Parametrically deformed free-form surfaces as part of a variational model

Abstract

A new approach is described which provides deformation methods for multi-patch tensor based free-form surfaces. The surface deformation generated is controlled by global geometric constraints. For example, the objective can be to deform a free-form surface until it becomes tangent to a pre-defined plane at a given point. This point can be fixed or free to slide on the surface. The parametric deformation of surfaces is dedicated to modifications of free-form surfaces within CAD software and to the design of objects submitted to aesthetic requirements. It is an alternative to previous approaches and it works with multiple surfaces through a simple mechanical model. The deformation method uses an analogy between the control polyhedron of each surface (based on a B-Spline model) and the mechanical equilibrium of a rigid bar network. The user can localize the surface deformation into an arbitrary shaped area through the selection of control polyhedron vertices spread over the entire surface. These vertices are used to automatically construct the associated bar network. The bar network equilibrium parameters are set up to achieve isotropic or anisotropic deformation as required by the designer. The surface deformation is then automatically carried through an optimization process which modifies mechanical parameters to agree with the global geometric constraint set up. The G 1 continuity across the different during the deformation process using a set of geometric constraints in addition to mechanical ones. Parametric free-form surface deformation can be subjected to non-linear geometric constraints such as the tangency of a surface to a pre-defined plane. The resolution of such a problem uses an optimization process which minimizes the variation of the parameters governing the equilibrium of the bar network, namely the external forces applied to the nodes of the network. Several examples illustrate basic deformation types with various sets of constraints.