Affinegeometria

Affine geometry is a branch of geometry that studies those properties of geometric figures that are preserved under affine transformations. An affine transformation is a function that preserves lines and parallelism, but not necessarily lengths or angles. This means that if two lines are parallel in the original figure, they will remain parallel after an affine transformation.

The fundamental objects in affine geometry are points, lines, and planes. Unlike Euclidean geometry, affine geometry does not initially define concepts like distance or angles. Instead, it focuses on relationships such as collinearity (points lying on the same line) and parallelism. A key concept is the affine space, which can be thought of as a vector space where the origin has been "forgotten".

In an affine space, any two points define a unique line. Parallel lines are defined as lines that lie in the same plane and do not intersect, or lines that coincide. Affine transformations can be understood as translations, rotations, reflections, scaling, and shearing. These transformations allow us to move and reshape figures while maintaining their essential affine properties.

Affine geometry provides a foundation for Euclidean geometry, as Euclidean geometry can be seen as a special case of affine geometry where distances and angles are also preserved. It is also crucial in areas like computer graphics, where it is used to model projections and transformations of objects in 3D space. The study of affine geometry helps to understand the underlying structure of geometric relationships that are invariant under a broader class of transformations than rigid motions.