24
$\begingroup$

I am curious about the following diagram:

enter image description here

The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and proven formally? Under what definition of 'circle' and 'line' does this hold?

Thanks!

  • 0
    Perhaps a "generalised circle" is what you are after (http://en.wikipedia.org/wiki/Generalized_circle)2011-11-15
  • 1
    Maybe it has to see with the fact that, under the stereo projection, great circles thru the north pole unwrap into lines of infinite length in the complex plane.2011-11-15
  • 0
    I find it amusing that robjohn's/J. M.'s answers (which are similar) contradict Gerry Meerson's answer, but I think both are useful.2011-11-15
  • 0
    @Ross, I don't know who Gerry Meerson is, but I note that my answer and those of robjohn and J. M. all refer to taking limits as the radius increases. I don't see any contradiction.2011-11-15
  • 0
    @GerryMyerson: Sorry for the typo. I had it wrong and tried to fix it. I was referring to the fact that you say there is no circle of infinite radius and the others accept it, but the sense is the same.2011-11-15

8 Answers 8