shape-space-dimension-kn-minus-4
IN premise — summaries/2026/08/24/wiki-Procrustes_analysis.md
Created 2026-08-24T17:11:21+00:00
The shape space of k landmark points in n dimensions is a manifold of dimension kn − 4, after removing 4 degrees of freedom (2 translation + 1 rotation + 1 uniform scaling in 2D)
Summary
If you describe a shape using k landmark points in 2D, you need exactly 2k − 4 independent numbers to fully capture what makes it "itself" — once you strip out where it sits, which way it's facing, and how big it is. This gives the system a precise count of how many distinct ways shapes can differ, which matters whenever you're comparing, classifying, or interpolating between geometric configurations.