This Article 
 Bibliographic References 
 Add to: 
D-NURBS: A Physics-Based Framework for Geometric Design
March 1996 (vol. 2 no. 1)
pp. 85-96

Abstract—This paper presents dynamic NURBS, or D-NURBS, a physics-based generalization of 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 common analytic shapes. Traditionally, 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. The conventional shape modification process can often be clumsy and laborious. 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, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces, and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of CAGD applications such as shape blending, scattered data fitting, and interactive sculpting.

[1] J. Baumgarte,"Stabilization of Constraints and Integrals of Motion in Dynamical Systems," Comp. Meth. in Appl. Mech. and Eng., vol. 1, pp. 1-16, 1972.
[2] M.I.J. Bloor and M.J. Wilson, “Representing PDE Surfaces in Terms of B-Splines,” Computer Aided Design, vol. 22, no. 6, pp. 324-331, 1990.
[3] M.I.J. Bloor and M.J. Wilson, “Using Partial Differential Equations to Generate Free-Form Surfaces,” Computer Aided Design, vol. 22, no. 4, pp. 202-212, 1990.
[4] B. Brunnett,H. Hagen, and P. Santarelli,"Variational Design of Curves and Surfaces," Surveys on Mathematics for Industry, vol. 3, pp. 1-27, 1993.
[5] G. Celniker and D. Gossard, “Deformable Curve and Surface Finite Elements for Free-Form Shape Design,” Proc. ACM SIGGRAPH, pp. 257-266, July 1991.
[6] G. Celniker and W. Welch, “Linear Constraints for Deformable B-Spline Surfaces,” Proc. Symp. Interactive 3D Graphics, pp. 165-170, 1992.
[7] W. Dahmen,C. Micchelli, and H.-P. Seidel,"Blossoming Begets B-Spline Bases Built Better by B-Patches," Mathematics of Computation, vol. 59, no. 199, pp. 97-115, 1992.
[8] C. de Boor,"On Calculating with B-Splines," J. Approximation Theory, vol. 6, no. 1, pp. 50-62, 1972.
[9] H. Muammar and M. Nixon, "Tristage Hough Transform for Multiple Ellipse Extraction," IEE Proc.-E, vol. 138, no. 1, pp. 27-35, Jan. 1991.
[10] G. Farin,Curves and Surfaces for Computer Aided Geometric Design, 3rd ed. New York: Academic Press, 1992.
[11] I.D. Faux and M.J. Pratt, Computational Geometry for Design and Manufacture, Ellis Horwood, Chichester, England, 1979, pp. 142.
[12] P. Fong and H.-P. Seidel,"An Implementation of Triangular B-Spline Surfaces Over Arbitrary Triangulations," Computer Aided Geometric Design, vol. 3-4, no. 10, pp. 267-275, 1993.
[13] D.R. Forsey and R.H. Bartels, “Hierarchical B-Spline Refinement,” Computer Graphics (SIGGRAPH '88 Proc.), vol. 22, no. 4, pp. 205-212, Aug. 1988.
[14] B.R. Gossick,Hamilton's Principle and Physical Systems.New York and London: Academic Press, 1967.
[15] G. Greiner,"Variational Design and Fairing of Spline Surfaces," Proc. EUROGRAPHICS'94, pp. 143-154, Blackwell, 1994.
[16] M. Halstead, M. Kass, and T. DeRose, “Efficient, Fair Interpolation Using Catmull-Clark Surfaces,” Proc., ACM SIGGRAPH, pp. 35-44, Aug. 1993.
[17] H. Kardestuncer,Finite Element Handbook.New York: McGraw-Hill, 1987.
[18] D. Metaxas and D. Terzopoulos, “Dynamic Deformation of Solid Primitives with Constraints,” Proc. ACM SIGGRAPH, pp. 309-312, July 1992.
