affiinin

Affiinin, in geometry and linear algebra, refers to concepts and transformations that are preserved under affine transformations. In Estonian mathematical terminology, affiin is the adjective for affine, and affiinin is used in phrases describing properties or objects related to affine geometry or affine maps.

An affine transformation in n-dimensional space is a map T: R^n -> R^m that can be written as T(x) = A x + b, where A is an m×n matrix and b is a vector in R^m. If A is invertible, T is a bijective affine transformation, and the set of all such maps forms the affine group Aff(n, R). Affine transformations are composed of a linear part (A) and a translation (b).

Key properties of affine transformations include: they map lines to lines and preserve parallelism, meaning that if two lines are parallel before the mapping, their images remain parallel. They preserve convex combinations: a point on a line segment between x and y maps to the corresponding point on the segment between T(x) and T(y). Consequently, they preserve ratios of segments along the same line. However, they do not in general preserve distances, angles, or areas.

Applications of affiin concepts are widespread in computer graphics, geometric modeling, and computer-aided design, where coordinate changes, projections, and alignment tasks are common. In data analysis and machine learning, affine transformations are used for normalization and linear feature preprocessing.

In Estonian usage, phrases like affiinin teisendus refer to affine transformations, while affiin olemus denotes the affine nature of a geometric or algebraic construct. See also affine space, affine geometry, projective geometry, and linear transformations for related concepts.

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