Effective Geometric Acceleration from Tangent-Direction Curvature

We investigate a geometric framework in which effective acceleration laws are associated with a curvature parametrization defined on tangent-direction space. Within this approach, particle motion is described through local geometric quantities involving a projective-curvature invariant and characteristic dynamical scales. In regimes of slowly varying curvature, the resulting acceleration laws recover the structure of standard inverse-square behaviour while providing an alternative geometric parametrization in which acceleration is associated with variations of tangent-direction geometry. Illustrative applications to spherically symmetric configurations and orbital motion are discussed. In particular, inverse-square scaling may be recovered for suitable curvature profiles, while circular motion admits a natural representation in terms of a characteristic evolution timescale. These results suggest that certain acceleration phenomena may admit an effective geometric description based on projective curvature in tangent-direction space.Keywords : geometric methods in physics --projective geometry --tangent-direction curvature --effective acceleration laws --orbital dynamics --scale-invariant dynamics}