One structural pair · five algorithmic lenses

The same problem looks different to every method.

Keep one two-domain hinge scenario fixed. Change the method, not the protein, and compare what each method requires, preserves, and misses.

Shared teaching scenario

teaching-hinge-r1 · 28 Cα-like points · two domains · one hinge · one perturbable tail

Method	Representation	Decision rule	Output	Strength	Blind spot
Kabsch 1976	Known corresponding 3D points	Solve the best rotation and translation for fixed pairs	Rotation, translation, and RMSD	A precise optimum that becomes the geometric core of later iterative methods	It does not discover residue matches or solve domain motion with one rigid body
DALI 1993	An internal residue-distance matrix	Compare compatible blocks in distance matrices	Locally similar blocks and their combination	Naturally insensitive to global translation and rotation	The full DALI search and statistics are much richer than this teaching view
CE 1998	Extendable fragment pairs and a monotone path	Extend stepwise under local and whole-path thresholds	An order-constrained structural correspondence	Connects local similarity to a global path	The heuristic can miss solutions not grown from the current best seed
FAST 2005	Candidate residue pairs and their compatibility graph	Build, vote, prune, then connect with DP and refinement	A paper-inspired correspondence and project scores	Shrinks the candidate space before path search	This is a clean-room teaching implementation, not a historical binary reproduction
TM-align 2005	A distance-weighted score matrix under the current fit	Superpose, score, run DP, and iterate	An alignment and a teaching view of length-normalized scoring	Accounts for global coverage better than RMSD alone	The site's TM~ is not the official TM-align TM-score

bibliography

Kabsch 1978 verifies bibliographic facts without inventing technical claims that cannot be checked from a lawfully reviewable full text.