[19] C.A. Micchelli,"On a Numerically Efficient Method for Computing with Multivariate B-Splines, Multivariate Approximation Theory, W. Schempp and K. Zeller, eds., pp. 211-248. Basel: Birkhauser, 1979.
[20] M. Minoux,Mathematical Programming.New York: Wiley, 1986.
[21] H. Moreton and C. Séquin,"Functional Optimization for Fair Surface Design," Computer Graphics, vol. 26, no. 2, pp. 167-176, 1992 (Siggraph 92).
[22] R. Pfeifle and H.-P. Seidel,"Fitting Triangular B-Splines to Functional Scattered Data," Proc. Graphics Interface'95, pp. 26-33.San Mateo, Calif.: Morgan Kaufmann, 1995.
[23] L. Piegl, "Modifying the Shape of Rational B-splines. Part 1: Curves," Computer Aided Design, Vol. 21, No. 8, Oct. 1989, pp. 509-518.
[24] L. Piegl,"Modifying the Shape of Rational B-Splines, Part 2: Surfaces," Computer-Aided Design, vol. 21, no. 9, pp. 538-546, 1989.
[25] L. Piegl, "On NURBS: A Survey," IEEE Computer Graphics and Applications, Vol. 11, No. 1, Jan. 1991, pp. 55-71.
[26] J. Platt,"A Generalization of Dynamic Constraints," CVGIP: Graphical Models and Image Processing, vol. 54, no. 6, pp. 516-525, 1992.
[27] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes, the Art of Scientific Computing. Cambridge, Mass.: Cambridge Univ. Press, 1986.
[28] H. Qin and D. Terzopoulos,"Dynamic NURBS Swung Surfaces for Physics-Based Shape Design," Computer Aided Design, vol. 27, no. 2, pp. 111-127, 1995.
[29] H. Qin and D. Terzopoulos, “Triangular NURBS and Their Dynamic Generalization,” Computer Aided Geometric Design, vol. 14, pp. 325-347, 1997.
[30] L.L. Schumaker,"Fitting Surfaces to Scattered Data," Approximation Theory II, G.G. Lorentz, C.K. Chui, and L.L. Schumaker, eds., pp. 203-267.New York: Academic Press, 1976.
[31] J. Snyder and J. Kajiya,"Generative Modeling: A Symbolic System for Geometric Modeling," Computer Graphics, vol. 26, no. 2, pp. 369-378, 1992.
[32] D. Terzopoulos, "Regularization of Inverse Visual Problems Involving Discontinuities," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 4, pp. 413-424, 1986.
[33] D. Terzopoulos and K. Fleischer,"Deformable Models," The Visual Computer, vol. 4, no. 6, pp. 306-331, 1988.
[34] D. Terzopoulos, J. Platt, A. Barr, and K. Fleischer, “Elastically Deformable Models,” Proc. ACM SIGGRAPH, pp. 205-214, 1987.
[35] D. Terzopoulos and H. Qin, “Dynamic NURBS with Geometric Constraints for Interactive Sculpting,” ACM Trans. Graphics, vol. 13, no. 2, pp. 103-136, Apr. 1994.
[36] W. Tiller,"Rational B-Splines for Curve and Surface Representation," IEEE Computer Graphics and Applications, vol. 3, no. 6, pp. 61-69, Sept. 1983.
[37] K. Versprille, Computer Aided Design Applications of the Rational B-Spline Approximation Form, doctoral dissertation, University of Syracuse, New York, 1975.
[38] W. Welch and A. Witkin, “Variational Surface Modeling,” Proc. ACM SIGGRAPH, pp. 157-166, July 1992.
[39] C. Woodward,"Cross-Sectional Design of B-Spline Surfaces," Computers and Graphics, vol. 11, no. 2, pp. 193-201, 1987.

Index Terms:
NURBS, geometric modeling, computer-aided design, computer graphics physics-based models, finite elements, dynamics.
Hong Qin, Demetri Terzopoulos, "D-NURBS: A Physics-Based Framework for Geometric Design," IEEE Transactions on Visualization and Computer Graphics, vol. 2, no. 1, pp. 85-96, March 1996, doi:10.1109/2945.489389
Usage of this product signifies your acceptance of the Terms of Use.