Problems in Geometry pp 30-34 | Cite as

# Affine-projective Relationship: Applications

## Abstract

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*.

## Keywords

Projective Space Disjoint Union Projective Geometry Conic Section Affine Space## Preview

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