Fast Surface Modelling Using a 6th Order PDE

Although the control-point based parametric approach is used most widely in free-form surface modelling, complementary techniques co-exist to meet various specialised requirements. The partial differential equation (PDE) based modelling approach is especially suitable for satisfying surface boundary constraints. They are also effective for the generation of families of free-form surfaces, which share a common base and differ in their secondary features. In this paper, we present a fast surface modelling method using a sixth order PDE. This PDE provides enough degrees of freedom not only to accommodate tangent, but also curvature boundary conditions and offers more shape control parameters to serve as user controls for the manipulation of surface shapes. In order to achieve real-time performance, we have constructed a surface function and developed a high-precision approximate solution to the 6th order PDE. Unlike some existing PDE-based techniques, this resolution method can satisfy the boundary conditions exactly, and is able to create free-form surfaces as fast and almost as accurately as the closed-form (analytical) solutions. Due to the fact that it has sufficient degrees of freedom to accommodate the continuity of 3-sided and 4-sided surface patches at their boundaries, this method is able to model complex surfaces consisting of multiple patches. Compared with existing PDE-based modelling methods, this method is both fast and can solve a larger class of surface modelling problems.