Affiineja

Affiineja are mathematical objects related to affine geometry. In English, the term is usually “affine transformations” or “affine maps.” They act on vector spaces or affine spaces and preserve straight lines and parallelism.

A common definition: Let V and W be vector spaces over a field F. An affine transformation f: V → W is of the form f(x) = Ax + b, where A is a linear map from V to W and b ∈ W is a fixed vector. Equivalently, f preserves affine combinations: f(t x + (1−t) y) = t f(x) + (1−t) f(y) for all x, y ∈ V and all scalars t ∈ F. This property ensures that lines map to lines and that parallelism is preserved.

Properties: Composition of affine maps is affine; the inverse of an invertible affine map is affine. The class of all affine maps from V to itself forms the affine general linear group, while invertible affine maps form the affine group. Affine transformations preserve ratios along parallel lines and the structure of affine spaces, providing a flexible framework for geometric operations without requiring a fixed origin.

Examples: Translations (f(x) = x + b), linear maps (b = 0), rotations, scalings, and shear maps are all affine. In three-dimensional space, these operations are widely used in computer graphics, robotics, and geometric modeling to reposition, reorient, or resize objects.

Affine spaces generalize Euclidean spaces by focusing on points and direction vectors rather than a distinguished origin. The affine hull of a set consists of all affine combinations with coefficients summing to one, forming a key construction in affine geometry.