Affine geometry

Affine geometry studies properties preserved by affine transformations, including collinearity, ratios along a line, and parallelism, while disregarding distances and angles.

Affine transformation: A map that preserves affine combinations of points, including lines and ratios along a line. These transformations define which geometric properties affine geometry preserves.
Homogeneous coordinates: Coordinates that represent points using one additional component, enabling affine maps to be written as matrix transformations. They package translations and linear changes into a single matrix operation.
Euclidean geometry: Geometry built around distance, angle, and the parallel postulate. Affine geometry retains parallelism but does not preserve Euclidean lengths or angles.
Affine combination: A weighted sum of points whose coefficients add to one. It supports constructions such as midpoints and centroids without a metric.
Felix Klein: A German mathematician whose Erlangen Program organized geometries by their transformation groups. His framework characterizes affine geometry through the transformations that preserve it.
Affine space: A space of points with a well-defined notion of displacement between points, but no distinguished origin. It formalizes geometry without requiring a chosen zero point.
Linear transformation: A map between vector spaces that preserves addition and scalar multiplication. The linear part of an affine transformation controls its action on directions.
Similarity transformation: A transformation that preserves angles and scales all distances by one common factor. Similarities preserve more metric structure than general affine transformations do.
Centroid: The center of mass of a body or, for a triangle, the intersection of its medians. A triangle's centroid is an affine combination of its vertices.
Erlangen Program: Felix Klein's 1872 proposal to classify geometries by the transformation groups that preserve their properties. It made affine geometry one member of a systematic hierarchy of geometries.

Linked from 17 pages