Affine Geometry - Core Definitions

Affine geometry studies properties that are preserved under affine transformations—transformations that preserve collinearity and ratios of distances along parallel lines. Unlike Euclidean geometry, affine geometry has no notion of distance or angle, making it a more general framework.

DefinitionAffine Space

An affine space over a vector space $V$ is a set $A$ together with a free and transitive action of $V$ on $A$ . Equivalently, for any two points $P, Q \in A$ , there exists a unique vector $\overrightarrow{PQ} \in V$ such that $Q = P + \overrightarrow{PQ}$ .

The vector space $V$ is called the direction space of $A$ . The dimension of $A$ equals $\dim V$ .

The key distinction from vector spaces is that affine spaces have no distinguished origin. Points in an affine space cannot be added, but their differences are vectors. This reflects the physical intuition that space itself has no preferred origin—only relative positions matter.

DefinitionAffine Transformation

An affine transformation $f: A \to A'$ between affine spaces is a map that preserves affine combinations:

f\left(\sum_{i=1}^n \lambda_i P_i\right) = \sum_{i=1}^n \lambda_i f(P_i)

for any points $P_i$ and scalars $\lambda_i$ with $\sum \lambda_i = 1$ .

In coordinates, affine transformations have the form:

f(\mathbf{x}) = A\mathbf{x} + \mathbf{b}

where $A$ is an invertible matrix and $\mathbf{b}$ is a translation vector.

Affine transformations include translations, rotations, reflections, shears, and scalings (including non-uniform scalings). The composition of affine transformations is affine, forming the affine group $\text{Aff}(n)$ . This group has structure:

\text{Aff}(n) = \mathbb{R}^n \rtimes GL(n, \mathbb{R})

ExampleAffine Coordinates

Given an affine space $A$ of dimension $n$ , choosing a point $O$ (the origin) and a basis $\{e_1, \ldots, e_n\}$ for the direction space establishes an affine coordinate system. Any point $P$ has coordinates $(x_1, \ldots, x_n)$ such that:

P = O + x_1 e_1 + \cdots + x_n e_n

Different choices of origin and basis give different coordinate systems, related by affine transformations.

Central concepts in affine geometry include parallelism (lines that don't intersect), ratios on lines (the ratio in which a point divides a segment), and barycenters (weighted averages of points). These notions make sense without reference to distances or angles.

Remark

Affine geometry emerges naturally in many contexts:

Parallel projection: Projecting one plane onto another along parallel lines preserves affine properties
Special relativity: Spacetime has affine structure before introducing the Minkowski metric
Computer graphics: Viewing transformations are often modeled as affine maps
Convex optimization: The feasible regions are typically affine or convex sets

The affine hull of a set $S$ is the smallest affine subspace containing $S$ . Points $P_0, \ldots, P_k$ are affinely independent if the vectors $\overrightarrow{P_0P_1}, \ldots, \overrightarrow{P_0P_k}$ are linearly independent. An affine space of dimension $n$ has affine bases consisting of $n+1$ affinely independent points (one more than the vector space dimension, accounting for the need to specify an origin).