Closing Trajectories: Equation‐Free Cyclic Animation via Koopman Surrogates

Abstract

Cyclic animation is widely used in computer graphics and interactive content. It supports seamless playback in games, VR, and interactive simulation, where short clips must repeat smoothly over long durations. Converting an observed non‐cyclic trajectory into a seamless loop is challenging because the endpoint states of the observed sequence rarely match exactly, and the governing equations of the underlying system are often unavailable. We therefore propose an equation‐free framework that identifies a Koopman surrogate from the observed trajectory and computes a cyclic trajectory by applying a Fourier‐parameterized, time‐varying control input under a hard temporal periodicity constraint. The resulting formulation reduces cyclic synthesis to a linearly constrained quadratic program that can be solved efficiently through a structured KKT system. Our method is applicable to a diverse range of examples, including N‐body systems, cloth, deformable objects, shallow water, etc.