Affine Geometry and Affine Transformations

Affine geometry is the study of geometric properties preserved by affine transformations — maps of the form

x \mapsto A x + b

that preserve straight lines, parallelism, and ratios of lengths along a line, but NOT lengths, angles, or absolute position. Key result: every affine map is an invertible linear map composed with a translation; it forms the affine group

Aff (n)

. Applies in: computer graphics, robotics, geometric modeling, projective geometry, linear algebra. Who/when: Euler and Möbius (19th c.); systematized via Klein's Erlangen Program (1872).

An affine space is a set of points on which a vector space acts freely and transitively by translation — informally, "a vector space that has forgotten where its origin is." An affine transformation is a map between affine spaces of the form $T (x) = A x + b$ , where $A$ is a linear map and $b$ is a translation vector. Affine geometry matters because it isolates exactly those geometric facts — collinearity, parallelism, midpoints, centroids, convexity — that do not depend on a choice of origin, units, or perpendicularity. The surprising fact: although affine maps can stretch, shear, and skew shapes beyond recognition, they preserve the ratio in which a point divides a segment, and they map the centroid of any point set to the centroid of its image — a rigidity hidden inside enormous flexibility.

The Core Insight

Geometry without an origin. In a vector space you can add vectors and scale them; in an affine space you cannot add two points, but you can subtract points to get a vector, and you can form affine combinations $\sum λ_{i} P_{i}$ when $\sum λ_{i} = 1$ . Affine transformations are precisely the maps that respect this structure: they preserve affine combinations. Everything else — that lines stay lines, parallels stay parallel, ratios stay fixed — follows from this single algebraic fact.

Formal Statement

Affine Space

An affine space modeled on a vector space $V$ over a field $K$ is a set $A$ together with a map (action)

A \times V \to A, (P, v) \mapsto P + v

⟳this is the formal "translate point

P

by vector

v

" operation; it is the primitive datum of the structure.

satisfying three axioms: (i) $P + 0 = P$ ; (ii) $(P + v) + w = P + (v + w)$ ; (iii) for every ordered pair of points $(P, Q)$ there exists a unique vector $v$ with $P + v = Q$ , written $v = P Q = Q - P$ . The dimension of $A$ is $dim V$ .

Affine Combination

Given points $P_{1}, \dots, P_{k} \in A$ and scalars $λ_{1}, \dots, λ_{k} \in K$ with $\sum_{i} λ_{i} = 1$ , their affine combination is the point

i = 1 \sum k λ_{i} P_{i} := P_{1} + i = 2 \sum k λ_{i} P_{1} P_{i}

⟳choose any base point (here

P_{1}

), express everything as translations of it, then verify the result is independent of the base point because

\sum λ_{i} = 1

makes the base-point terms cancel.

If additionally all $λ_{i} \geq 0$ , the combination is a convex combination, and the set of all such points is the convex hull.

Affine Transformation

A map $T : A \to B$ between affine spaces is affine if there exists a linear map $L : V \to W$ (the linear part, or differential $d T$ ) such that

T (P + v) = T (P) + L (v) for all P \in A, v \in V

⟳differentiate the definition — the linear part

L

is forced to be unique because the vector-space action is transitive, so

L (v) = T (P + v) - T (P)

is well defined independent of

P

In coordinates (after choosing origins so $A = K^{n}$ , $B = K^{m}$ ), every affine map takes the canonical form

T (x) = A x + b, A \in K^{m \times n}, b \in K^{m}

⟳set

b = T (0)

and

A = L

; then

T (x) = T (0 + x) = T (0) + L (x) = A x + b

by equation [3].

Characterization of Affine Maps

For a map $T : A \to B$ , the following are equivalent:

$T$ is affine (satisfies [3]).
$T$ preserves all affine combinations: $T (\sum λ_{i} P_{i}) = \sum λ_{i} T (P_{i})$ whenever $\sum λ_{i} = 1$ .
$T$ has the form $x \mapsto A x + b$ in coordinates.
(Over $R$ , $dim \geq 2$ , $T$ bijective) $T$ maps lines to lines — the Fundamental Theorem of Affine Geometry.

