Affine Geometry - Key Properties

The properties preserved by affine transformations define the content of affine geometry. These invariants reveal the structure that remains when metric notions are stripped away.

DefinitionAffine Invariants

Properties preserved by all affine transformations include:

Collinearity: If three points lie on a line, their images lie on a line
Parallelism: Parallel lines map to parallel lines (or the same line)
Ratios of parallel segments: If $AB \parallel CD$ , then $|AB|/|CD|$ equals $|f(A)f(B)|/|f(C)f(D)|$
Ratios on a line: The ratio $P$ divides $\overline{AB}$ equals the ratio $f(P)$ divides $\overline{f(A)f(B)}$
Barycentric combinations: Weighted averages of points transform predictably

Unlike Euclidean geometry, affine geometry does not preserve distances (except for translation and rotation), angles (except for similarities), or areas (except for equi-affine transformations). A circle can be transformed into an ellipse, demonstrating the loss of metric structure.

DefinitionAffine Subspaces

An affine subspace of an affine space $A$ is a subset of the form $P + U$ where $P \in A$ and $U$ is a linear subspace of the direction space. Equivalently, it's a set closed under affine combinations.

Affine subspaces of various dimensions have special names:

Dimension 0: points
Dimension 1: lines
Dimension 2: planes
Dimension $n-1$ : hyperplanes

The intersection of affine subspaces is an affine subspace (possibly empty). Two affine subspaces of dimension $k$ and $\ell$ in $n$ -dimensional space generically intersect in dimension $k + \ell - n$ (if this is non-negative). This generalizes the fact that two lines in a plane intersect at a point, and two planes in 3-space intersect in a line.

ExampleParallel Projection

Consider parallel projection from one plane $\pi_1$ to another $\pi_2$ along a direction $\mathbf{d}$ . Each point $P \in \pi_1$ maps to the intersection of $\pi_2$ with the line through $P$ in direction $\mathbf{d}$ .

This map is affine, preserving ratios and parallelism. It explains why railroad tracks appear to converge in photographs (though physically parallel)—perspective projection is not affine, but parallel projection is.

Affine combinations generalize linear combinations to affine spaces. A point is an affine combination $\sum \lambda_i P_i$ if $\sum \lambda_i = 1$ . The set of all affine combinations of points in $S$ is the affine hull of $S$ . When $\lambda_i \geq 0$ , we get convex combinations, whose set forms the convex hull.

Remark

The constraint $\sum \lambda_i = 1$ in affine combinations reflects coordinate invariance. In an affine space with no canonical origin, only combinations with this constraint are well-defined. This distinguishes affine geometry from vector space geometry where arbitrary linear combinations make sense.

Affine maps between affine spaces correspond bijectively to linear maps between their direction spaces (once origins are chosen). If $f: A \to A'$ is affine, then the associated linear map $\vec{f}: V \to V'$ satisfies:

\overrightarrow{f(P)f(Q)} = \vec{f}(\overrightarrow{PQ})

This relationship allows us to transfer linear algebra tools to affine geometry, analyzing affine transformations via their linear parts.