Parallel Curves

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.

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/   

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 
(third grade) got into a discussion about 'what if' scenarios. 
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)

Thanks in advance for your help!
Kim Given

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/