The Point $\O$ at Infinity

The graphs of the previous section are each missing a point at infinity. They are graphs in the plane $\mathbb{R}^2$. The plane is a subset of the projective plane $\P ^2$. The ``closure'' of the graph of $y^2 = x^3 +ax+b$ in $\P ^2$ has exactly one extra point $\O$, which has rational coordinates, and which we sometimes call ``the point at infinity''.

Definition 2.1   The projective plane $\P ^2$ is the set of triples $(a,b,c)$, with $a,b,c$ not all 0, modulo the equivalence relation

$$(a,b,c)\sim (\lambda a ,\lambda b , \lambda c)$$

for any nonzero $\lambda$. We denote by $(a\!:\!b\!:\!c)$ the equivalence class of $(a,b,c)$.

The ``closure'' in $\P ^2$ of the graph of $y^2 = x^3 +ax+b$ is the graph of

$$y^2 z = x^3 + axz^2 + bz^3$$

and the extra point is $\O =(0\!:\!1\!:\!0)$. All finite points are of the form $(a\!:\!b\!:\!1)$.

For more about the projective plane, see page 28 of [Kato et al.].