Automatic Reconstruction and Web Visualization of Complex PDE Shapes

Various Partial Differential Equations (PDE) have been used in computer graphics for approximating surfaces of geometric shapes by finding solutions to PDEs subject to suitable boundary conditions. The PDE boundary conditions are defined as 3D curves on the surface of the shapes. We propose how to automatically derive these curves as boundaries of curved patches on the surface of the original polygon mesh. The analytic solution to the PDE used throughout this work is fully determined by finding a set of coefficients associated with parametric functions according to the particular set of boundary conditions. The PDE coefficients require an order of magnitude smaller space compared to the original polygon data and can be interactively rendered with different level of detail. It allows for an efficient exchange of the PDE shapes in 3D Cyber worlds and their web visualization. In this paper we analyze and formulate the requirements for extracting suitable boundary conditions, describe the algorithm for the automatic deriving of the boundary curves, and present its implementation as a part of the function-based extension of VRML and X3D.