The Affine Group

The invertible affine transformations of $A ≅ K^{n}$ (those with $A \in GL (n, K)$ ) form a group under composition, the affine group $Aff (n, K)$ . Composition obeys

(A_{2} x + b_{2}) \circ (A_{1} x + b_{1}) = (A_{2} A_{1}) x + (A_{2} b_{1} + b_{2})

⟳substitute the inner map into the outer and collect the linear coefficient of

x

and the constant term separately.

There is a short exact sequence $1 \to (K^{n}, +) \to Aff (n, K) \to GL (n, K) \to 1$ , exhibiting $Aff (n)$ as a semidirect product $K^{n} ⋊ GL (n, K)$ .

Why It Holds / Derivation

Affine maps preserve ratios on a line

We show that if $T$ is affine and $C = (1 - t) A + tB$ divides segment $A B$ in ratio $t$ , then $T (C)$ divides $T (A) T (B)$ in the same ratio.

Step 1. Write

C

as an affine combination:

C = (1 - t) A + tB

, and check

(1 - t) + t = 1

, so this is a valid affine combination by [2]. Step 2. Apply

T

and use preservation of affine combinations (Theorem, item 2):

T (C) = T ((1 - t) A + tB) = (1 - t) T (A) + t T (B)

⟳this is exactly property (2) of the characterization theorem applied with

λ_{1} = 1 - t

λ_{2} = t

Step 3. The right side of [6] is the point dividing

T (A) T (B)

in ratio

t

. Hence

\frac{T ( A ) T ( C )}{T ( C ) T ( B )} = \frac{t}{1 - t} = \frac{A C}{C B}

⟳subtract

T (A)

and

T (B)

from [6] to read off the two directed sub-segments, then take their ratio; the linear part

A

cancels.

The ratio is invariant. Note lengths themselves are not preserved (the linear part rescales them), but the ratio is. $■$

Preservation of parallelism

Let lines $ℓ_{1}, ℓ_{2}$ be parallel with common direction $v \in V$ , i.e. $ℓ_{i} = {P_{i} + s v : s \in K}$ . Apply $T$ with linear part $L$ :

T (ℓ_{i}) = {T (P_{i}) + s L (v) : s \in K}

⟳substitute the parametrization of

ℓ_{i}

into the defining property [3] of

T

Both image lines have the same direction vector $L (v)$ , hence are parallel (or coincide). If $T$ is invertible, $L (v) \neq = 0$ , so they are genuine parallel lines. $■$

An affine transformation (shear + scale) maps a unit square to a parallelogram while preserving parallelism and the diagonal's midpoint

Building Intuition

intuition Think of an affine map as what happens to a drawing on a sheet of rubber that you stretch uniformly, shear, rotate, and slide — but never crumple or fold. Straight pencil lines stay straight; parallel lines stay parallel; if a point was the midpoint of a segment, it stays the midpoint; the centroid of a triangle stays the centroid. What is destroyed is anything requiring a ruler or protractor: distances change, angles change, circles become ellipses, squares become parallelograms.

A second mental model: an affine map is a linear map plus a shift. The linear part $A$ does all the rotating/stretching/shearing about the origin; the vector $b$ then slides the whole result. Equation [4] literally says "transform directions linearly, then move." Crucially, affine geometry refuses to single out the origin as special — that is the whole point of "forgetting where the origin is."

August Ferdinand Möbius | 1790–1868

Though "affine" was coined by Euler, it was Möbius in Der barycentrische Calcul (1827) who built the first systematic theory by representing points through barycentric coordinates — weights $λ_{i}$ on reference points whose center of mass is the point in question. This is exactly the affine-combination idea of equation [2]. Möbius realized that the ratios of weights carry geometric meaning independent of any origin, anticipating both affine and projective geometry. His barycentric calculus turned synthetic geometric statements (collinearity, concurrence) into clean algebraic conditions and directly inspired the group-theoretic viewpoint Felix Klein later formalized.

