Scalar- and planar-valued curve fitting using splines under tension

ABSTRACT

The spline under tension was introduced by Schweikert in an attempt to imitate cubic splines but avoid the spurious critical points they induce. The defining equations are presented here, together with an efficient method for determining the necessary parameters and computing the resultant spline. The standard scalar-valued curve fitting problem is discussed, as well as the fitting of open and closed curves in the plane. The use of these curves and the importance of the tension in the fitting of contour lines are mentioned as application.

REFERENCES

• 1: Ahlberg, J.H., Nilson, E.N., and Walsh, J.L. The Theory of Splines arm Their Applications. Academic Press, New York, 1967.
• 2: A. K. Cline, Six subprograms for curve fitting using splines under tension, Communications of the ACM, v.17 n.4, p.220-223, April 1974 [doi>10.1145/360924.360974]
• 3: Schweikert, D.G. An interpolation curve using a spline in tension. J. Math. and Physics 45 (1966), 312 317.

INDEX TERMS

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

Additional 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.2 DISCRETE MATHEMATICS

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

Keywords:
contouring, curve fitting, interpolation, splines