Data dependent thin plate energy and its use in interactive surface modeling

When modeling spline surfaces of complex shape, one has to deal with an overwhelming number of control points. Modeling by direct manipulation of the control points is a tedious task. In particular, it is very difficult to maintain a generally pleasant looking surface shape. It becomes therefore increasingly important to build tools that allow the designer to specify only a few geometric constraints while automatically determining the explicit representation of the surface. The basic concept of such a tool is simple. In a first step one has to somehow measure the "fairness" (=quality) of a surface. Once this is achieved, an optimization process selects the one surface with optimal fairness from all surfaces satisfying the user specified geometric constraints. To measure the fairness, thin plate energy functionals ∫ κ² ₁ + κ² ₂+ 2(1 - b)κ¹κ₂ are a good choice. However, for interactive use these functionals are far too complex. We will present appropriate approximations to these functionals that allow an optimization nearly in real time. The functionals are obtained by introducing reference surfaces thus leading to data dependent, quadratic approximations to the exact thin plate energy functionals. We apply the method to interactive surface manipulations based on energy constraints.