Geometry Foundations¶
Recall (doc/postdoc). Not a first course.
Job: incidence structure of Euclidean geometry; primitives before metric/order axioms (Hilbert).
Core¶
| Primitive | Role |
|---|---|
| Point / line / plane | \(0\)/\(1\)/\(2\)-flats in affine \(3\)-space |
| Segment / ray / line | convex hull; half-line; affine line |
| Betweenness | order on a line |
| Congruence | isometry equivalence (segments/angles) |
Incidence: two distinct points determine unique line; three non-collinear points determine unique plane.
Euclid I–IV + Playfair (V): through \(P\notin\ell\), exactly one line through \(P\) parallel to \(\ell\).
Model: \((\mathbb{R}^n,\|\cdot\|_2)\) with affine subspaces — completeness of \(\mathbb{R}\) is analytic input, not pure Euclid.
Used later for¶
- Parallel lines: Euclidean \(\Leftrightarrow\) similar triangles scale
- Non-Euclidean: drop/alter V
- Topology: manifolds locally Euclidean
- Linear algebra: affine spaces over \(K\)
Check¶
Does incidence alone determine the parallel postulate?
No — affine planes over any field share incidence patterns of parallels (Playfair-type); curvature/metric axioms distinguish geometries; hyperbolic plane has different parallel behavior.