Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are.
To illustrate the difference (over the real numbers), a parabola in the affine plane intersects the line at infinity, whereas an ellipse does not.
So a parabola and ellipse are the same when thought of projectively, but different when regarded as affine objects.
Somewhat less intuitively, over the complex numbers, an ellipse intersects the line at infinity in a pair of points while a parabola intersects the line at infinity in a single point.
So, for a slightly different reason, an ellipse and parabola are inequivalent over the complex affine plane but remain equivalent over the (complex) projective plane.
Since any two affine spaces of the same dimension are isomorphic, in some situations it is appropriate to identify them with
Then an affine space is a set A together with a simple and transitive action of V on A.
Another way is to define a notion of affine combination, satisfying certain axioms.
The algebra of polynomials in the affine functions on A defines a ring of functions, called the affine coordinate ring in algebraic geometry.
This ring carries a filtration, by degree in the affine functions.
For another example, suppose that X is a two-dimensional vector space over the complex numbers.
An analogous construction applies to the solution of first order linear ordinary differential equations.
The solutions of the homogeneous differential equation is a one-dimensional linear space, whereas the set of solutions of the inhomogeneous problem is a one-dimensional affine space A.
Another example is the set of solutions of a second-order inhomogeneous linear ordinary differential equation (over the complex numbers).
A complex affine space A has a canonical projective completion P(A), defined as follows.
Form the vector space F(A) which is the free vector space on A modulo the relation that affine combination in F(A) agrees with affine combination in A.
The stabilizer of the hyperplane at infinity is a parabolic subgroup, which is the automorphism group of A.
It is isomorphic (but not naturally isomorphic) to a semidirect product of the group GL(V) and V. The subgroup GL(V) is the stabilizer of some fixed reference point o (an "origin") in A, acting as the linear automorphism group of the space of vector emanating from o, and V acts by translation.
In contrast, the automorphism group of the affine space A as an algebraic variety is much larger.
The Jacobian determinant of such an algebraic automorphism is necessarily a non-zero constant.
It is believed that if the Jacobian of a self-map of a complex affine space is non-zero constant, then the map is an (algebraic) automorphism.
So in the Zariski topology, a subset of A is closed if and only if it is the zero set of some collection of complex-valued polynomial functions on A.
A subbase of the Zariski topology is the collection of complements of irreducible algebraic sets.
A metric can be defined on a complex affine space, making it a Euclidean space, by selecting an inner product on V. The distance between two points p and q of A is then given in terms of the associated norm on V by The open balls associated to the metric form a basis for a topology, which is the same as the analytic topology.
The family of holomorphic functions on a complex affine space A forms a sheaf of rings on it.
By definition, such a sheaf associates to each (analytic) open subset U of A the ring
The uniqueness of analytic continuation says that given two holomorphic functions on a connected open subset U of Cn, if they coincide on a nonempty open subset of U, they agree on U.
Oka's coherence theorem states that the structure sheaf
This is the fundamental result in the function theory of several complex variables; for instance it immediately implies that the structure sheaf of a complex-analytic space (e.g., a complex manifold) is coherent.