[RETRACTED] Geodesic Path Planning on Configuration Space Manifolds: Shorter, Smoother Trajectories for Humanoid Robots

RETRACTION NOTICE (March 2026) This paper has been retracted by the author. The methodology compares against baseline methods that have been superseded by current state-of-the-art approaches in the field. The improvements reported are against methods that are no longer in common use, making the claimed contributions not practically relevant. The author takes full responsibility for this error and apologises for any confusion caused. The mathematical content may be technically correct but the framing and baseline comparisons do not reflect current practice. This retraction notice replaces the original publication.