Affine Geometry

Affine Geometry is not concerned with the notions of circle, angle and distance. It’s a known dictum that in Affine Geometry all triangles are the same. In this context, the word affine was first used by Euler (affinis). In modern parlance, Affine Geometry is a study of properties of geometric objects that remain invariant under affine transforma-tions (mappings).

Affine transformations preserve collinearity of points: if three points belong to the same straight line, their images under affine transformations also belong to the same line and, in addition, the middle point remains between the other two points. As further examples, under affine transformations

-Parallel lines remain parallel,-Concurrent lines remain concurrent (images of intersecting lines intersect),-The ratio of length of line segments of a given line remains constant,-The ratio of areas of two triangles remains constant (and hence the ratio of any areas remain constant),-Ellipses remain ellipses and the same is true for parabolas and hyperbolas.barycenters of triangles (and other shapes) map into the corresponding barycenters