Smoothing of polyhedral models

ABSTRACT

This paper is divided in two parts of which the first describes a method to obtain solid models with free-form geometry from polyhedral models. This is achieved by replacing the edges of the original model with curved faces, i.e. sharp edges are replaced by rounds or fillets. The “radii” of these rounds and fillets are controlled by weight values assigned to the original edges. The value of each weight can vary in the interval from 0 to 1, which gives increased possibilities for the user to control the radius of a round or fillet without changing the topology or shape of the polyhedral model.Several of the curved faces will be non-rectangular and in the second part of the paper a method to determine surfaces to these faces is described. To do this we can either split these faces into rectangular subfaces and fit rectangular surfaces to the subfaces or use non-rectangular surfaces. I have tried the second alternative and used a scheme that is inspired by the methods presented by Gregory & Charrot, (1980), (1983) and (1984). However, my method differs from theirs in some respects:Their methods are based on a convex combination of Boolean sum surfaces which interpolate position and slope along to adjacent boundaries. The Boolean sum surfaces are based on linear Taylor interpolants. Instead of Boolean sum surfaces I use convex combinations of Taylor interpolants.I have also found that linear Taylor interpolants may not always give a satisfactory shape of the interior of the surface. A considerable improvement can be obtained by using higher degree interpolants.

REFERENCES

• 1: Robert E. Barnhill , Richard F. Riesenfeld, Computer Aided Geometric Design; Proceedings of a Conference Held at the University of Utah, Salt Lake City, Utah, March 18-21, 1974, Academic Press, Inc., Orlando, FL, 1974
• 2: Barnhill, R.E., Brown, J.H. and KLucewicz, I.M. (1978), A new twist in CAGD, Computer Graphics and Image Processing 8, 78-79.
• 3: Catmull, E.E. and Clark, J.H. (1978), Recursively generated B-spline surfaces on arbitrary topological meshes, Computer-aided Design 10, 350-355.
• 4: Charrot, P. and Gregory, J.A. (1984), A pentagonal surface patch for computer aided geometric design, Computer Aided Geometric Design 1, 87-94.
• 5: Hiroaki Chiyokura , Fumihiko Kimura, Design of solids with free-form surfaces, Proceedings of the 10th annual conference on Computer graphics and interactive techniques, p.289-298, July 25-29, 1983, Detroit, Michigan, United States
• 6: Fj~llstr~m, P.O. (1985), Integration of Free-Form Surfaces and Solid Modelling, Dissertation.
• 7: I. D. Faux , M. J. Pratt, Computational Geometry for Design and Manufacture, Halsted Press, New York, NY, 1979
• 8: GPM, An Internordic Joint Project (1982), GPM Report No. 25
• 9: Gregory, J.A. (1974), Smooth Interpolation without Twist Constraints, in: (Barnhill et al. (1974)), 71-88.
• 10: Gregory, J.A. and Charrot, P. (1980), A C:~ triangular interpolation patch for computer-aided geometric design, Computer Graphics and Image Processing 13, 80-87.
• 11: Gregory, J.A. (1983), C~ rectangular and non-rectangular surface patches, in: (Barnhill et al. (1983)), 25-33.
• 12: Sabin, M.A. (1983), Non-Rectangular Surface Patches Suitable for Inclusion in a B-Spline Surface, Eurographics '83.

INDEX TERMS

Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Smoothing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory

  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations
  I.4 IMAGE PROCESSING AND COMPUTER VISION
      I.4.3 Enhancement
          Subjects: Smoothing

General Terms:
Algorithms, Theory, Verification