Affine space

The 2- and 3-dimensional point spaces studied in elementary Euclidean geometry are examples of affine spaces, A₂ and A₃, respectively.

Assume that any two points P and Q in a space A can be connected by a line segment in A; this possibility is Axiom 1 of Euclidean geometry. If we order P and Q (we say that P comes before Q), then the line segment obtains a direction (points from P to Q) and has become a vector, the difference vector ${\overrightarrow {PQ}}$. When all difference vectors can be mapped onto vectors of the same n-dimensional vector space V_n, we may call the point space A an affine space of dimension n, written A_n.

Usually one takes as a difference space an inner product space. Its elements have a well-defined length, namely, the square root of the inner product of the vector with itself. The distance between points P and Q in an affine space is then defined as the length of the image of ${\overrightarrow {PQ}}$ in V_n. Difference vectors that are mapped onto the same element of V_n are said to be parallel, they differ from each other by translation.