Constructive modeling of FRep solids using spline volumes

ABSTRACT

We present an approach to constructive modeling of FRep solids [2] defined by real-valued functions using 4D uniform rational cubic Bspline volumes as primitives. While the first three coordinates are used to represent the spatial component of the volume to be sculpted, the fourth coordinate is used as a scalar, which corresponds to a function value or a volume density. Thus, the shape can be manipulated by changing the scalar control coefficients of the spline volume. This modeling process is interactive as the isosurface can be polygonized and visualized in real time. The distance property we obtain, combined with the properties of the spline volumes, allow us to use the resulting 3D solid as a leaf of a constructive tree and to apply to it set-theoretic, blending and other operations defined using R-functions [2]. Additional deformations can be achieved by moving arbitrary points in the coordinate space and applying space mapping at any level of the constructive tree. The final constructive solid is defined by a single real-valued function evaluated by the tree traversing procedure.

REFERENCES

• 1: David R. Forsey , Richard H. Bartels, Hierarchical B-spline refinement, ACM SIGGRAPH Computer Graphics, v.22 n.4, p.205-212, Aug. 1988
• 2: Pasko A., Adzhiev V., Sourin A., Savchenko V., Function representation in geometric modeling: concepts, implementation and applications", The Visual Computer, vol.11, No.8, 1995, pp.429-446
• 3: Pasko A., Pilyugin V., Pokrovskiy V., Geometric modeling in the analysis of trivariate functions, Computers and Graphics, vol. 12, Nos.3/4, 1988, pp.457.465. Original in Russian: Communications of JINR P10-86-310, Joint Institute of Nuclear Research, Dubna, 1986.
• 4: Raviv A., Elber G. Three dimensional freeform sculpting via zero sets of scalar trivariate functions. Technical Report CIS9903, Computer Science Department, Technion, 1999.
• 5: Schmitt B., gaTkov M., Pasko A., Savchenko V. Volume sculpting with 4D spline volumes, CISST2000, Las Vegas, pp. 475-483, 2000.
• 6: Eric J. Stollnitz , Tony D. Derose , David H. Salesin, Wavelets for computer graphics: theory and applications, Morgan Kaufmann Publishers Inc., San Francisco, CA, 1996

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, Performance, Theory