affiinigeometria

Affinigeometria, often translated as affine geometry, is a branch of geometry that studies properties of figures that are invariant under affine transformations. An affine transformation is a geometric transformation that preserves parallel lines and ratios of distances along parallel lines. It can be thought of as a combination of translation, rotation, scaling, and shearing. Unlike Euclidean geometry, affine geometry does not necessarily preserve lengths or angles.

The fundamental objects in affine geometry are points, lines, and planes. A line is defined by two distinct points, and three points are collinear if they lie on the same line. Affine geometry focuses on concepts such as parallelism, midpoints, and ratios of lengths along parallel lines. For example, the property that the diagonals of a parallelogram bisect each other is an affine invariant.

A key concept is the affine space, which is a set of points together with a vector space. The difference between any two points in an affine space is a vector. Affine transformations in an affine space are functions that map points to points such that for any three points P, Q, R, if the vector PQ is related to the vector PR by a scalar multiple, then the corresponding transformed vectors P'Q' and P'R' will also be related by the same scalar multiple.

Affine geometry provides a framework for understanding transformations that are more general than rigid motions (translations and rotations) but less general than projective transformations. It is a foundational concept in fields such as linear algebra, computer graphics, and physics, where understanding how shapes change under various transformations is crucial.