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.