LESSON: Affine Invariants

Felix Klein's Erlangen programme (1872) defines a geometry by its group of symmetries. Affine geometry is the study of properties invariant under the affine group. This lesson systematically catalogs what is and isn't preserved by affine transformations, giving you a complete picture of what "affine geometry" actually studies.

1. Klein's Erlangen Programme

Principle. A geometry is a pair (X, G) where X is a set and G is a group of transformations of X. The geometric properties are exactly those properties invariant under all transformations in G.

The hierarchy of classical geometries, from most to least restrictive:

Euclidean geometry (G = isometries): preserves distances, angles, areas.
Similarity geometry (G = similarities): preserves angles and ratios of distances.
Affine geometry (G = affine maps): preserves parallelism, ratios on lines, convexity.
Projective geometry (G = projective maps): preserves cross-ratio, incidence, conics.

                        💡 Each group contains the previous one, so each geometry has fewer
                        invariants but more symmetries. Affine geometry sits exactly in the middle
                        of the classical hierarchy.
                    

2. Collinearity and Concurrence

Theorem. Affine transformations preserve collinearity: if P, Q, R are collinear, then f(P), f(Q), f(R) are collinear.

Proof. If R = (1−t)P + tQ, then f(R) = (1−t)f(P) + tf(Q), so f(R) lies on the line through f(P) and f(Q). □

As a consequence, affine maps also preserve concurrence: if three lines meet at a point, their images under an affine map also meet at a point.

Application. The medians of any triangle are concurrent at the centroid. Since any triangle is the affine image of any other triangle, and affine maps preserve concurrence and midpoints, this theorem needs to be proved only for one specific triangle (e.g., the equilateral triangle), and it automatically holds for all triangles.

                        💡 This "affine reduction" trick is powerful: prove a theorem for one
                        convenient triangle, then invoke affine invariance to conclude it holds for all.
                    

3. Parallelism and Midpoints

Theorem. Affine transformations preserve parallelism: if ℓ₁ ∥ ℓ₂, then f(ℓ₁) ∥ f(ℓ₂).

Proof. If dir(ℓ₁) = dir(ℓ₂) = span{v}, then the images have direction span{L(v)} where L is the linear part of f. Both images share the same direction, so they are parallel. □

Corollary. Midpoints are preserved. If M is the midpoint of PQ, then f(M) is the midpoint of f(P)f(Q), because M = ½P + ½Q and affine maps preserve affine combinations.

Together, these imply that parallelograms are an affine concept: a quadrilateral ABCD is a parallelogram if and only if the diagonals bisect each other, and both bisection and the resulting parallelism are affine invariants.

4. Ratios of Lengths and Areas

Theorem. Affine transformations preserve the signed ratio of collinear points: if C divides AB in ratio t:(1−t), then f(C) divides f(A)f(B) in the same ratio.

Theorem (Area ratios). If f is an affine transformation with linear part A, then areas scale by |det(A)|. In particular:

The ratio of areas of two figures is preserved (the det(A) factor cancels).
If det(A) = 1, the transformation is equi-affine (area-preserving).
If det(A) = −1, areas are preserved but orientation is reversed.

The equi-affine group (affine maps with |det| = 1) is an important subgroup. Shears and SL(n) rotations are equi-affine; scalings are not.

                        💡 Any two triangles in R2 have the same area ratio to any
                        fixed region, regardless of the affine coordinate system used.
                    

5. What is NOT Affinely Invariant

Understanding what's not preserved is just as important:

Not affine invariants:

Distance: the shear (x,y) ↦ (x+y, y) changes |PQ|.
Angles: the same shear changes angles between lines.
Perpendicularity: "right angle" is not an affine concept.
Circles: a circle can be mapped to any ellipse.
Area (absolute): scaling by 2 doubles all areas.
The cross-ratio: this is a projective invariant, not merely affine.

Key consequence. In affine geometry, there is no distinction between:

A circle and an ellipse (both are non-degenerate conics).
An equilateral triangle and a scalene triangle (both are non-degenerate triangles).
A square and a parallelogram (both are parallelograms).

In affine geometry, all triangles are "the same" and all ellipses are "the same."

                        💡 "All triangles are affinely equivalent" is perhaps the most striking
                        fact of affine geometry. It means any theorem about triangles that uses only
                        affine properties can be proved for the equilateral case alone.
                    

6. Classic Affine Theorems

Armed with the invariant toolkit, here are theorems that belong purely to affine geometry:

Theorem (Medial Triangle). The medial triangle (formed by the midpoints of the sides) has area exactly 1/4 of the original. Since area ratios are affine invariants, this needs only be verified for one specific triangle.

Theorem (Varignon). The midpoints of the sides of any quadrilateral form a parallelogram. This is an affine theorem: it uses only midpoints and parallelism.

Proof. Let ABCD be a quadrilateral with midpoints M₁, M₂, M₃, M₄ of AB, BC, CD, DA. Then M₁M₂ is parallel to AC (midpoint theorem in triangle ABC), and M₃M₄ is also parallel to AC (midpoint theorem in triangle ACD). So M₁M₂ ∥ M₃M₄. Similarly M₂M₃ ∥ M₁M₄. □

                        💡 The beauty of the affine viewpoint: Varignon's theorem follows from
                        just two applications of the midpoint theorem plus the transitivity of parallelism.
                    