Shape Analysis of Euclidean Curves under Frenet-Serret Framework
Existing curve shape analysis methods rely solely on first-order geometric information (e.g., tangent vectors), leading to registration artifacts and interpolation distortions. To address this, we propose a Riemannian geometric modeling framework grounded in the Frenet–Serret apparatus. By incorporating generalized curvature—comprising both curvature and torsion—and constructing the Square Root Velocity Transformation (SRVT), we embed open and closed curves from Euclidean space into a Riemannian manifold endowed with a well-defined metric structure. This representation is the first to jointly exploit curvature and torsion within an invariant framework, thereby fully capturing higher-order differential geometric features. The approach significantly improves physical interpretability and visual fidelity in geodesic distance computation, shape averaging, and interpolation. Extensive evaluation on synthetic data and real-world sign language motion trajectories demonstrates superior modeling accuracy and robustness over conventional first-order methods.