Biharmonic Bézier Surfaces

An interesting possibility is emerging from bringing together two rather different ideas in geometric modelling. Bézier surfaces give designers an intuitive way to control shape through familiar control points, while biharmonic methods provide a mathematical mechanism for producing smooth and naturally varying surfaces. Combining these ideas could give us Biharmonic Bézier surfaces, where the accessibility of classical Bézier modelling is retained but the behaviour of the surface is influenced by the smoothness properties of the biharmonic equation.

The mathematical interest lies in finding new basis functions and blending schemes that behave like Bézier representations while also reflecting the properties of solutions to biharmonic problems. This could provide greater control over fairness, curvature and the way changes propagate across a surface. Rather than treating PDE modelling and conventional parametric modelling as separate techniques, it suggests that the two can be connected within a common mathematical framework.

Such surfaces could have applications wherever smooth free form geometry is important, including computer aided design, animation, aircraft and automotive surfaces, medical modelling and character generation. Perhaps more importantly, they suggest a way of making PDE based geometry more compatible with existing design systems. If the mathematical advantages of biharmonic modelling can be expressed through tools that designers already understand, PDE methods could move from being mainly a mathematical curiosity towards becoming a practical part of geometric design.