Trajectoryyn

Trajectoryyn is a theoretical construct used in discussions of dynamic trajectories. It denotes a functional or descriptor that encodes how a path in space changes shape over time, often in relation to control inputs, environmental fields, or optimization objectives. The term is not a standard object in mainstream mathematics, but appears in exploratory writings and in some robotics and data-analysis discussions as a way to compare trajectory shapes.

Definitions vary. For a smooth trajectory γ: [0, T] → R^n, a trajectoryyn T[γ] may be a scalar functional such as a weighted integral of curvature, T[γ] = ∫0^T w(t) κ(t) dt, where κ is curvature and w(t) a weight. Some authors also define trajectoryyn as a vector that aggregates curvature and torsion across the path. There is no canonical form, and different proposals reflect different modeling priorities or application contexts.

Properties. Many formulations aim for a form of reparameterization invariance under proper weighting; values depend on sampling, discretization, and noise, and different definitions yield different numbers for the same geometric path. The concept is typically sensitive to the chosen representation of the trajectory and to the weighting scheme used in the definition.

Relation to related concepts. Trajectoryyn relates to total curvature, path length, and energy-based descriptors used in motion planning and animation. It can serve as a compact descriptor to compare trajectories produced by different controllers, environments, or strategies, especially in exploratory or comparative studies.

Examples and interpretation. For a planar unit-speed curve with constant curvature, a trajectoryyn defined as ∫ κ dt grows linearly with the duration, illustrating cumulative bending of the path. In practice, trajectoryyn is used as a feature or descriptor in trajectory databases and in clustering or classification tasks.

See also. curvature; total curvature; path length; Frenet-Serret framework; motion planning.