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Dynamic NURBS with Geometric Constraints
for Interactive Sculpting

Demetri Terzopoulos and Hong Qin
Department of Computer Science, University of Toronto Toronto

Published in: ACM Transactions on Graphics, 13(2), April, 1994, 103-136.


This article develops a dynamic generalization of the nonuniform rational B-spline (NURBS) model. NURBS have become a de facto standard in commercial modeling systems because of their power to represent free-form shapes as well as common analytic shapes. To date, however, they have been viewed as purely geometric primitives that require the user to manually adjust multiple control points and associated weights in order to design shapes. Dynamic NURBS, or D-NURBS, are physics-based models that incorporate mass distributions, internal deformation energies, and other physical quantities into the popular NURBS geometric substrate. Using D-NURBS, a modeler can interactively sculpt curves and surfaces and design complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. D-NURBS move and deform in a physically intuitive manner in response to the user's direct manipulations. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that automatically evolve the control points and weights in response to the applied forces and constraints. To derive these equations, we employ Lagrangian mechanics and a finite-element-like discretization. Our approach supports the trimming of D-NURBS surfaces using D-NURBS curves. We demonstrate D-NURBS models and constraints in applications including the rounding of solids, optimal surface fitting to unstructured data, surface design from cross sections, and free-form deformation. We also introduce a new technique for 2D shape metamorphosis using constrained D-NURBS surfaces.

CR Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling -- Curve, Surface, solid, and object representations; Physically based modeling; splines; I.3.6 [Computer Graphics]: Methodology and Techniques -- Interaction techniques.

General Terms: Algorithms, Design, Theory

Additional Keywords and Phrases: CAGD, NURBS, Deformable Models, Dynamics, Constraints, Finite Elements, Trimming, Solid Rounding, Optimal Curve and Surface Fitting, Cross-Sectional Shape Design, Free-Form Deformation, Shape Metamorphosis.

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