Least-Squares B-Spline Curve Approximation with Arbitary End Derivatives

Abstract.An algorithm for least-squares approximation of data with end derivative constraints is presented. The approximating curve minimises the error in the least-squares sense, while at the same time assumes the derivatives specified. For degree p approximation, up to p 2 1 derivatives may be specified, resulting in C p−1 continuous curve approximants. The method is useful to piece individual curve segments together, or to create closed curves with various degrees of smoothness.

[1]  Elaine Cohen,et al.  A data dependent parametrization for spline approximation , 1989 .

[2]  Tom Lyche,et al.  Mathematical methods in computer aided geometric design , 1989 .

[3]  G. E. Hölzle,et al.  Knot placement for piecewise polynomial approximation of curves , 1983 .

[4]  D. F. Rogers.  Constrained B-spline curve and surface fitting , 1989 .

[5]  M. Grossman,et al.  Parametric Curve Fitting , 1971, Comput. J..

[6]  Maureen Stone,et al.  Curve-fitting with piecewise parametric cubics , 1983, SIGGRAPH.

[7]  E. T. Y. Lee,et al.  Choosing nodes in parametric curve interpolation , 1989 .

[8]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[9]  J. G. Hayes,et al.  Numerical Approximations to Functions and Data. , 1971 .

[10]  J. Mason,et al.  Algorithms for approximation , 1987 .

[11]  M. Epstein.  On the Influence of Parametrization in Parametric Interpolation , 1976 .

[12]  M. Cox.  Curve Fitting with Piecewise Polynomials , 1971 .

[13]  Gábor Renner,et al.  Method of shape description for mechanical engineering practice , 1982 .

[14]  J. R. Manning.  Continuity Conditions for Spline Curves , 1974, Comput. J..

[15]  D. J. McConalogue.  A Quasi-Intrinsic Scheme for Passing a Smooth Curve Through a Discrete Set of Points , 1970, Comput. J..

[16]  Josef Hoschek,et al.  Intrinsic parametrization for approximation , 1988, Comput. Aided Geom. Des..

[17]  J. G. Hayes,et al.  Numerical Approximations to Functions and Data. , 1971 .

[18]  M. G. Cox,et al.  A knot placement strategy for least squares spline fitting based on the use of local polynomial approximations , 1990 .

[19]  Hiroshi Akima,et al.  A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures , 1970, JACM.

[20]  C. D. Boor,et al.  Good approximation by splines with variable knots. II , 1974 .

[21]  P. J. Hartley,et al.  Parametrization and shape of B-spline curves for CAD , 1980 .

[22]  Gregory M. Nielson,et al.  Knot selection for parametric spline interpolation , 1989 .

[23]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[24]  C. D. Boor,et al.  Least Squares Cubic Spline Approximation I | Fixed Knots , 1968 .

[25]  Josef Hoschek,et al.  Spline conversion for trimmed rational Bézier- and B-spline surfaces , 1990, Comput. Aided Des..

[26]  A. Robin Forrest,et al.  Interactive interpolation and approximation by Bézier polynomials , 1990, Comput. Aided Des..

[27]  C. D. Boor,et al.  Least Squares Cubic Spline Approximation, II - Variable Knots , 1968 .