affine transformation

Thân thiện

An affine transformation can rotate and translate a geometric shape on a coordinate plane.

Definition

1. Noun:
   • (Mathematics) A geometric transformation that preserves points, straight lines, and planes, and the parallelism between lines. An affine transformation is a function that maps points to points, combining operations like translation, rotation, scaling, reflection, and shearing. Crucially, it keeps parallel lines parallel after the transformation.

Usage Examples

• Noun:
  • Applying an affine transformation to the image corrected the perspective distortion.
  • In computer graphics, an affine transformation is used to rotate and scale a 3D model.
  • The geometric proof required showing the figure was invariant under a specific affine transformation.

Advanced Usage

• "To be affine-invariant": A property that remains unchanged under any affine transformation.
  • The ratio of areas of two triangles is an affine-invariant property.
• "To undergo an affine transformation": To be changed or mapped by such a function.
  • The coordinate system underwent an affine transformation to align with the new data.

Variants and Related Words

• Affine (adjective): Pertaining to or having the properties of an affine transformation.
  • Affine geometry studies properties preserved under affine transformations.
• Affinity (noun): In a general sense, a close similarity or connection. (Note: In mathematics, this is distinct from "affine").
• Linear Transformation (noun): A more restrictive type of transformation (a subset of affine transformations) that fixes the origin.

Synonyms

• Geometric transformation: A general term for a mapping between geometric spaces.
• Linear map (with translation): Describes the mathematical composition of an affine transformation.

Related Phrases

• Affine combination: A linear combination where the coefficients sum to 1.
• Affine space: A geometric structure that generalizes the properties of Euclidean space without a fixed origin, the setting for affine transformations.

Học thuật

Noun

1. (mathematics) a transformation that is a combination of single transformations such as translation or rotation or reflection on an axis