This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Volume-Preserving Free-Form Solids
March 1996 (vol. 2 no. 1)
pp. 19-27

Abstract—Some important trends in geometric modeling are the reliance on solid models rather than surface-based models and the enhancement of the expressive power of models, by using free-form objects in addition to the usual geometric primitives and by incorporating physical principles. An additional trend is the emphasis on interactive performance. In this paper we integrate all of these requirements in a single geometric primitive by endowing the tri-variate tensor product free-form solid with several important physical properties, including volume and internal deformation energy. Volume preservation is of benefit in several application areas of geometric modeling, including computer animation, industrial design and mechanical engineering. However, previous physics-based methods, which usually have used some forms of "energy," have neglected the issue of volume (or area) preservation. We present a novel method for modeling an object composed of several tensor-product solids while preserving the desired volume of each primitive and ensuring high-order continuity constraints between the primitives. The method utilizes the Uzawa algorithm for non-linear optimization, with objective functions based on deformation energy or least squares. We show how the algorithm can be used in an interactive environment by relaxing exactness requirements while the user interactively manipulates free-form solid primitives. On current workstations, the algorithm runs in real-time for tri-quadratic volumes and close to real-time for tri-cubic volumes.

[1] K. Arrow,L. Hurwicz, and H. Uzawa,Studies in Linear and Nonlinear Programming.Stanford, Calif.: Stanford Univ. Press, 1958.
[2] G. Aumann,"Two Algorithms for Volume-Preserving Approximations of Surfaces of Revolution," Computer-Aided Design, vol. 24, no. 12, pp. 651-657, 1992.
[3] R. Barzel and A.H. Barr, "A Modeling System Based on Dynamic Constraints," Computer Graphics (Proc. Siggraph), Vol. 22, No. 4, Aug. 1988, pp. 179-188.
[4] M. Bercovier,Y. Hasbani,Y. Gilon, and K.J. Bathe,"A Finite Element Procedure for Non-Linear Incompressible Elasticity," invited Paper, Symp. Hybrid and Mixed Methods, S. Atluri, ed.New York: Wiley, 1981.
[5] M. Bercovier and A. Yacoby,"Minimization, Constraints and Composite Bézier Curves," Leibniz Center for Research in Computer Science, Mar. 1993.
[6] P. Borrel and D. Bechmann,"Deformation of n-Dimensional Objects," Intl. J. of Computational Geometry and Applications, vol. 1, no. 4, 1991. Also in ACM Symp. Solid Modeling, Austin, Texas, pp. 351-370, June5-7, 1991.
[7] P. Borrel and A. Rappoport,"Simple Constrained Deformations for Geometric Modeling and Interactive Design," ACM Trans. Graphics, vol. 13, no. 2, pp. 137-155, 1994.
[8] G. Celniker and D. Gossard, “Deformable Curve and Surface Finite Elements for Free-Form Shape Design,” Proc. ACM SIGGRAPH, pp. 257-266, July 1991.
[9] J.E. Chadwick,D.R. Haumann,, and R.E. Parent,“Layered construction for deformable animated characters,” Computer Graphics (SIGGRAPH’89 Proceedings), vol. 23, no. 3, pp. 243-252, 1989.
[10] P.G. Ciarlet,Introduction to Numerical Linear Algebra and Optimization.Cambridge, England: Cambridge Univ. Press, 1988.
[11] S. Coquillart, “Extended Free-Form Deformation: A Sculpturing Tool for 3D Geometric Modeling,” Computer Graphics, vol. 24, no. 4, pp. 187-196, Aug. 1990.
[12] G. Elber,"Symbolic and Numeric Computation in Curve Interrogation," Computer Graphics Forum, to appear.
[13] L. Fang and D.C. Gossard,"Reconstruction for Smooth Parametric Surfaces From Unorganized Data Points," SPIE vol. 1830, Curves and Surfaces in Computer Vision and Graphics III, pp. 226-236, 1992.
[14] G. Farin,Curves and Surfaces for Computer Aided Geometric Design, 3rd ed. New York: Academic Press, 1992.
[15] R. Farouki and J. Hinds,"A Hierarchy of Geometric Forms," IEEE Computer Graphics and Applications, vol. 5, no. 5, pp. 51-78, 1985.
[16] G.A. Gibson,Advanced Calculus.London: Macmillan, 1944.
[17] G. Greiner and H.-P. Seidel,"Curvature Continuous Blend Surfaces," IFIP Conf. Geometric Modeling in Computer Graphics,Genova, Italy, June 1993, B. Falcidieno and T.L. Kunii, eds., Geometric Modeling in Computer Graphics.New York: Springer, pp. 309-317, 1993.
[18] J. Griessmair and W. Purgathofer,"Deformation of Solids With Tri-Variate B-Splines," Eurographics 89, pp. 137-148, 1989.
[19] W.M. Hsu,J.F. Hughes, and H. Kaufman,"Direct Manipulation of Free-Form Deformations," Computer Graphics, vol. 26, no. 2, pp. 177-184, 1992 (Siggraph 92).
[20] T.J.R. Hughes,The Finite Element Method.Englewood Cliffs, N.J.: Prentice-Hall, 1987.
[21] K. Joy,"Utilizing Parametric Hyperpatch Methods for Modeling and Display of Free Form Solids," ACM Symp. Solid Modeling,Austin, Texas, pp. 245-254, June5-7, 1991.
[22] M. Kallay,"Constrained Optimization in Surface Design," B. Falcidieno and T.L. Kunii, eds., Modeling in Computer Graphics.New York: Springer-Verlag, pp. 85-94, 1993.
[23] D. Liu,"Algorithms for Computing Area and Volume Bounded by Bézier Curves and Surfaces," Mathematica Numerica Sinica, vol. 9, pp. 327-336, 1987.
[24] H. Moreton and C. Séquin,"Functional Optimization for Fair Surface Design," Computer Graphics, vol. 26, no. 2, pp. 167-176, 1992 (Siggraph 92).
[25] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes in C.Cambridge, England: Cambridge Univ. Press, 1988.
[26] A. Rappoport,Y. Hel-Or, and M. Werman,"Interactive Design of Smooth Objects Using Probabilistic Point Constraints," ACM Trans. Graphics, vol. 13, no. 2, pp. 156-176, 1994.
[27] T.W. Sederberg and S.R. Parry, “Free-Form Deformations of Solid Geometric Models,” Computer Graphics, vol. 20, no. 4, pp. 151-160, Aug. 1986.
[28] L. Shapiro Brotman and A.N. Netravali,"Motion Interpolation by Optimal Control," Computer Graphics, vol. 22, no. 4, pp. 309-316, 1988 (Siggraph 88).
[29] 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.
[30] W. Welch and A. Witkin, “Variational Surface Modeling,” Proc. ACM SIGGRAPH, pp. 157-166, July 1992.

Index Terms:
Free-form solids, free-form deformations (FFD), volume preservation, energy constraints, continuity constraints, physics-based modeling, Uzawa's algorithm.
Citation:
Ari Rappoport, Alla Sheffer, Michel Bercovier, "Volume-Preserving Free-Form Solids," IEEE Transactions on Visualization and Computer Graphics, vol. 2, no. 1, pp. 19-27, March 1996, doi:10.1109/2945.489383
Usage of this product signifies your acceptance of the Terms of Use.