Simple constrained deformations for geometric modeling and interactive design

ABSTRACT

Deformations are a powerful tool for shape modeling and design. We present a new model for producing controlled spatial deformations, which we term Simple Constrained Deformations (Scodef). The user defines a set of constraint points, giving a desired displacement and radius of influence for each. Each constraint point determines a local B-spline basis function centered at the constraint point, falling to zero for points beyond the radius. The deformed image of any point in space is a blend of these basis functions, using a projection matrix computed to satisfy the constraints. The deformation operates on the whole space regardless of the representation of the objects embedded inside the space. The constraints directly influence the final shape of the deformed objects, and this shape can be fine-tuned by adjusting the radius of influence of each constraint point. The computations required by the technique can be done very efficiently, and real-time interactive deformation editing on current workstations is possible.

REFERENCES

• 1: Alan H. Barr, Global and local deformations of solid primitives, ACM SIGGRAPH Computer Graphics, v.18 n.3, p.21-30, July 1984
• 2: BARR, A. AN{) WITKIN, A. EI)S. 1989. Topics in physically based modeling. In ACM Siggraph 89. Course notes 30. ACM, New York.
• 3: B()RREI,, P. AN{) BE('.iIMANN, D. 1991. Deformation of n-dimensional objects. Int. J. Cornput. Geom. Appl. 1, 4.
• 4: BotTI.I,I()N, T. L. AND OI)EIA~, P.L. 1971. Generalized Inverse Matrices. Wiley, New York.
• 5: George Celniker , Dave Gossard, Deformable curve and surface finite-elements for free-form shape design, ACM SIGGRAPH Computer Graphics, v.25 n.4, p.257-266, July 1991
• 6: J. E. Chadwick , D. R. Haumann , R. E. Parent, Layered construction for deformable animated characters, ACM SIGGRAPH Computer Graphics, v.23 n.3, p.243-252, July 1989
• 7: Sabine Coquillart, Extended free-form deformation: a sculpturing tool for 3D geometric modeling, ACM SIGGRAPH Computer Graphics, v.24 n.4, p.187-196, Aug. 1990
• 8: GRENVI I,{ ,E, T. N.E. 1960. Some applications of the pseudo-inverse of a matrix. SIAM Revue II, 15 22.
• 9: (}i~{){'TKBAY, U. AND OZt;tR', B. 1990. Free form solid modeling using deformation. Comput. Graph. 14, 3/4, 491-500.
• 10: William M. Hsu , John F. Hughes , Henry Kaufman, Direct manipulation of free-form deformations, ACM SIGGRAPH Computer Graphics, v.26 n.2, p.177-184, July 1992
• 11: Kenneth I. Joy, Utilizing parametric hyperpatch methods for modeling and display of free-form solids, Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications, p.245-254, June 05-07, 1991, Austin, Texas, United States [doi>10.1145/112515.112547]
• 12: LANCASTER, P. ANI) SAI.KAI:SKAS, K. 1986. Curve and Surface Fitting: An Introduction. Academic Press, New York.
• 13: John C. Platt , Alan H. Barr, Constraints methods for flexible models, ACM SIGGRAPH Computer Graphics, v.22 n.4, p.279-288, Aug. 1988
• 14: RAPPOPORT, A. 1991. About deformations. Int. IBM manuscript. IBM T. J. Watson Research Center, Yorktown Heights, N.Y.
• 15: Thomas W. Sederberg , Scott R. Parry, Free-form deformation of solid geometric models, ACM SIGGRAPH Computer Graphics, v.20 n.4, p.151-160, Aug. 1986
• 16: Demetri Terzopoulos , Andrew Witkin, Physically Based Models with Rigid and Deformable Components, IEEE Computer Graphics and Applications, v.8 n.6, p.41-51, November 1988 [doi>10.1109/38.20317]

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

Additional Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Splines

General Terms:
Design

Keywords:
B-splines, constraints, deformation, geometric design, geometric modeling, interpolation, spatial deformation