Canonical injection of $\mathbb{R}^n$ into ${\cal P}^n$ and Points at infinity

• Affine space $\mathbb{R}^n$ can be embedded in ${\cal P}^n$ by
  $$(x_1,\ldots,x_n) \rightarrow (x_1,\ldots,x_n,1)$$

• Affine points can be recovered from projective points with $x_{n+1} \neq 0$ by
  $$(x_1,\ldots,x_n) \sim (\frac{x_1}{x_{n+1}},\ldots,\frac{x_n}{x_{n+1}},1) \rightarrow (\frac{x_1}{x_{n+1}},\ldots,\frac{x_n}{x_{n+1}})$$

• A projective point with $x_{n+1} = 0$ corresponds to a point at infinity.
• The ray $(x_1,\ldots,x_n,0)$ can be viewed as an additional ideal point as $(x_1,\ldots,x_n)$ recedes to infinity in a certain direction. For example, in ${\cal P}^2$ ,
  $$\lim_{T \rightarrow 0} (X/T,Y/T,1) = \lim_{T \rightarrow 0} (X,Y,T) = (X,Y,0)$$

It is important to note that these are affine concepts.