Geometry & Topology 2024, 30: Affine Geometry

Affine geometry takes place on an n-dimensional vector space with an inhomogeneous linear structure. Inhomogeneous linear maps allow transformations of affine space, dilations and shear maps, however, they can't define an origin, distances between two points or angles that holds with invariance under them. Affine space has a set of points maps one-to-one onto position vectors. This relationship is not fixed. A choice of basis of an n-dimensional subspace maps to a coordinate system onto the affine space. Vectors and points function as in regular n-dimensional Euclidean spaces. An affine linear subspace is a nonempty subset E = P + U = {P + v: v ∈ U}, with P as a point of the Affine space, and U a vector from an n-dimensional vector subspace. This does not define an origin in E. PQ = {P + λPQ: λ ∈ ℝ}. If an affine linear subspace E = P + U, then U is uniquely defined by E through {PQ: P, Q ∈ E}. If E is affine and a nonempty subset, then PQ is contained in E. An affine linear combination is

For the dimension of intersections, given the formula for vector subspaces, U, W, derive for affine subspaces E, F

Transformations follow the construction T(x) = Ax + b, where A is an n ⨉ n matrix with nonzero determinant, and b an n-dimensional vector. Suitable affine transformations map all points to all other point. The absence of an origin means that no point is only ever mapped onto itself. Linear dependency of vectors sees a nontrivial relation between the position vectors.