refers to two paths that are identical but
offset. Two circles of the same size but with different centers would
qualify. Another use refers to two paths that always maintain the same
distance from each other. Closest points between the paths always have
the same distance. In this case, concentric circles are parallel.
Objects traveling along parallel lines will always be moving in the same
direction as each other. Objects traveling along concentric circles
only maintain same directions if they are both traveling at the same
number of revolutions per second.
Dr. Ken Mellendorf
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Update: June 2012