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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

\begin{displaymath}(x_{1},\ldots, x_{n+1})

A projective point with xn+1=0 corresponds to an ideal ``point at infinity'' in the (x1, ..., xn) direction in affine space. The set of all such ``infinite'' points satisfying the homogeneous linear constraint xn+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 xn+1 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.

Bill Triggs