Adam wrote: << Oh, sorry about the lack of explanation. Areals (also known as barycentric coordinates) are a way of assigning a triple of values to any point in the (projective) plane, based on some arbitrary reference triangle. The vertices of the triangle (A, B and C) are assigned the areals (1,0,0), (0,1,0) and (0,0,1). All scalar multiples are considered equivalent, so (x,y,z) = (kx,ky,kz). The centroid of the triangle (G) is (1,1,1). The line at infinity is defined by x + y + z = 0. When the projective plane is extended so that (x,y,z) are complex instead of real, it can be shown that all circles pass through two points on the line at infinity, known as 'circular points'. Indeed, a circle can be defined as any conic passing through those two points; note that the degrees of freedom are reduced from five to three.
If I'm understanding, you are referring (in 2nd paragraph) to what topologists call CP^2, the complex projective plane. For any homogeneous polynomial P in C[x,y,z], the equation P(x,y,z) = 0 will define a surface in CP^2 (or "curve" to algebraic geometers), which is smooth under certain natural nonsingularity conditions. Assuming P(x,y,z) = 0 is smooth, then for deg(P) = d, the genus of that surface will be g = (d-1)(d-2)/2. The polynomial P(x,y,z) := x+y+z gives a smooth surface of genus g = 0, i.e., topologically the sphere S^2. CP^2 is kind of tricky to visualize, but one way to think of it is a 4-ball D^4 with an identification on its boundary bd(D^4) == S^3 == {(z,w) in C^2 : |z|^2 + |w|^2 = 1} via the Hopf map S^3 -> S^2, i.e., identifying (z,w) and (z',w') in S^3 if and only if (z',w') = (uz,uw) where u is a complex number with |u| = 1. (In fact no matter which point Q of CP^2 you start with, the points L(r) at a distance r > 0 from Q will form a 3-sphere S^3 as r increases, until it reaches the maximum distance from Q (= the diameter of CP^2), upon which L(r) will be a 2-sphere S^2. So I think what Adam (as well as maybe algebraic geometers in general) calls the "line at infinity" is topologically a 2-sphere S^2. But I'm puzzled by the comment that all circles pass through the "line at infinity" in two points, since one example of a circle (in the usual sense) in CP^2 is the set {[exp(it), 0, 1] in CP^2 : t in [0,2pi)}, which intersects x+y+z in only one point, [-1,0,1]. Or am I missing something? --Dan ________________________________________________________________________________________ It goes without saying that .