affiinimuunnoksen

Affiinimuunnos, often translated as affine transformation, is a geometric transformation that preserves lines and parallelism, but not necessarily lengths or angles. In essence, it's a combination of a linear transformation and a translation. A linear transformation, such as rotation, scaling, or shearing, can be represented by matrix multiplication. A translation shifts every point by a fixed vector. When these two operations are combined, the result is an affine transformation.

In a two-dimensional space, an affine transformation can be described by the equation:

x' = Ax + b

where x is the original point (represented as a vector), x' is the transformed point, A is a 2x2 matrix representing the linear part of the transformation, and b is a 2-dimensional vector representing the translation. For higher dimensions, A would be an nxn matrix and b would be an n-dimensional vector.

Affine transformations are fundamental in computer graphics, image processing, and various fields of mathematics and physics. For example, they are used to resize, rotate, and move objects in 2D and 3D environments. In image processing, they can be employed for image registration, where one image is transformed to match another. The preservation of lines and parallelism makes affine transformations particularly useful for maintaining the structural integrity of geometric shapes under transformation.