Is there such a thing as "parallel circles?" Or can only lines be parallel?

Question

Our tech ed teacher says that circles withing circles are parallel. Our math techer says that only lines can be parallel and that our tech teacher probably should use the term "concentric circles." Who is right?

Answer 1

This is a purely semantic question.

The way I learned the definition of parallel in math class, I'd say your math teacher is right.

Your tech ed teacher can be right, too, though, provided that everybody understands that he's using a different definition of parallel than most mathematicians use (ie using it to describe circles a mathematician calls concentric). I wouldn't get too wrapped around the axle about who's right and who's wrong so long as you can understand what each teacher means he uses the terms in his own class.

Answer 2

In Euclidian geometry, parallel is for straight lines only

Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space:

Every point on line m is located exactly the same minimum distance from line l ('equidistant lines', including the degenerate case where m = l).
Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity in either direction).
Lines m and l are both intersected by a third straight line (a transversal) in the same plane, and the corresponding angles of intersection with the transversal are equal.
Parallel lines have to lie in the same plane. This is implied by the first characterization, and is a prerequisite for the latter two. In other words, parallel lines must be located in the same plane, and parallel planes must be located in the same three-dimensional space. A parallel combination of a line and a plane may be located in the same three-dimensional space. Lines parallel to each other have the same gradient.

Answer 3

Coplanar circles sharing the same center could be considered parallel, but concentric is a much better term.

Congruent circles on parallel planes with colinear centers could be considered parallel. You could argue whether they would still be parallel if their radii differed.

I can't believe your math teacher said only lines can be parallel, since planes can also be parallel.

Answer 4

No, there is not such a thing as "parallel circles". Circles within circles, where they have the same center point, are called "concentric circles".

Your math teacher is right.

Answer 5

In a strictly mathematical sense, parallel only applies to straight lines. It is the foundation of Euclid's fifth postulate, which must be obeyed by all Euclidian geometries. A mathematician would understand parallel in terms of this postulate.

But in vernacular it can be applied to other concept, such as concenticity.

Answer 6

technically speaking, lines have to be straight to be parallel. However, in a 3 dimensional sense, you could make the plane of two circles be parallel.

Answer 7

The word 'parallel' is derived from Greek words 'para'
(beside) and 'allElOn' (of one another). The meaning
'everywhere equidistant, or equidistant at all points
and not meeting' applies to concentric circles and
concentric spheres.

http://www.m-w.com/dictionary/parallel

The word can be used for designating curves or surfaces
which are everywhere equidistant.

http://www.bartleby.com/61/97/P0059700.html

The word is also used for 'parallel of latitude'
in Geography even while refering to them on
a globe. The word can be applied to curves which
are parallel or 'a road parallels the river'.

http://dictionary.reference.com/browse/parallel

"Parallel is one of a number of imaginary lines around
the Earth always at the same distance from the
equator".

http://dictionary.cambridge.org/define.asp?key=57477&dict=CALD

It applies to all lines, planes, or curved surfaces
in mathematics.

http://encarta.msn.com/encnet/features/dictionary/DictionaryResults.aspx?refid=1861723685
http://www.askoxford.com/concise_oed/parallel?view=uk
http://en.wikipedia.org/wiki/Parallel_%28curve%29

In Euclidean geometry it is more common to talk about
geodesics than (straight) lines. A geodesic is the path
that a particle follows if no force is applied to it.
In non-Euclidean geometry (spherical or hyperbolic) the
above three definitions are not equivalent; only the
second one is useful in other geometries. In general,
equidistant lines are not geodesics; so the equidistant
definition can "not" be used.

In general geometry, it is useful to distinguish the
three definitions above as three different types of
lines, respectively equidistant lines, parallel
geodesics and geodesics sharing a common perpendicular.

http://en.wikipedia.org/wiki/Parallel_%28geometry%29

Now, you tell me whether the word 'parallel'
can be used for the follwing:

(1) The railway line that runs by the side of road.
http://en.wiktionary.org/wiki/parallel

(2) A trench cut in the ground before a fortress, by the
side of its defenses, for the purpose of covering a
besieging force.

http://www.infoplease.com/dictionary/parallel

(3) A jointed system of links, rods, or bars, by which the
motion of a reciprocating piece, as a piston rod, may
be guided exactly in a straight line.

http://machaut.uchicago.edu/cgi-bin/WEBSTER.sh?WORD=parallel

Answer 8

Circles within circles are called concentric. This is true if they have the same center.

Answer 9

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