Direct manipulations of NURBS surfaces subjected to geometric constraints

Shape design is a complex process and requires interactive manipulations of curves and surfaces. This paper presents a deformable model for the direct manipulations of NURBS surfaces. A NURBS surface is modeled as a deformable plate subjected to specified constraints. Formulations of a deformable model results in a set of linear equations for a non-rational B-spline surface and a set of non-linear equations for a NURBS surface. A perspective mapping is introduced to convert NURBS formulations to B-spline formulations and linearize the deformable model defined for a NURBS surface. The final equations for a deformable model are similar to those obtained for the bending of a plate problem. These equations are defined by a stiffness matrix that is formulated in terms of the blending functions of a parametric surface and a force vector that is associated with the user-defined constraints. The solution of these equations results in the new locations of control points so that the deformed surface satisfies user-defined constraints. Several examples are presented to demonstrate the capabilities of this approach.