A General Scheme for Shape Preserving Planar Interpolating Curves

This paper describes the application of the so-called Abstract Schemes (AS) for the construction of shape preserving interpolating planar curves. The basic idea behind AS is given by observing that when we interpolate some data points by a spline, we can dispose of several free parameters d0,d1,...,dN (di∈Rq), which are associated with the knots. If we now express shape constraints as conditions relative to each interval between two knots, they can be rewritten as a sequences of inclusion conditions: ({d}i,di+1)∈Di⊂R2q, where the sets Di are the corresponding feasible domains. In this setting the problems of existence, construction and selection of an optimal solution can be studied with the help of Set Theory in a general way. The method is then applied for the construction of shape preserving, planar interpolating curves.

[1]  Francesca Pelosi,et al.  Shape-Preserving Approximation by Space Curves , 2004, Numerical Algorithms.

[2]  Panagiotis D. Kaklis,et al.  Shape-preserving interpolation in R3 , 1997 .

[3]  Paolo Costantini.  A General Method for Constrained Curves with Boundary Conditions , 1993 .

[4]  J. W. Schmidt.  On shape preserving spline interpolation: existence theorems and determination of optimal splines , 1989 .

[5]  Paolo Costantini.  An algorithm for computing shape-preserving interpolating splines of arbitrary degree , 1988 .

[6]  W. Cheney,et al.  Proximity maps for convex sets , 1959 .

[7]  Tim N. T. Goodman,et al.  Automatic interpolation by fair, shape-preserving, G2 space curves , 1998, Comput. Aided Des..

[8]  J. W. Schmidt.  Staircase algorithm and construction of convex spline interpolants up to the continuity C3 , 1996 .

[9]  D. Schweikert.  An Interpolation Curve Using a Spline in Tension , 1966 .

[10]  Rossana Morandi,et al.  A Geometric Approach for Knot Selection in Convexity-Preserving Spline Approximation , 2000 .

[11]  Panagiotis D. Kaklis,et al.  Convexity-Preserving Polynomial Splines of Non-uniform Degree , 1990 .

[12]  Bert Jüttler.  Shape preserving least-squares approximation by polynomial parametric spline curves , 1997, Comput. Aided Geom. Des..

[13]  P. Costantini,et al.  Boundary-valued shape-preserving interpolating splines , 1997, TOMS.

[14]  P. Costantini.  Algorithm 770: BVSPIS—a package for computing boundary-valued shape-preserving interpolating splines , 1997, TOMS.

[15]  Stuart E. Dreyfus,et al.  Applied Dynamic Programming , 1965 .

[16]  Paolo Costantini.  Abstract schemes for functional shape-preserving interpolation , 1996, Advanced Course on FAIRSHAPE.

[17]  Jochen W. Schmidt,et al.  Composition based staircase algorithm and constrained interpolation with boundary conditions , 2000, Numerische Mathematik.

[18]  R. E. Carlson,et al.  Monotone Piecewise Cubic Interpolation , 1980 .

[19]  Horst Nowacki,et al.  Experiences in Curve Fairing , 1996, Advanced Course on FAIRSHAPE.

[20]  T. Goodman,et al.  Shape preserving interpolation by parametrically defined curves , 1988 .

[21]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[22]  Jochen W. Schmidt,et al.  Convex interval interpolation using a three-term staircase algorithm , 1999, Numerische Mathematik.

[23]  Paolo Costantini,et al.  On monotone and convex spline interpolation , 1986 .