On the construction of surfaces interpolating curves. I. A method for handling nonconstant parameter curves

ABSTRACT

In industrial design, the tool of choice for constructing surfaces that interpolate curves is the Boolean sum surface technique. However, if curves do not lie on constant parameter lines, reparametrizations will be needed, and this may introduce derivative discontinuities. A new technique which shows promise in overcoming this problem is described here. The method is based on describing the interpolation problem directly as a system of linear equations rather than as a curve-blending problem. The resulting system of equations is usually underdetermined and can be solved using numerical linear algebra methods without the a priori determination of certain parameters. The “free” parameters can be used to control the shape of the resulting surface. Two examples of the procedure are given.

REFERENCES

• 1: S. A. Coons, SURFACES FOR COMPUTER-AIDED DESIGN OF SPACE FORMS, Massachusetts Institute of Technology, Cambridge, MA, 1967
• 2: DE BOOR, C. A Practical Guide to SpIines, Springer-Verlag, New York, 1978.
• 3: I. D. Faux , M. J. Pratt, Computational Geometry for Design and Manufacture, Halsted Press, New York, NY, 1979
• 4: GORDON, W. J. Distributive lattices and the approximation of multivariate functions, in Approximation with Special Emphasis on Splines, I. E. Schoenberg, Ed. Academic Press, Orlando, Fla., 1969, pp. 223-277.
• 5: GREGORY, J.A. Smooth interpolation without twist constraints. In Computer Aided Geometric Design, R. E. Barnhill and R. F. Riesenfeld, Ed. Academic Press, Orlando, Fla., 1974, pp. 71-87.
• 6: GREGORY, J.A. C1 rectangular and non-rectangular surface patches. In Surfaces in Computer Aided Geometric Design, R. E. Barnhill and W. Boehm, Ed. North-Holland, Amsterdam, 1983, pp. 25-33.
• 7: LINPACK Users' Guide. SIAM, Philadelphia, Pa., 1979.
• 8: WIXOM, J.A. Blending--Function interpolation over non-rectangular domains. General Motors Research Rep. GMR-1957, 1975.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)

  J. Computer Applications
  J.6 COMPUTER-AIDED ENGINEERING
      Subjects: Computer-aided design (CAD)

General Terms:
Algorithms, Design