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Affine Geometry and What It Sees

Drop the right to measure length and angle, keep the right to draw straight lines and call lines parallel — and you land in affine geometry. We name its transformation group, list exactly what survives its blurring eye, and watch where it sits one step below projective and one step above Euclidean.

A Geometry Defined by What It Refuses to Measure

By now Klein's slogan from the start of this rung should feel like second nature: a geometry is not a list of figures but the invariants of a transformation group. Choose your group, and the geometry chooses itself — it is exactly the collection of properties that survive every motion in that group. In the previous guide you climbed the hierarchy of geometries from the top down, watching each rung gain new invariants by shrinking its group. This guide stops to live on one particular rung and asks, slowly and concretely, what it actually sees.

Affine geometry is the rung you reach by granting yourself one freedom and surrendering another. The freedom you keep is generous: you may stretch, squash, shear, rotate, slide, and even reflect the plane, as long as you never bend a straight line into a curve. The thing you surrender is the ruler and the protractor. In affine geometry there is no fact of the matter about how long a segment is or how wide an angle opens — those numbers are not yours to know. What is left is a surprisingly rich world, and learning its exact boundaries is the work ahead.

The Affine Group, in Coordinates

To pin a geometry down we name its group precisely, and for affine geometry the formula is clean. An affine transformation of the plane sends each point P with coordinates (x, y) to a new point P' by first applying an invertible linear map — multiply by a 2-by-2 matrix with nonzero determinant — and then sliding everything by a fixed translation vector. The linear part is what allows stretching and shearing; the translation part is what lets the map move the origin. Every such map is reversible, and composing two of them gives a third of the same kind, so they form a genuine group.

affine map:   (x, y)  ->  (x', y')

  x' = a x + b y + e
  y' = c x + d y + f

  with determinant  a d - b c  not equal to 0

  linear part = [a, b; c, d]   (stretch / shear / rotate / reflect)
  translation = (e, f)         (slide the whole plane)
An affine map is an invertible linear part followed by a translation; the nonzero-determinant condition is exactly what keeps it reversible and stops the plane from collapsing onto a line.

Notice how this group sits relative to the others. If you force the matrix to be a rotation or reflection that preserves length, you drop down to the rigid motions of Euclidean geometry; allow a uniform scaling on top of that and you get similarity geometry. The affine group is strictly larger than both because it permits the matrix to be any invertible matrix at all — including the lopsided shears that tilt a square into a slanted parallelogram. Larger group, fewer invariants: that is the trade the whole hierarchy runs on.

What Survives the Blur: The Affine Invariants

Now the central question of Klein's program for this rung: which properties does no affine map ever disturb? First, straightness — an affine map carries every line to a line, never to a curve, since the formula is linear in x and y. Second, parallelism — if two lines never met before the map, they never meet after it, because an affine map cannot create or destroy an intersection. So 'is a line' and 'are these two lines parallel' are honest affine facts, true or false independently of any ruler.

The subtler survivor is the one that gives affine geometry its real power: the ratio in which a point divides a segment. Take three collinear points A, B, C. An affine map will change the length |AB| and the length |BC| unpredictably — those numbers are gone — but it preserves their ratio |AB| / |BC| exactly. The most famous special case is the midpoint: the midpoint of a segment maps to the midpoint of the image segment, every single time. Halfway stays halfway, even when the two halves are stretched to wildly different physical lengths.

One more invariant rounds out the picture: the ratio of two areas. A single area is meaningless in affine geometry, because a shear or stretch scales every area by the same factor — the determinant of the matrix part. But because every region is scaled by that one common factor, the quotient of two areas is untouched: if one triangle had twice the area of another before the map, it still has exactly twice after. So area-ratios live in the affine world even though absolute areas do not.

