Circle With Radius of Zero

Date: 12/28/2004 at 15:22:59
From: Jack
Subject: A circle with a radius = 0?

Is it possible for a circle to have a radius that equals zero?  Is it
possible for a set of points (e.g., multiple points) to occupy the
same location?

One textook defines a circle as the set of points that is equidistant 
from a center point and that the radius is greater than or equal to 
zero.  I dispute this.  Am I wrong?

Date: 12/28/2004 at 22:48:10
From: Doctor Peterson
Subject: Re: A circle with a radius = 0?

Hi, Jack.

If the radius is zero, then it isn't really a circle, but might be
called a degenerate circle--that is, what you get if you slightly 
stretch the definition of a circle by using the same equation but
taking it to extremes by making the radius zero.  The point is (no pun
intended!) that many things you can say about a circle will still be
true if the radius is zero (making a single point), and they have for
some reason chosen to allow that.  I wouldn't do so, because there are
too many other things that would no longer work in that case.  I hope,
for example, that in theorems about tangents to a circle they specify
that the radius has to be greater than zero.  If not, then they are
inconsistent in their use of the definition, which is not uncommon in
textbooks.

By the way, there is nothing wrong in talking about a set of points
that consists only of one point; nothing in that wording should be
taken to imply that there are multiple points.  The problem with this
definition lies in the difficulty of writing theorems based on it, not
on how many points there are.

If you have any further questions, feel free to write back.

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

Date: 12/29/2004 at 02:14:15
From: Jack
Subject: Thank you (A circle with a radius = 0?)

Dear Dr. Peterson:

Thank you very much for your thoughtful reply.  Unfortunately, I lost
an argument, but I appreciate knowing the truth even more than being
"right."  :-)

- Jack