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.

Formal definition

We will restrict the definition to vector spaces over the field $\mathbb {R}$ of real numbers. Let V be an n-dimensional vector space and A a set of elements that we will call points. Assume that a relation between points and vectors is defined in the following way:

1. To every ordered pair P, Q of A there is assigned a vector of V, called the difference vector, denoted by ${\overrightarrow {PQ}}\in V$.
2. To every point P of A and every vector ${\vec {v}}$ of V there exists exactly one point Q in A, such that ${\overrightarrow {PQ}}={\vec {v}}$
3. If P, Q, and R are three arbitrary points in A, then

${\overrightarrow {PQ}}+{\overrightarrow {QR}}={\overrightarrow {PR}}.$

If these three requirements are fulfilled, the set A is called an n-dimensional affine space with the difference space V.