Congruence and Similarity¶
Recall (doc/postdoc). Not a first course.
Job: isometry vs homothety orbits of figures; Euclidean group \(E(n)=\mathbb{R}^n\rtimes O(n)\).
Core¶
| Relation | Meaning | Group action |
|---|---|---|
| Congruent \(\cong\) | there is isometry \(f\) with \(f(F)=G\) | \(E(n)\) |
| Similar \(\sim\) | isometry \(\circ\) homothety (uniform scale) | similarities |
Criteria (triangles): congruence SSS/SAS/ASA/AAS/HL; similarity AA, SSS (sides prop.), SAS (two sides prop. + included angle).
Scale factor \(k\): lengths \(\times k\), areas \(\times k^2\), volumes \(\times k^3\).
Angles preserved under similarities; orientation-reversing if improper isometries (reflections).
Used later for¶
- Linear algebra: orthogonal maps, \(O(n)\)
- Complex analysis: conformal maps (local similarity)
- Differential geometry: isometries of Riemannian manifolds
- Triangles: AA similarity in parallels
Check¶
Are all isometries of \(\mathbb{R}^n\) linear?
No — affine: \(x\mapsto Ox+b\), \(O\in O(n)\). Linear iff fix origin (after choosing origin).