Talk:Circular points at infinity

Where do these points lie?
The article states "These points are called circular points at infinity because they lie at infinity, on that line, and they also lie on all circles." This phrase is ambiguous at best, and probably incorrect. The line at infinity is a CP1 in other words, an S2. No point on this sphere lies on all of its circles! The correct statement appears to be that the circular points lie on the complexification of every real circle. Tkuvho (talk) 12:25, 21 April 2010 (UTC)
 * They lie on all circles in the sense that they lie on the curves when extended to the complex projective plane. The the classical theory of algebraic curves it's usually implicitly assumed that a curve lives in the complex projective plane, otherwise you don't get Bézout's theorem which is fundamental. In this point of view, the real curve is just a particularly interesting slice of the larger object.--RDBury (talk) 13:28, 26 April 2010 (UTC)