affine geometry

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Definition

Noun: - A branch of geometry that studies properties of geometric figures which remain unchanged under affine transformations. These properties include parallelism, ratios of lengths along parallel lines, and concurrency, but not absolute distances or angles.

Usage
  • Affine geometry provides the foundational framework for concepts like parallel lines and ratios, which are essential in linear algebra and computer graphics.
  • In affine geometry, the concept of a midpoint is meaningful, but the concept of a circle is not, as circles are not preserved under affine transformations.
Examples
Advanced Usage
  • Affine Space: The fundamental setting for affine geometry. It is a mathematical structure that generalizes the properties of Euclidean space without a fixed origin or the concept of absolute distance.
    • Vectors in an affine space represent translations rather than positions.
  • Affine Combination: A linear combination where the coefficients sum to 1. This concept is central to defining lines and planes in affine geometry.
    • The point P = αA + βB is an affine combination of points A and B if α + β = 1.
Variants and Related Words
  • Affine Transformation (n): A geometric transformation that preserves parallel lines and ratios of distances along them. It includes operations like translation, scaling, rotation, and shearing.
    • Applying an affine transformation to a square can turn it into any parallelogram.
  • Affinely (adv): In a manner relating to or by means of affine geometry or transformations.
    • The two shapes are affinely equivalent.
Synonyms
  • Geometry of Parallelism: Emphasizes the core invariant of affine transformations.
  • Linear Geometry (in a specific context): Sometimes used informally, though linear algebra often operates on vector spaces with a fixed origin, while affine geometry does not.
Related Concepts (Analogous to Idioms/Phrasal Verbs)
  • Invariant under Affine Transformations: A property that does not change when an affine transformation is applied.
    • Collinearity of points is invariant under affine transformations.
  • Affine Hull: The smallest affine subspace containing a given set of points, analogous to the concept of a span in linear algebra.
    • The affine hull of two distinct points is the line through them.
Noun
  1. the geometry of affine transformations