# Affine geometry Affine geometry is what remains of Euclidean [[Geometry]] when you surrender the ruler and the protractor but keep the straightedge: points, lines, parallelism, and ratios along parallel lines survive; lengths, angles, and circles do not. Its transformations are exactly the maps x ↦ Ax + b with invertible A — the affine group, sitting between the rigid motions of Euclid and the full [[Projective_geometry|projective]] collineations in the hierarchy Felix Klein organized in his Erlangen program (1872), which redefined a geometry as a [[Group_theory|group]] together with its invariants. The affine viewpoint is the daily dialect of [[Linear_algebra]] applications: [[Computer_vision]] approximates distant cameras affinely, [[Robotics]] composes affine transforms in homogeneous coordinates, [[Dynamical_system|dynamical systems]] linearize into affine ones near equilibrium, and [[Isaac_Newton|Newtonian]] spacetime itself — before [[General_relativity]] curved it — is an affine space with no preferred origin. ## Klein's lens: a geometry is a group Klein's Erlangen program ranks [[Geometry|geometries]] by how much their transformation [[Group_theory|groups]] forget. Rigid motions (rotations, translations) preserve distance; add uniform scaling and you get similarity geometry; admit all invertible linear parts and you reach the affine group GA(n), a [[Lie_group]] of dimension n² + n; pass to the [[Projective_geometry|projective]] group and even parallelism dissolves. Each descent trades structure for generality, and each level's theorems are exactly the statements its group cannot destroy — the same invariance-first logic by which [[Mathematical_physics]] classifies laws under symmetry transformations and [[Control_theory]] normalizes [[Linear_time-invariant_system|state-space models]] by change of [[Linear_algebra|basis]]. Affine geometry is the sweet spot where "parallel," "midpoint," "centroid," and "ratio along a line" are all meaningful, but "perpendicular" and "unit length" are not even wrong — they are simply not affine words. ## Points versus vectors: the torsor discipline An affine space is a vector space that has forgotten its origin. Points cannot be added — what would Minneapolis plus Chicago mean? — but their differences are honest vectors, and a point plus a vector is a point. The bookkeeping runs through affine combinations Σλᵢpᵢ with Σλᵢ = 1, the barycentric calculus Möbius introduced in 1827: the centroid of a triangle is the equal-weights case, and it cuts each median in the affine-invariant ratio 2:1. Restrict the weights to λᵢ ≥ 0 and you get convexity — every [[Polytope]] is an affine-combination object, which is why the feasible regions of linear programs in [[Mathematical_optimization]] and [[Operations_research]] are affine geometry at work. The distinction between points and displacement vectors, enforced by type rather than by taste, is precisely the discipline [[Computer_programming|graphics code]] and physics [[Simulation|simulations]] adopt to keep [[Cartesian_coordinate_system|coordinate]] bugs from compiling. ## What affine maps preserve — and what they destroy An affine map preserves collinearity, parallelism, ratios along parallel lines, midpoints, tangency, and barycenters; it multiplies every area by the same factor — the [[Linear_algebra|determinant]] |det A|. It does not preserve length, angle, or the difference between a circle and an ellipse — and that failure is a tool. Map the unit circle by (x, y) ↦ (ax, by): the image is the ellipse with semi-axes a and b, and since areas scale by ab, the ellipse area πab falls out of the circle's π with no [[Calculus]] at all. Because any triangle maps affinely onto any other, a theorem proved for one convenient triangle — Ceva's concurrency criterion (1678), Menelaus's collinearity test — holds for all. The same invariance underwrites practice: [[Statistics|statistical]] procedures are prized when affine-equivariant, immune to units and offsets, and data augmentation in [[Machine_learning]] applies random affine maps precisely because class identity should be an affine invariant of an [[Computer_vision|image]]. ## Coordinates without a ruler A [[Cartesian_coordinate_system]] imports more structure than affine geometry owns — its right angles and unit lengths are Euclidean luxuries. The native frame is n + 1 points in general position, with barycentric coordinates locating every other point. In homogeneous form, an affine map packs into one (n+1)×(n+1) matrix, so composition becomes [[Linear_algebra|matrix]] multiplication: this is the 4×4 pipeline that [[Robotics]] uses for kinematic chains and that every [[Graphics_processing_unit]] executes billions of times per frame — rotation, scale, shear, translation, all one multiply. [[Computer_vision]] leans on the affine camera as a calibrated shortcut when depth variation is small relative to distance, and affine image registration — in medical imaging and [[Geographic_information_system|GIS]] georeferencing alike — aligns scenes before any [[Nonlinear_system|nonlinear]] warp is attempted, because the affine part is the low-dimensional, well-conditioned share of the deformation. ## Finite affine planes: geometry you can count Drop continuity and keep the incidence axioms — two points determine a line; through a point off a line passes exactly one parallel — and affine planes become finite [[Combinatorics|combinatorial]] objects. A plane of order n has n² points and n² + n lines in n + 1 parallel classes; coordinates come from the finite [[Field_(mathematics)|field]] GF(q) whenever the order is a prime power, tying the subject to [[Finite_geometry]] and [[Discrete_mathematics]]; adjoin a line at infinity and the [[Projective_geometry|projective]] completion reappears. The order-3 plane AG(2,3) has 9 points and 12 lines, and its 4-dimensional cousin AG(4,3) is exactly the card game SET: 81 cards, with "sets" as affine lines. Counting line-free subsets there — the cap-set problem — was cracked by Ellenberg and Gijswijt (2016) with a polynomial method now exported across [[Combinatorics]] and [[Graph_theory|extremal graph theory]], a reminder that austere axioms breed hard questions. ## Affine structure in physics and dynamics Newtonian kinematics is affine geometry wearing physics' clothes: spacetime is an affine space, simultaneity slices it into parallel sheets, and [[Isaac_Newton|Newton]]'s first law says free particles trace affine lines — inertial frames are just affine coordinate choices agreeing on time, as Hermann Weyl's 1918 axiomatization made explicit. Near an equilibrium, a smooth [[Dynamical_system]] is approximated by an affine one, ẋ = Ax + b; the whole apparatus of [[Linear_algebra|eigenvalue]] stability analysis in [[Control_theory]] lives in that approximation, and control-affine models ẋ = f(x) + Σgᵢ(x)uᵢ keep the affine dependence on inputs even when the drift is [[Nonlinear_system|nonlinear]]. Detrending a time series in [[Statistics]] removes its affine part; and when geometry finally needs curvature, the affine idea survives as the affine connection of [[Differential_geometry]] — the device [[General_relativity]] uses to say "straight" without ever saying "distance." **On the spine:** [[Projective_geometry]] · [[Linear_algebra]] · [[Finite_geometry]] · [[Cartesian_coordinate_system]] · [[Differential_geometry]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Affine_geometry) : [Wikitube](https://en.wikitube.io/wiki/Affine_geometry) ## Previous hub tags Hubs: `Systems`. Portals: [[PORTAL_Dynamical_system]], [[PORTAL_Graph_theory]], [[PORTAL_Information_theory]], [[PORTAL_Control_theory]], [[PORTAL_Operations_research]]. --- *Repopulated 2026-08-12 · redlink fill · 0 deletions.*