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.
I do not know exactly what you mean by the "outside" and "inside" lines on a
lock, but I can tell you that curves can be considered to be parallel. Before
talking about these curves, let us review what two parallel lines are: two
lines in a plane that do not intersect. These lines are straight and this
definition is for Euclidean geometry (the kind we study most in high school).
In non-Euclidean geometry, we can talk about "curved lines" called geodesics,
in curved spaces. In two-dimensions, consider a curved "plane" like the
surface of a sphere. A great circle has the circumference of the sphere and
divides the sphere into two equal hemispheres. Even though the earth is not
a perfect sphere, you can consider the equator to be a great circle. A
segment on a great circle is called a geodesic. Two geodesics are considered
parallel if they do not intersect anywhere on the curved surface on which they
That may be a lot more information than you wanted, but without seeing your lock,
I think there is a pretty good chance that you and your teacher may not have
clarified whether you were speaking about Euclidean or non-Euclidean geometry.
From a non-Euclidean perspective, your two curves may indeed have been parallel.
Dr. Eric Hagedorn
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Update: June 2012