Dynamic Non-Uniform Rational B=Splines

Non-uniform rational B-splines (NURBS) have become a de facto standard in commercial modeling systems because of their power to represent both free-form shapes and some common analytic shapes. To date, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom (DOFs )-control points and associated weights-to achieve desired shapes. Despite modem interactive devices, this conventional shape modification process can be clumsy and laborious when it comes to designing complex, real-world objects. This thesis paves the way for NURBS to achieve their full potential, by putting the laws of physics on their side. The thesis proposes, develops, and applies dynamic NURBS (D-NURBS), a physics-based generalization of the NURBS representation. D-NURBS unify the features of the industry-standard g~ometry with the many demonstrated conveniences of interaction through physical dynamics. D-NURBS are physics-based models that incorporate mass distributions, internal deformation energies, forces, and other physical quantities into the NURBS geometric substrate. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that automatically evolve the geometric DOFs in response to the applied forces and constraints to produce physically meaningful, hence highly intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required .spedfications not only in the traditional indirect fashion, by adjusting control points, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. An important advantage of this physics-based framework is that existing geometric toolkits continue to be applicable at the basic geometry level, while it also affords designers new force-based toolkits that support dynamic manipulation and interactive sculpting at the physics level. The mathematical development consists of four related parts: (i) D-NURBS curves, (ii) tensor product D-NURBS surfaces, (iii) swung D-NURBS surfaces, and (iv) triangular DNURBS surfaces. We use Lagrangian mechanics to formulate the equations of motion for all four varieties, and finite element analysis to reduce these equations to efficient algorithms that can be simulated at interactive rates using standard numerical techniques. We implement a prototype modeling environment based on D-NURBS, demonstrating that D-NURBS are effective tools in a wide range of applications in CAD and graphics. We· demonstrate shape blending, scattered data fitting, surface trimming, cross-sectional shape design, shape metamorphosis, and free-form deformation with geometric and physical constraints. Thus, D-NURBS provide a systematic and unified approach for a variety of CAD and graphics modeling problems such as constraint-based optimization, variational parametric design, automatic weight selection, shape approximation, user interaction, etc. They also support direct manipulation and interactive sculpting through the use of forcebased manipulation tools, the specification of geometric constraints, and the adjustment of physical parameters such as mass, damping, and elasticity.