# affine geometry

Affine geometry is the study of properties of geometric objects
that remain unchanged after parallel projection from one plane to another.
During such a projection, first studied by Leonhard Euler,
each point (*x*, *y*) is mapped to a new point (*ax* + *cy* + *e*, *bx* + *dy* + *f*). Circles, angles, and distances
are altered by affine transformations and so are of no interest in affine
geometry. Affine transformations do, however, 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. Similarly, under affine transformations,
parallel lines remain parallel, concurrent lines remain concurrent (images
of intersecting lines intersect), the ratio of lengths of line segments
of a given line remains constant, the ratio of areas of two triangles remains
constant, and ellipses, parabolas, and hyperbolas continue to be ellipses,
parabolas, and hyperbolas.