Affiinimuunnos

Affinimuunnos, often translated as affine transformation, is a fundamental concept in geometry and computer graphics. It describes a type of geometric transformation that preserves collinearity, meaning that points lying on a straight line before the transformation will still lie on a straight line after the transformation. However, it does not necessarily preserve parallelism or angles.

An affine transformation can be represented as a combination of translation, scaling, rotation, and shearing. Mathematically, it is typically expressed using matrix multiplication and vector addition. For a 2D affine transformation, this is often represented by a 3x3 matrix, where the last column handles the translation, and the top-left 2x2 submatrix handles the linear transformations (scaling, rotation, shearing). In 3D, a 4x4 matrix is used.

Affine transformations are widely used in various fields. In computer graphics, they are essential for manipulating objects in 2D and 3D scenes, allowing for movement, resizing, and tilting. Image editing software relies heavily on affine transformations for tasks like scaling, rotating, and skewing images. In computer vision, they are used for tasks such as image registration and object recognition. The preservation of straight lines makes them particularly useful for maintaining the structural integrity of geometric shapes during transformations.