Canonical Injection of $I\!\!R^{n}$ into ${I\!\!P}^{n}$

Affine space $I\!\!R^{n}$ can be embedded isomorphically in ${I\!\!P}^{n}$ by the standard injection $(x_1,\ldots,x_n)\longmapsto(x_1,\ldots,x_n,1)$. Affine points can be recovered from projective ones with $x_{n+1}\not=0$by the mapping

$$(x_{1},\ldots, x_{n+1}) \sim (\frac{x_{1}}{x_{n+1}},\ldots,\f... ...ongmapsto (\frac{x_{1}}{x_{n+1}},\ldots,\frac{x_{n}}{x_{n+1}})$$

A projective point with x_n+1=0 corresponds to an ideal ``point at infinity'' in the (x<sub>1</sub>, ..., x<sub>n</sub>) direction in affine space. The set of all such ``infinite'' points satisfying the homogeneous linear constraint x_n+1=0 behaves like a hyperplane, called the hyperplane at infinity.

However, these mappings and definitions are affine rather than projective concepts. They are only meaningful if we are told in advance that $(x_1,\ldots,x_n)$ represents ``normal'' affine space and x<sub>n+1</sub> is a special homogenizing coordinate. In a general projective space any coordinate (or linear combination) can act as the homogenizing coordinate and all hyperplanes are equivalent -- none is especially singled out as the ``hyperplane at infinity''. These issues will be discussed more fully in chapter 4.