Cyclic animation using partial differential equations

Abstract

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 consists of using attaching the set of curves to a skeletal system holding the animation for cyclic motions linked to a set mathematical expressions, the second one exploits the spine associated with the analytic solution of the PDE as a driving mechanism to achieve cyclic animation, which is also manipulated mathematically. The first of these approaches is implemented within a framework related to cyclic motions inherent to human-like characters, whereas the spine-based approach is focused on 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.

Physical Cyclic Animations

“…These approaches guarantee cyclic animations, at the cost of producing less physical motion. Some approaches solve partial differential equations with periodic boundary conditions using spectral methods [Castro et al 2010].…”

Modelling character motions on infinite-dimensional manifolds

“…The problem of modelling cyclic animations has been covered in, among others, [15] and [6]. The authors of [15] use a timeseries representation and Fourier analysis techniques, while the authors of [6] apply PDE methods to generate cyclic animations for humans as well as fish locomotion. In Sect.…”

“…5 Also known as elastic metric. 6 They are parametrized by weights balancing the influences of bending vs. stretching forces.…”

“…In every iteration, the projection calculates a correction vector along the normal space (6). See Algorithm 1 and [19] for more details.…”