Affine Space - Relation To Projective Spaces

Relation To Projective Spaces

Affine spaces are subspaces of projective spaces: an affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines.

Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformations, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.

However, one cannot take the projectivization of an affine space, so projective spaces are not naturally quotients of affine spaces: one can only take the projectivization of a vector space, since the projective space is lines through a given point, and there is no distinguished point in an affine space. If one chooses a base point (as zero), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.