Affine Geometry - Examples and Constructions

Affine geometry provides natural settings for various geometric constructions and applications. Understanding what can be accomplished without reference to distances or angles clarifies the essential structure of space.

ExampleConstructing the Centroid

The centroid of a triangle $\triangle ABC$ is the point:

G = \frac{1}{3}(A + B + C)

This is an affine combination (the coefficients sum to 1), so $G$ is well-defined in any affine space. The centroid is the intersection point of the medians, and this construction uses only affine notions—no distances or angles required.

Affine transformations map centroids to centroids, so this construction is affinely invariant.

Barycentric coordinates provide a powerful coordinate system for affine spaces. Given affinely independent points $P_0, \ldots, P_n$ in $n$ -dimensional affine space, any point $P$ can be uniquely expressed as:

P = \sum_{i=0}^n \lambda_i P_i \quad \text{where } \sum_{i=0}^n \lambda_i = 1

The coefficients $(\lambda_0, \ldots, \lambda_n)$ are the barycentric coordinates of $P$ . They generalize mass distributions— $\lambda_i$ represents the "mass" at $P_i$ needed to balance at $P$ .

DefinitionAffine Independence

Points $P_0, \ldots, P_k$ in affine space $A$ are affinely independent if whenever:

\sum_{i=0}^k \lambda_i P_i = \sum_{i=0}^k \mu_i P_i \quad \text{and} \quad \sum \lambda_i = \sum \mu_i = 1

we have $\lambda_i = \mu_i$ for all $i$ .

Equivalently, the vectors $\overrightarrow{P_0P_1}, \ldots, \overrightarrow{P_0P_k}$ are linearly independent. The maximum number of affinely independent points in an $n$ -dimensional affine space is $n+1$ .

Parallelism plays a fundamental role. In affine geometry, parallel lines remain parallel under all affine transformations. This allows us to define parallel transport—moving a vector from one point to another—which is automatic in affine spaces (since the direction space is the same everywhere) but requires additional structure (like a connection) in more general geometries.

ExampleAffine Transformations in Computer Graphics

In computer graphics, object transformations are typically affine:

Translation: $T(\mathbf{x}) = \mathbf{x} + \mathbf{t}$ moves objects
Rotation: $R(\mathbf{x}) = A\mathbf{x}$ with $A$ orthogonal
Scaling: $S(\mathbf{x}) = \text{diag}(s_1, \ldots, s_n)\mathbf{x}$ changes size
Shear: $H(\mathbf{x}) = (x + ay, y)$ in 2D

Composition of these gives general affine transformations. Using homogeneous coordinates, affine transformations become linear:

\begin{pmatrix} \mathbf{x}' \\ 1 \end{pmatrix} = \begin{pmatrix} A & \mathbf{b} \\ \mathbf{0}^T & 1 \end{pmatrix} \begin{pmatrix} \mathbf{x} \\ 1 \end{pmatrix}

Affine hulls and convex hulls are key constructions. The affine hull of points $P_1, \ldots, P_k$ is:

\text{aff}(P_1, \ldots, P_k) = \left\{ \sum_{i=1}^k \lambda_i P_i : \sum_{i=1}^k \lambda_i = 1 \right\}

The convex hull adds the constraint $\lambda_i \geq 0$ :

\text{conv}(P_1, \ldots, P_k) = \left\{ \sum_{i=1}^k \lambda_i P_i : \lambda_i \geq 0, \sum_{i=1}^k \lambda_i = 1 \right\}

Remark

Affine geometry underlies linear programming and convex optimization. Feasible regions defined by linear inequalities are convex polyhedra (intersections of half-spaces), which are convex sets in affine space. The simplex method exploits the affine structure of these regions.

In higher dimensions, affine geometry governs the behavior of hyperplanes (codimension-1 affine subspaces). A hyperplane in $\mathbb{R}^n$ has equation $\mathbf{a} \cdot \mathbf{x} = c$ , dividing space into two half-spaces. Hyperplane arrangements—collections of hyperplanes—have rich combinatorial and topological structure, studied in algebraic combinatorics.