procrustes-3d-rotation-via-svd-kabsch

IN premise — summaries/2026/08/24/wiki-Procrustes_analysis.md

Created 2026-08-24T17:11:21+00:00

In 3D Procrustes analysis, the optimal rotation is a 3×3 rotation matrix found via Singular Value Decomposition of the cross-covariance matrix, where the constraint det(R) = 1 (no reflection) is known as the Kabsch algorithm

Summary

Finding the best rotation to align one 3D shape to another has a clean closed-form solution: decompose the cross-covariance matrix, pull out the rotation, and correct for any accidental mirror flip. This means the system can align point clouds, protein structures, or other 3D objects in a single deterministic step without needing iterative optimization.