Dynamic manipulation of triangular B-splines

Abstract

Abstract: Triangular B-splines provide a unified representation scheme for all piecewise polynomials. They are useful for modeling a broad class of complex objects defined over arbitrary, nonrectangular domains. To date, however, they have been viewed as purely geometric primitives requiring the manual adjustment of multiple control points to design shapes. This indirect design process can be laborious and often clumsy. As an alternative, we develop a new model based on the elegant triangular B-spline geometry and principles of physical dynamics. The dynamic behavior of our model, resulting from the numerical integration of differential equations of motion, produces physically meaningful and highly intuitive shape variation. The equations govern the evolution of control points in response to applied forces and constraints. We use Lagrangian mechanics to formulate the equations of motion and finite element analysis to reduce these equations to efficient numerical algorithms. Dynamic triangular B-splines provide a systematic and unified approach for a variety of solid modeling problems including shape blending, constraint-based design, and parametric design. They also support direct manipulation and interactive sculpting of shapes using force-based tools. We demonstrate several applications of dynamic triangular B-splines, including interactive sculpting using forces and physical parameters, the fitting of unstructured data, and solid rounding with geometric and physical constraints.