Worked Examples

Mapping three points (solving for an affine map in the plane)

Given: Find the affine map

T (x) = A x + b

R^{2}

sending

P_{1} = (0, 0) \to Q_{1} = (1, 1)

P_{2} = (1, 0) \to Q_{2} = (3, 2)

P_{3} = (0, 1) \to Q_{3} = (2, 4)

. Step 1 — find $b$ . Since

T (0, 0) = A \cdot 0 + b = b

, we get

b = Q_{1} = (1, 1)

. Step 2 — find columns of $A$ . Because

P_{2} = e_{1}

T (e_{1}) = A e_{1} + b = (col_{1} A) + b

. So

col_{1} A = Q_{2} - b = (3, 2) - (1, 1) = (2, 1)

. Similarly

col_{2} A = Q_{3} - b = (2, 4) - (1, 1) = (1, 3)

. Step 3 — assemble.

A = (2113), b = (11), det A = 5 \neq = 0

⟳three non-collinear point-images uniquely determine a planar affine map; columns of

A

come from images of the standard basis after subtracting the image of the origin.

Check:

T (1, 0) = (2, 1) + (1, 1) = (3, 2) = Q_{2}

. ✓

T (x, y) = (2 x + y + 1, x + 3 y + 1)

Area scales by

∣ det A ∣ = 5

: a unit triangle of area

0.5

maps to area

2.5

(square units).

Ratio preservation under a shear

Given: The shear

T (x, y) = (x + 2 y, y)

(so

A = (1021)

b = 0

). Let

A = (0, 0)

B = (4, 2)

, and

C

divide

A B

in ratio

t = \frac{1}{4}

, i.e.

C = \frac{3}{4} A + \frac{1}{4} B = (1, 0.5)

. Step 1 — image of endpoints.

T (A) = (0, 0)

;

T (B) = (4 + 2 \cdot 2, 2) = (8, 2)

. Step 2 — image of $C$ .

T (C) = (1 + 2 \cdot 0.5, 0.5) = (2, 0.5)

. Step 3 — verify ratio. The point dividing

T (A) T (B)

t = \frac{1}{4}

\frac{3}{4} (0, 0) + \frac{1}{4} (8, 2) = (2, 0.5)

, which equals

T (C)

. ✓

\frac{T ( A ) T ( C )}{T ( A ) T ( B )} = \frac{( 2 , 0.5 )}{( 8 , 2 )} = \frac{1}{4} = t

⟳confirm both numerator and denominator are parallel (they must be, since the image is a line) and compute the scalar ratio of magnitudes; it equals the original

t

as predicted by equation [6].

Ratio t = \frac{1}{4} is preserved; lengths are not (∣ A B ∣ = 20 \to ∣ T (A) T (B) ∣ = 68) .

Common Misconceptions

Affine maps do NOT preserve distances, angles, or areas in general. Only special affine maps do:

$A$ orthogonal ( $A^{T} A = I$ ) and $b$ arbitrary → isometry (preserves distance & angle).
$det A = \pm 1$ → equiareal / special affine (preserves area but maybe not distance).
General invertible $A$ → preserves only collinearity, parallelism, and ratios along a line.

A frequent error: assuming "preserves ratios" means "preserves all ratios of lengths." It does not — it preserves the ratio in which a point divides a segment of the same line, and ratios of parallel segments. It does NOT preserve the ratio of two non-parallel lengths (e.g. base-to-height of a triangle), because the linear part stretches different directions differently.

Claim (false): "An affine map preserves the ratio of the sides of a rectangle."

Take the unit square with vertices at $(0, 0), (1, 0), (1, 1), (0, 1)$ and the affine map $T (x, y) = (3 x, y)$ — pure horizontal scaling. The horizontal side $1 \to 3$ , the vertical side $1 \to 1$ . The aspect ratio changes from $1 : 1$ to $3 : 1$ . The square becomes a $3 \times 1$ rectangle.

Worse, $T (x, y) = (x + y, y)$ turns the square into a parallelogram: right angles are destroyed entirely. Angles and aspect ratios are affine-invariant only for isometries (and similarities for angles), not for general affine maps.

Claim (false): "Every line-preserving bijection of the plane is affine — that's the Fundamental Theorem."

Over $R$ this is true. But over $C$ it fails: the map $z \mapsto \overset{z}{ˉ}$ (complex conjugation, viewed appropriately) and more dramatically maps built from a wild field automorphism of $C$ send lines to lines yet are not affine over $C$ — they are only semi-affine, affine up to a field automorphism. The Fundamental Theorem of Affine Geometry requires the field's only automorphism fixing the structure to be the identity (true for $R$ , false for $C$ ).

