Same shape, two ways of seeing it
Take a square. In Euclidean geometry, it's a 4-sided figure with equal sides and four right angles — and any rigid motion (rotation, reflection, translation) gives you another square. Anything else "deforms" it. Affine geometry is more permissive: it considers a square and a parallelogram to be the same object, because one can be sent to the other by an affine map. The difference is what each geometry chooses to preserve.
Euclidean RIGID
Allowed: rotation only. Square → rotated square. Distances and angles preserved.
Affine FLEXIBLE
Allowed: shear, scale, rotation. Square → parallelogram. Parallels preserved; angles & distances are not.
The cheat sheet
Here is the compact view. Every Euclidean transformation is also affine — but not every affine transformation is Euclidean. The further down the table you read, the more "structure" survives.
| Property | Affine | Euclidean |
|---|---|---|
| Straight lines stay straight | ✓ Yes | ✓ Yes |
| Parallel lines stay parallel | ✓ Yes | ✓ Yes |
| Midpoints / ratios on a line | ✓ Yes | ✓ Yes |
| Convexity | ✓ Yes | ✓ Yes |
| Angles between lines | ✗ No | ✓ Yes |
| Distances / lengths | ✗ No | ✓ Yes |
| Areas (in absolute terms) | ✗ No | ✓ Yes |
| Ratios of areas | ✓ Yes | ✓ Yes |
| Circles vs ellipses | ✗ No (same class) | ✓ Yes |
The key entry is the third row from the bottom: ratios of areas survive affine maps even when areas themselves don't. That fact alone makes affine reasoning powerful for the centroid and many classical theorems.
Euclidean is a subgroup of Affine
Each geometry comes with a group of allowed transformations. The Euclidean group is strictly contained in the affine group: every Euclidean motion is affine, but most affine maps are not Euclidean.
Euclidean = Affine with the extra constraint that A is orthogonal:
An orthogonal matrix has ATA = I. This forces columns to be unit length and perpendicular — exactly what's needed to preserve distances. Drop that constraint and you're back in the full affine world.
Counting parameters: in 2D, the affine group has 6 free numbers (4 in the matrix, 2 in the translation). The Euclidean group has only 3 (rotation angle + 2 translation components). The "extra" 3 in affine encode shear and non-uniform scaling.
Where affine sits among the geometries
Felix Klein's "Erlangen program" (1872) organized geometries by which transformations they allow. The more transformations a geometry permits, the less structure it preserves. Each rung loosens the rules.
Topology
Continuous deformation. Lines can bend, but a coffee cup is a donut. Preserves only: connectedness, holes.
Projective Geometry
Camera-style perspective allowed. Parallel lines can meet at a "point at infinity." Preserves: lines, incidence, cross-ratio.
Affine Geometry ← YOU ARE HERE
Linear maps + translations. Parallels preserved. Preserves: lines, parallelism, ratios on lines, ratios of areas.
Similarity Geometry
Affine + uniform scaling only (no shear). Preserves: angles, ratios of distances. Same shape, different size.
Euclidean Geometry
Rigid motions only. Preserves everything: lines, parallels, angles, distances, areas. The strictest of the standard geometries.
Each row inherits everything below it. So Euclidean theorems are a subset of similarity theorems, which are a subset of affine, which are a subset of projective. Working at a higher level (more permissive) often gives cleaner proofs — which is exactly why we'll spend the rest of this unit at the affine rung.
Two languages for two kinds of question
Not every geometric question needs every tool. The right framework is the weakest one that still has all the structure your problem needs. Use this rule of thumb:
Use Affine when... FLEXIBLE
- You're proving a theorem about ratios, midpoints, or collinearity (e.g. centroid, Menelaus, Ceva).
- The result should be invariant under stretching or shearing.
- You're building a 2D/3D rendering pipeline.
- Convexity or ratios of areas is the core idea.
Use Euclidean when... RIGID
- The problem mentions an actual distance or angle.
- Pythagoras, trig, dot products, or perpendicularity is essential.
- Circles (as opposed to ellipses) matter.
- You need a length to come out to a specific number.