Affine-projective Relationship: Applications

We have shown, at the beginning of chapter 4 why it is necessary to go beyond the framework of affine spaces, adjoining to them points at infinity This is now possible in the following way: we associate to the affine space X its universal space \( \hat X\) (cf. 3.D), which is a vector space in which X is embedded as an affine hyperplane whose direction is a vector hyperplane of \(\hat X\). Considering now the projectivization \(\tilde X = P\left( {\hat X} \right)\) , we see that \(\tilde X\) is the disjoint union of two sets: P( X), which is canonically identified with X, and P (\(\vec X\)), which, being the space of lines in \(\vec X\), is also the space of directions of lines in X. We denote it by \({\infty _X} = P\left( {\vec X} \right)\) We write \( \tilde X = X \cup {\infty _X}\), and say that ∞x is the hyperplane at infinity in X.