Connections

Affine geometry sits between Linear Algebra and Projective Geometry in Klein's hierarchy. Drop the origin from a vector space and you get an affine space; add a "line at infinity" and you get projective space — there, parallelism disappears (parallels meet at infinity) and even ratios are replaced by the cross-ratio, the projective invariant. Restrict affine maps by demanding $A$ orthogonal and you recover Euclidean Geometry (isometries, the rigid-motion group). The relationship to Convexity and Convex Sets is intimate: convex combinations are affine combinations with nonnegative weights, so affine maps send convex sets to convex sets and preserve the structure of convex hulls. The barycentric viewpoint links directly to Centroids and Centers of Mass.

Real-World Applications

Computer graphics & GPUs. Every translation, rotation, scale, and shear of a 2D/3D model is an affine map; graphics pipelines use $4 \times 4$ homogeneous matrices so that translation becomes a matrix multiplication (composing via [5]). "Affine texture mapping" was a hallmark of early 3D consoles.
Robotics & kinematics. Rigid-body poses are affine (in fact Euclidean) transformations; forward kinematics chains them by composition [5]. The special Euclidean group $S E (3) \subset Aff (3)$ describes every robot end-effector pose.
Image registration / medical imaging. Aligning two MRI or satellite images often uses an affine model $x \mapsto A x + b$ to correct for stretch, shear, and shift between scans before finer nonlinear warping.
Computer-aided geometric design (CAD). Bézier and B-spline curves are affine invariant: transforming the control points and re-evaluating gives the same result as transforming the curve — a direct consequence of affine combinations [2].
Data science — affine layers. A neural-network "linear layer" $x \mapsto W x + b$ is literally an affine map; nonlinearity is added only by the activation function.
Crystallography. Affine (and subgroup Euclidean) transformations classify crystal lattice symmetries and lattice-to-lattice deformations.
Typography & font rendering. Italic slant is a shear; condensing/expanding fonts is anisotropic scaling — both affine.

Historical Development

c. 1748 — Euler introduces the term affine (affinis, "related") in Introductio in analysin infinitorum for curves related by coordinate scaling.
1827 — Möbius publishes Der barycentrische Calcul, founding barycentric coordinates and the algebraic theory of affine combinations.
1832 — Plücker develops analytic methods linking affine and projective coordinates.
1844 — Grassmann (Ausdehnungslehre) builds an abstract calculus of points and vectors anticipating affine spaces.
1872 — Felix Klein's Erlangen Program defines each geometry by its transformation group; affine geometry = invariants of the affine group $Aff (n)$ , sandwiched between projective and Euclidean.
1899 — Hilbert (Grundlagen der Geometrie) gives a rigorous axiomatic treatment isolating affine (incidence + parallel) axioms from metric ones.
1918 — Weyl (Raum, Zeit, Materie) gives the now-standard axiomatic definition of an affine space as a set acted on by a vector space, used in relativity.
20th c. — Veblen & Young, Artin (Geometric Algebra, 1957) prove the Fundamental Theorem of Affine/Projective Geometry, characterizing line-preserving maps.
1960s–present — affine transformations become foundational in computer graphics (Sutherland's Sketchpad, 1963) and in homogeneous-coordinate rendering pipelines.

Subtleties & Limits

Where affine geometry runs out. Affine structure cannot express anything metric — there is no canonical notion of "perpendicular," "unit length," or "circle" without extra data (an inner product, lifting you to Euclidean geometry). It also cannot capture incidence at infinity: parallel lines genuinely never meet in an affine plane, which is why projective completion is needed for a uniform theory of conics and perspective. A subtler issue is the field dependence of the Fundamental Theorem: over fields with nontrivial automorphisms (like

C

), line-preserving bijections need only be semi-affine, and classifying these "collineations" is governed by the field's automorphism group. In infinite dimensions, affine spaces over topological vector spaces raise questions about which affine maps are continuous and how the semidirect-product structure interacts with the topology — active terrain in functional analysis and the theory of affine schemes in algebraic geometry, where "affine" takes on a related but generalized meaning.