Cyclic animation using partial differential equations

This work presents an efficient and fast method for achieving cyclic animation using partial differential equations (PDEs). The boundary-value nature associated with elliptic PDEs offers a fast analytic solution technique for setting up a framework for this type of animation. The surface of a given character is thus created from a set of pre-determined curves, which are used as boundary conditions so that a number of PDEs can be solved. Two different approaches to cyclic animation are presented here. The first of these approaches consists of attaching the set of curves to a skeletal system, which is responsible for holding the animation for cyclic motions through a set mathematical expressions. The second approach exploits the spine associated with the analytic solution of the PDE as a driving mechanism to achieve cyclic animation. The spine is also manipulated mathematically. In the interest of illustrating both approaches, the first one has been implemented within a framework related to cyclic motions inherent to human-like characters. Spine-based animation is illustrated by modelling the undulatory movement observed in fish when swimming. The proposed method is fast and accurate. Additionally, the animation can be either used in the PDE-based surface representation of the model or transferred to the original mesh model by means of a point to point map. Thus, the user is offered with the choice of using either of these two animation representations of the same object, the selection depends on the computing resources such as storage and memory capacity associated with each particular application.