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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:

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]:

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.


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Next: 2 Background Up: Dynamic NURBS with Geometric Previous: Contents

Demetri Terzopoulos | Source Reference