<p><pre><code>  &#62; Actually, having an infinity at the end of
  &#62; every line on the complex plane just gives
  &#62; you a simple disc.
</code></pre>
Er, judging from the rest of your comment you must simply have misunderstood me.  For every line through 0 in the complex plane, take a point and define it to be at both ends of the line.  That is doing as you say, associating antipodal points.<p>And I do do calculus (well, equivalent procedures) on such objects.  It's clear from your comments that you know about these things - trust me that I do as well.<p>So we're agreed that the concept of infinite sized sets and the concept of "points at infinity" are different, and for teaching about "infinity" it might be useful to separate them explicitly.<p>Interesting observation.  Thanks.