1 Introduction

In 1975 Versprille [40] proposed the Non-Uniform Rational B-Splines or NURBS. This shape representation for geometric design generalized Riesenfeld's B-splines. NURBS quickly gained popularity and were incorporated into several commercial modeling systems [23]. The NURBS representation has several attractive properties. It offers a unified mathematical formulation for representing not only free-form curves and surfaces, but also standard analytic shapes such as conics, quadrics, and surfaces of revolution. By adjusting the positions of control points and manipulating associated weights, one can design a large variety of shapes using NURBS [10, 12, 24, 21, 22, 23, 39].

Because NURBS are a purely geometric representation, however, their extraordinary flexibility has some drawbacks:

• The designer is faced with the tedium of indirect shape manipulation through a bewildering variety of geometric parameters; i.e., by repositioning control points, adjusting weights, and modifying knot vectors. Despite the recent prevalence of sophisticated 3D interaction devices, the indirect geometric design process remains clumsy and time consuming in general.
• Shape design to required specifications by manual adjustment of available geometric degrees of freedom is often elusive, because relevant design tolerances are typically shape-oriented and not control point/weight oriented. The geometric ``redundancy'' of NURBS tends to make geometric shape refinement ad hoc and ambiguous; for instance, to adjust a shape should the designer move a control point, or change a weight, or move two control points, etc.?
• Typical design requirements may be stated in both quantitative and qualitative terms, such as ``a fair and pleasing surface which approximates scattered data and interpolates a cross-section curve.'' Such requirements impose both local and global constraints on shape. The incremental manipulation of local shape parameters to satisfy complex local and global shape constraints is at best cumbersome and often unproductive.

Physics-based modeling provides a means to overcome these drawbacks. Free-form deformable models, which were introduced to computer graphics in [37] and further developed in [36, 26, 33, 20, 34, 17] are particularly relevant in the context of modeling with NURBS. Important advantages accrue from the deformable model approach [36]:

• The behavior of the deformable model is governed by physical laws. Through a computational physics simulation, the model responds dynamically to applied simulated forces in a natural and predictable way. Shapes can be sculpted interactively using a variety of force-based ``tools.''
• The equilibrium state of the dynamic model is characterized by a minimum of the potential energy of the model subject to imposed constraints [35]. It is possible to formulate potential energy functionals that satisfy local and global design criteria, such as curve or surface (piecewise) smoothness, and to impose geometric constraints relevant to shape design.
• The physical model may be built upon a standard geometric foundation, such as free-form parametric curve and surface representations. This means that while shape design may proceed interactively or automatically at the physical level, existing geometric toolkits are concurrently applicable at the geometric level.

In this article, we propose Dynamic NURBS, or D-NURBS. D-NURBS are physics-based models that incorporate mass distributions, internal deformation energies, and other physical quantities into the NURBS geometric substrate. Time is fundamental to the dynamic formulation. The models are governed by dynamic differential equations which, when integrated numerically through time, continuously evolve the control points and weights in response to applied forces. The D-NURBS formulation supports interactive direct manipulation of NURBS curves and surfaces, which results in physically meaningful hence intuitively predictable motion and shape variation.

Using D-NURBS, a modeler can interactively sculpt complex shapes not merely by kinematic adjustment of control points and weights, but, dynamically as well--by applying forces. Additional control over dynamic sculpting stems from the modification of physical parameters such as mass, damping, and elastic properties. Elastic functionals allow the imposition of qualitative ``fairness'' criteria through quantitative means. Linear or nonlinear constraints may be imposed either as hard constraints that must not to be violated, or as soft constraints to be satisfied approximately. The latter may be interpreted intuitively as simple forces. Optimal shape design results when D-NURBS are allowed to achieve static equilibrium subject to shape constraints. All of these capabilities are subsumed under an elegant formulation grounded in physics.

Section 2 discusses the similarities and distinctive features of D-NURBS relative to prior models. Section 3 briefly reviews NURBS geometry and its properties. In Section 4, we formulate D-NURBS and derive their equations of motion. Section 5 discusses the application of forces and constraints for physics-based design. We discuss the numerical simulation of D-NURBS in Section 6. Section 7 describes our prototype D-NURBS modeling system and presents applications and results. Section 8 concludes the article.