Affine

An affine transformation is a geometric operation that preserves points, straight lines, and planes. It maintains the collinearity of points and ratios of distances along parallel lines, but may alter the size, shape, and angles of objects. Affine transformations are widely used in computer graphics, image processing, and various fields of mathematics and engineering due to their simplicity and flexibility.

Mathematically, an affine transformation can be represented as a combination of linear transformation and translation. In two-dimensional space, this can be expressed as a function F(x) = Ax + b, where x is a point represented as a vector, A is a linear transformation matrix (such as rotation, scaling, or shearing), and b is a translation vector. In three dimensions, the same concept applies with appropriate matrix and vector dimensions.

Affine transformations include basic operations such as translation, scaling, rotation, shearing, and reflection. They form a group under composition, meaning that applying multiple affine transformations sequentially results in another affine transformation. This property makes them particularly useful for modeling and manipulating objects in a consistent and predictable manner.

In the context of computer vision, affine transformations are useful for image registration, perspective correction, and object recognition, where the object may undergo changes in position, size, or orientation. In addition, many coordinate transformations in robotics and spatial analysis are affine due to their computational efficiency and the preservation of relevant geometric properties.

Overall, affine transformations provide a versatile and fundamental tool for geometric modeling, allowing for efficient and predictable manipulation of spatial data across numerous scientific and technological applications.