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Parallel Curves

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Date: 05/21/97 at 19:27:01
From: Melodie A. Bernhard's students
Subject: Geometry

We know that line segments can be parallel, but can curves be
parallel?

We looked up the definition of parallel in our math book and in
several others. We still aren't sure.
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Date: 05/27/97 at 17:03:04
From: Doctor Rob
Subject: Re: Geometry

Well, that depends!  It depends on exactly what the wording is when
you define the word "parallel."

If you mean that the curves never intersect, then there can be
parallel curves (can you think of an example?) If you mean that they
intersect "at infinity," but nowhere else, then there can be parallel
curves.  If you mean that the curves are similar (have the same shape)
and never intersect, that is a more difficult problem, but there is
the case of concentric circles.

A meaning I find interesting is that there is a certain minimum
distance between any point on one curve and any point on the other,
call it d, and, given any point on one curve, there is a point on the
other curve with distance d between them. Again the concentric circle
example serves, but now you can create more examples.

I hope this helps.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Date: 05/30/97 at 10:38:52
From: matt
Subject: Latitude and Parallel Lines

Dear Dr. Math,

Are the lines of latitude parallel?

Thanks!
Chris and Matt

Teacher's note:

We were discussing parallel lines in class today and the students
We learned about longitudinal lines in non-Euclidean geometry, but we
would like to know if concentric circles are considered parallel in
Euclidean geometry (I'm assuming that they are in non-Euclidean
geometry)

Kim Given
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Date: 05/31/97 at 08:14:50
From: Doctor Jerry
Subject: Re: Latitude and Parallel Lines

Hi Chris and Matt

Mathematicians tend to be fussy about language; this annoys some
people but does save a lot of argument.

Euclid talks about lines and by that term means "straight lines."
Two lines in a plane are parallel if they are identical or never
intersect.

In space, it's more complicated. Lines can fail to intersect in space
but can still be non-parallel. Can you give an example of this?

In space two planes are parallel if they are identical or never
intersect.

One often speaks of "parallels of latitude." I suppose that the term
parallel was used since the planes containing the circles of equal
latitude are parallel. The term "lines of latitude" may mean the
curves on a map that correspond to the parallels of latitude.
Depending on the mathematical theory on which the map is based, the
curves corresponding to the parallels of latitude may be parallel
lines in the plane of the map (Mercator projection, for example) or
curves or lines that intersect (polar projection, for example, in
which all lines corresponding to the parallels of latitude meet at the
pole).

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Middle School Definitions
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Two-Dimensional Geometry

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