Can Two Curves Be Parallel?

Date: 12/19/2007 at 10:33:25
From: Debbie
Subject: Is there such a thing as parallel graphs?

Straight lines are parallel if they are equally distant and never 
intersect.  Can the graphs of quadratic or cubic equations be 
considered parallel if they are equally distant and never intersect? 

The graphs of the quadratic or cubic equations meet the criteria of 
parallel lines - equally distant and do not intersect--but I've never 
heard of them described as "parallel graphs".  One of my bright 8th 
grade algebra students asked this question and I want to give him an 
accurate answer.

Date: 12/19/2007 at 19:44:50
From: Doctor Wallace
Subject: Re: Is there such a thing as parallel graphs?

Dear Debbie,

What an interesting question!

Although I think the most common use of the word "parallel" is lines 
in Euclidean geometry that lie in a plane yet never intersect, the 
word can be applied to other ideas.

Merriam Webster's dictionary says that parallel can mean

  (1) similar, analogous, or interdependent in tendency or 
      development

  (2) exhibiting parallelism in form, function, or development
      <parallel evolution>

  (3) extending in the same direction, everywhere equidistant, and not 
      meeting <parallel rows of trees> b: everywhere equally distant 
      <concentric spheres are parallel>

Note that last example they give:  concentric spheres are parallel.

I also consulted the Harper Collins Dictionary of Mathematics (Third 
Printing), which lists, in addition to the usual Euclidean meaning:

  parallel (of a set of curves):  remaining a constant distance apart

as one of its entries for this word.

So, even though I've personally not heard the word applied to a set of 
curves, say quadratic or cubic, that remain a constant distance apart 
and never intersect, or to objects such as spheres, these two sources 
say it is a correct use of the term.

Thanks for writing to Dr. Math!

- Doctor Wallace, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/21/2007 at 20:14:02
From: Doctor Rick
Subject: Re: Is there such a thing as parallel graphs?

Hi, Debbie.

May I add something to what Dr. Wallace told you in answer to your 
question:

The following well-respected math Web site defines parallel curves 
as Dr. Wallace found in another source:

  Wolfram MathWorld: Parellel Curves
    http://mathworld.wolfram.com/ParallelCurves.html 

However, note that the distance between curves is measured 
perpendicular to the curves.  I don't know for sure what your student 
was picturing for quadratic equations that are a constant distance 
apart, but I suspect it was incorrect.  The graphs of, say, y = x^2 
and y = x^2 + 4 are NOT parallel curves, because the constant 
"distance" of 4 is measured vertically, not perpendicular to either 
curve.

My gut feeling is that a curve parallel to a quadratic will not be a 
quadratic, and probably not a polynomial at all.  Curves parallel to 
a circle are indeed concentric circles, but curves parallel to a 
(non-circular) ellipse are not ellipses, as you can see in the 
MathWorld figure.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/