Affine

Mathematical concept in linear algebra and geometry

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Affine refers to a type of transformation in linear algebra and geometry that preserves points, straight lines, and planes. Affine transformations include translations, scalings, rotations, and shears. These transformations are fundamental in various fields such as computer graphics, robotics, and computer vision.

Affine Space

An affine space is a geometric structure that generalizes the properties of Euclidean space but without a fixed origin. It consists of a set of points and a vector space that defines the directions and distances between these points. The concept of affine space is crucial in projective geometry and differential geometry.

Affine Transformation

An affine transformation is a function between affine spaces which preserves the structure of the space. Mathematically, an affine transformation can be represented as: \[ f(\mathbf{x}) = A\mathbf{x} + \mathbf{b} \] where \( A \) is a linear transformation matrix and \( \mathbf{b} \) is a translation vector. Affine transformations are used extensively in computer graphics to manipulate images and shapes.

Properties

Affine transformations have several important properties:

• They preserve collinearity: Points that lie on a straight line before the transformation will still lie on a straight line after the transformation.
• They preserve ratios of distances: The ratio of lengths of two segments on a line remains constant.
• They do not necessarily preserve angles or lengths.

Applications

Affine transformations are widely used in various applications:

• In computer graphics, they are used for image transformations, including rotation, scaling, and translation of images.
• In robotics, they are used to describe the movement and orientation of robots.
• In computer vision, they are used for object recognition and image registration.

Related Concepts

• Linear transformation
• Euclidean space
• Projective geometry
• Differential geometry
• Shear mapping
• Translation (geometry)
• Scaling (geometry)
• Rotation (geometry)

See Also

• Affine space
• Affine transformation
• Linear algebra
• Geometry
• Computer graphics
• Robotics
• Computer vision