Triangular NURBS and their dynamic generalizations

Triangular B-splines are a new tool for modeling a broad class of objects defined over arbitrary, nonrectangular domains. They provide an elegant and unified representation scheme for all piecewise continuous polynomial surfaces over planar triangulations. To enhance the power of this model, we propose triangular NURBS, the rational generalization of triangular B-splines, with weights as additional degrees of freedom. Fixing the weights to unity reduces triangular NURBS to triangular B-splines. Conventional geometric design with triangular NURBS can be laborious, since the user must manually adjust the many control points and weights. To ameliorate the design process, we develop a new model based on the elegant triangular NURBS geometry and principles of physical dynamics. Our model combines the geometric features of triangular NURBS with the demonstrated conveniences of interaction within a physics-based framework. The dynamic behavior of the model results from the numerical integration of differential equations of motion that govern the temporal evolution of control points and weights in response to applied forces and constraints. This results in physically meaningful hence highly intuitive shape variation. We apply Lagrangian mechanics to formulate the equations of motion of dynamic triangular NURBS and finite element analysis to reduce these equations to efficient numerical algorithms. We demonstrate several applications, including direct manipulation and interactive sculpting through force-based tools, the fitting of unorganized data, and solid rounding with geometric and physical constraints.