What Gets Erased: Length, Angle, and Circles

Being honest about a geometry means being just as clear about what it cannot see. Length is the first casualty: a shear can turn a one-centimetre segment into a ten-centimetre one while leaving a parallel segment untouched, so no affine map respects |AB| as a number. Angle goes next. Apply the shear x' = x + y, y' = y to the corner of a square and the right angle tilts open; perpendicularity is not affine, and neither is any specific angle measure m(angle ABC). Saying 'these two lines are perpendicular' is simply not a sentence affine geometry can evaluate.

A vivid casualty is the circle. To affine eyes a circle is not special at all — it is merely one ellipse among many, and an affine map can squash it into any ellipse you like, or stretch a thin ellipse back out into a perfect circle. 'Circle' is a similarity notion, requiring equal scaling in all directions, so it lives one rung lower. What affine geometry can say is the weaker statement that the figure is an ellipse rather than, say, a hyperbola — that distinction survives, because it is about how the curve meets the line at infinity, which is a projective-and-affine fact, not a metric one.

Equiaffine Geometry: Putting Area Back

Klein's program is not a single ladder with fixed rungs; it is a recipe for building a geometry from any group you fancy. To see the recipe in action, let us bolt one extra condition onto the affine group and watch a new geometry appear. Demand that the matrix part have determinant exactly 1 (or exactly plus-or-minus 1 if you allow reflections). This rules out every map that scales area, leaving only those that preserve it. The resulting smaller group defines equiaffine geometry, the geometry of area-preserving affine maps.

Because we shrank the group, a new invariant must appear — and it does: absolute area itself, not merely the ratio of two areas. In equiaffine geometry a triangle of area 5 stays a triangle of area 5 no matter how you shear it, so 'area' is finally a meaningful number again, even though length and angle are still invisible. This is the Erlangen idea working in miniature: tighten the group by one constraint, gain exactly one new thing you are allowed to measure. The whole hierarchy is built by repeating this move.

  1. Start from the full affine group: all maps (x, y) -> (a x + b y + e, c x + d y + f) with a d - b c not equal to 0.
  2. Impose the single extra rule that the determinant a d - b c equals 1 (or plus-or-minus 1 with reflections allowed).
  3. Check that this smaller collection is still a group: the determinant of a product is the product of the determinants, so composing two determinant-1 maps gives another, and inverses keep determinant 1.
  4. Read off the new invariant: anything the smaller group preserves is now a legal measurement — here, absolute area joins straightness, parallelism, and midpoints as something equiaffine geometry can see.

Affine Geometry's Place on the Ladder

Step back and see the neighbours. Just above affine geometry sits projective geometry, whose group is larger still: a projective map may send some finite points off to the line at infinity and pull infinity back into view, so it does not even preserve parallelism — two parallel lines can be mapped to two lines that meet. Affine geometry is exactly the slice of projective geometry you recover by protecting the line at infinity, forbidding any map from moving it. That single restriction is what hands parallelism back to you, which is why affine geometry, and not projective, is where 'parallel' becomes a stable word.

Just below affine geometry sits similarity geometry, and below that Euclidean geometry, each reached by shrinking the matrix part further. Restrict the matrix to a scaled rotation or reflection — equal stretching in every direction — and you recover angles, and the notion of 'same shape', giving similarity. Forbid the scaling too, locking the stretch factor at 1, and lengths come back: that is Euclidean geometry, the most refined and most invariant-rich rung. The full chain projective ⊃ affine ⊃ similarity ⊃ Euclidean is a chain of shrinking groups and, inverted, of growing invariants.

It is worth saying plainly that none of these geometries is the 'real' or 'correct' one. The Erlangen program reframes the question entirely: each rung is true to its own group, and the right one to use is whichever matches the symmetries of your problem. Studying parallel projections of shadows, or the centroid of a region, or theorems about midpoints and ratios? That is affine territory, and reaching for Euclidean tools there would only burden you with structure the problem does not respect. Choosing the geometry is choosing what you agree to ignore.