Partial Differential Equations for Function Based Geometry Modelling within Visual Cyberworlds

We propose the use of partial differential equations (PDEs) for shape modelling within visual cyberworlds. PDEs, especially those that are elliptic in nature, enable surface modelling to be defined as boundary-value problems. Here we show how the PDE based on the Biharmonic equation subject to suitable boundary conditions can be used for shape modelling within visual cyberworlds. We discuss an analytic solution formulation for the Biharmonic equation which allows us to define a function based geometry whereby the resulting geometry can be visualised efficiently at arbitrary levels of shape resolutions. In particular, we discuss how function based PDE surfaces can be readily integrated within VRML and X3D environments.

1. Introduction

Cyberworlds within the cyberspace can be considered as entities where information modelling can be undertaken through the integration of both spatial and temporal models. There is no doubt that visualisation within cyberworlds is an important aspect which requires special attention. As the gap between the cyberworlds and the real world gets narrower, the demand placed for realism within these shared virtual worlds, both in terms of the complexity of geometric models as well as their appearance, is ever increasing.