The branch of geometry that studies properties invariant under affine transformations (compositions of linear maps and translations), emphasizing parallelism, ratios of directed segments on parallel lines, barycentric relations and collinearity while excluding metric notions such as angles and lengths.

Principle

Organize geometric reasoning by invariance under the affine group: preserve straightness and parallelism but not distances or angles; treat figures up to nondegenerate linear change of coordinates plus translation.

Demonstration

Consider the affine plane over a field: parallelograms remain parallelograms under any affine map, midpoints of segments on the same direction are mapped to midpoints, and the ratio in which a line parallel to a side of a triangle cuts the other two sides is preserved; barycentric coordinates give an affine description of points inside a triangle.

Misapplication

Using Euclidean angle chasing or assuming orthogonality is preserved by arbitrary affine maps; treating circles as invariant objects under general affine transformations leads to incorrect conclusions because circles typically map to ellipses.

Consequence

When applied correctly one can classify conic sections up to affine equivalence, reduce many planar problems to linear algebra in coordinates, and reason about incidence and division of segments without introducing a metric.

Reversal

Invert to metric (Euclidean or Riemannian) geometry where inner products and distances are primary and parallelism is not the central invariant; alternatively move to projective geometry where parallelism is absorbed into points at infinity.

Boundary

Applies to vector spaces or affine spaces over fields or rings and to combinatorial affine structures; excludes any argument that requires an inner product, lengths, angles, or curvature, and excludes transformations that are nonlinear in coordinates.

Semantic Tension

Tension arises with the term 'linear'—linear geometry often assumes an origin while affine allows translation; tension also appears versus Euclidean geometry where lengths/angles are primary invariants.

Synthesis

Affine geometry is the coordinate-free study of straightness, parallelism and proportional division under the affine group: it captures those structural relations that survive all nonsingular linear changes and translations, but not metric information.