Affine space

An affine space is a set of points whose pairwise differences are vectors in an associated vector space. It has no distinguished origin, so points can be subtracted to obtain vectors, but points cannot be added as vectors.

Vector space: A set with vector addition and scalar multiplication satisfying the vector space axioms. The associated vector space supplies the differences between affine points.
Displacement vector: A vector describing the change from one point to another. It is the result of subtracting two affine points.
Affine geometry: The study of properties preserved by affine transformations. It studies geometric structure using the operations available in affine spaces.
Metric space: A set equipped with a distance function satisfying the metric axioms. A metric supplies distances, which affine structure alone does not determine.
Vector: An element of a vector space that can represent a direction and magnitude. Differences of points in an affine space are vectors.
Barycentric coordinates: Coordinates that express a point as an affine combination of reference points. They describe affine points while keeping the weights’ sum equal to one.
Projective geometry: A geometry in which points and lines are considered without fixed metric distances or angles. Affine spaces embed into projective spaces by adding points at infinity.
Projective space: The space of one-dimensional subspaces of a vector space. Projective geometry forgets parallelism that affine spaces retain.
Scalar multiplication: The operation that multiplies a vector by a scalar to rescale it. It lets vectors scale displacements between affine points.
Affine transformation: A map that preserves affine combinations of points. Such maps preserve the structure defining an affine space.

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