affine geometry

Imagine looking at a flat drawing through a lens that can stretch it unevenly and slant it, but never bends straight lines and never makes parallel lines cross. Distances and angles get scrambled, yet some things stubbornly survive: lines stay lines, parallel stays parallel, and midpoints stay midpoints. Affine geometry is the geometry of exactly those survivors — the study of what remains true when you forget all about size and angle but still respect straightness and parallelism.

In the Erlangen scheme, affine geometry is the geometry whose transformation group is the affine group: all maps of the plane built from a linear part (which can rotate, scale unequally in different directions, and shear) followed by a translation, with the linear part required to be reversible. Such a map sends every line to a line, sends parallel lines to parallel lines, and preserves ratios of lengths measured along any one line — so it preserves midpoints, points that divide a segment in a given ratio, and the centroid of a triangle. What it does not preserve is absolute distance, angle, or the comparison of lengths in different directions: a square can become any parallelogram under an affine map, and a circle can become any ellipse. The affine invariants are thus parallelism, collinearity, betweenness, ratios along a line, and ratios of areas; the things lost are length and angle.

Affine geometry sits one level below projective geometry and one above similarity geometry in the hierarchy. It is the right home for any fact that uses parallelism and ratio but never length or angle — the concurrence of a triangle's medians, the fact that the midpoints of a quadrilateral's sides form a parallelogram, the centroid dividing each median in ratio 2 to 1. A common misunderstanding is that affine maps can do 'anything', but they cannot: they always preserve straightness and parallelism, so they can never turn a triangle into a four-sided figure, and they can never make two parallel lines meet.