AffineTransformation

An affine transformation is a geometric transformation that preserves lines and parallelism but not necessarily lengths or angles. It is a combination of linear transformations and translations. In essence, an affine transformation maps straight lines to straight lines and keeps parallel lines parallel.

Affine transformations are fundamental in computer graphics, image processing, and computer vision. They are used for tasks such as resizing, rotating, shearing, and translating images and objects. A key property is that they can be represented using matrix multiplication. For instance, in two dimensions, a point (x, y) can be transformed to (x', y') by multiplying a vector containing x and y (often with an added constant 1 for translation) by a 3x3 matrix.

The matrix for an affine transformation typically includes parameters for scaling, rotation, shearing, and translation. If a transformation only involves scaling, rotation, and shearing, it is considered a linear transformation, and the origin remains fixed. The addition of a translation component shifts the entire transformed object without altering its shape or orientation. Understanding affine transformations is crucial for manipulating geometric data in various applications.