---
tag: K5C968HMSN
kind: concept
name: Affine differential geometry
msc: [53A15]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_differential_geometry
same_as:
  wikidata: Q4688939
  wikipedia: Affine differential geometry
---

## Summary

Affine differential geometry studies curves and surfaces through the properties that do not change under volume-preserving affine maps of space.
Lengths and angles are not preserved by such maps, so the Euclidean curvature of a curve or surface is not an invariant here.
Instead the theory builds its own invariants.
Examples are the affine arc length of a curve and the affine normal and Blaschke metric of a hypersurface.
It is an instance of the idea that each group of transformations has its own geometry.

## Explanations

### L1 human

Stretch a drawing in one direction and squeeze it in another so that areas stay the same.
Circles turn into ellipses, and lengths change, but some features of curves survive.
Affine differential geometry studies exactly these surviving features.
For example, every parabola looks the same as every other parabola from this point of view, just as every circle looks the same in Euclidean geometry up to size.