Affine Geometry - Main Theorem

TheoremFundamental Theorem of Affine Geometry

Let $A$ and $A'$ be affine spaces of dimension $n \geq 2$ , and let $f: A \to A'$ be a bijection. Then $f$ is an affine transformation if and only if $f$ preserves collinearity: whenever points $P$ , $Q$ , $R$ are collinear, so are $f(P)$ , $f(Q)$ , $f(R)$ .

Moreover, the group of affine transformations $\text{Aff}(A)$ is generated by translations and linear transformations.

This remarkable theorem shows that affine structure is completely determined by the notion of collinearity. No reference to distances, angles, or even ratios is needed—just the combinatorial property of which triples of points lie on lines.

The proof uses the fact that preserving collinearity forces preservation of parallelism and ratios on lines. The dimension restriction $n \geq 2$ is necessary: in dimension 1, any bijection of a line preserves collinearity, but not all are affine.

Remark

The Fundamental Theorem has profound implications:

Axiomatization: Affine geometry can be axiomatized using only incidence relations (which points lie on which lines)
Characterization: Affine transformations are precisely the collineation-preserving bijections
Simplification: To verify a map is affine, we need only check the simple condition of preserving collinearity

The theorem generalizes to higher-dimensional subspaces: an affine transformation preserves the property of points lying in any affine subspace. This follows from collinearity preservation via induction on dimension.

ExampleApplication to Perspective

In perspective drawing, artists use vanishing points to create the illusion of depth. Mathematically, this corresponds to projection from a 3D scene onto a 2D canvas.

Parallel lines in 3D that are not parallel to the canvas appear to converge at a vanishing point. This is because perspective projection is not affine—it doesn't preserve parallelism. Affine transformations (like orthogonal projection) preserve the parallel structure, which is why engineering drawings often use orthogonal rather than perspective projection.

The structure of the affine group has the form:

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

This is a semidirect product: every affine transformation uniquely decomposes as a translation followed by a linear transformation. The subgroup $GL(n, \mathbb{R})$ acts on $\mathbb{R}^n$ by the standard action, giving the semidirect product structure.

DefinitionAffine Equivalence

Two geometric figures in affine spaces are affinely equivalent if there exists an affine transformation mapping one to the other.

For example:

All triangles are affinely equivalent (given two triangles, we can map three non-collinear points to three non-collinear points)
All ellipses are affinely equivalent to circles
All parallelograms are affinely equivalent

This shows that many shape distinctions in Euclidean geometry collapse in affine geometry.

The theorem also applies to affine algebraic geometry, where varieties defined by polynomial equations in affine space are studied up to affine equivalence. The affine coordinate ring of a variety encodes its geometric properties in a way compatible with affine transformations, making it a fundamental tool in algebraic geometry.