Local Lagrange Interpolation on Powell-Sabin Triangulations and Terrain Modelling

Local Lagrange interpolation schemes for quadratic C l—splines on arbitrary triangulations with Powell—Sabin splits are constructed. By using the concept of weak interpolation, it is proved that the interpolation method yields optimal approximation order. We test our method by interpolating scattered data and show how the method can be applied for terrain modelling. We compare the interpolating splines on fine and coarse triangulations obtained from thinning strategies and analyze the data reduction.

[1]  Frank Zeilfelder,et al.  Developments in bivariate spline interpolation , 2000 .

[2]  D. W. Scharpf,et al.  The TUBA Family of Plate Elements for the Matrix Displacement Method , 1968, The Aeronautical Journal (1968).

[3]  Frank Zeilfelder,et al.  Cubic Spline Interpolation on Nested Polygon Triangulations , 2000 .

[4]  Günther Nürnberger,et al.  Lagrange and hermite interpolation by bivariate splines , 1992 .

[5]  W. Schempp,et al.  Multivariate Approximation Theory IV , 1989 .

[6]  L. R. Scott,et al.  A nodal basis for ¹ piecewise polynomials of degree ≥5 , 1975 .

[7]  G. Nürnberger,et al.  Lagrange interpolation by splines on triangulations , 1998 .

[8]  P. Lancaster.  Curve and surface fitting , 1986 .

[9]  Graeme Fairweather.  An investigation of Romberg quadrature , 1978, TOMS.

[10]  Günther Nürnberger,et al.  Multivariate Approximation and Splines , 1997 .

[11]  Mohammed Laghchim-Lahlou.  The Cr-fundamental splines of Clough–Tocher and Powell–Sabin types for Lagrange interpolation on a three direction mesh , 1998, Adv. Comput. Math..

[12]  Paul Sablonnière,et al.  Cr-finite elements of Powell-Sabin type on the three direction mesh , 1996, Adv. Comput. Math..

[13]  Frank Zeilfelder,et al.  Local Lagrange Interpolation by Cubic Splines on a Class of Triangulations , 2001 .

[14]  G. Alexits.  Approximation theory , 1983 .

[15]  Gerhard Heindl,et al.  Interpolation and Approximation by Piecewise Quadratic C1 — Functions of Two Variables , 1979 .

[16]  Frank Zeilfelder,et al.  Approximation order of bivariate spline interpolation for arbitrary smoothness , 1998 .

[17]  Larry L. Schumaker,et al.  On the approximation power of bivariate splines , 1998, Adv. Comput. Math..

[18]  Paul Sablonnière,et al.  Error Bounds for Hermite Interpolation by Quadratic Splines on an α-Triangulation , 1987 .

[19]  Frank Zeilfelder,et al.  Interpolation by spline spaces on classes of triangulations , 2000 .

[20]  Günther Nürnberger,et al.  Error analysis in interpolation by bivariate C1-splines , 1998 .

[21]  Günther Nürnberger,et al.  Interpolation by C 1 splines of degree q ≥4 on triangulations , 2000 .

[22]  Larry L. Schumaker,et al.  Macro-elements and stable local bases for splines on Powell-Sabin triangulations , 2003, Math. Comput..

[23]  Frank Zeilfelder,et al.  Lagrange Interpolation by C1 Cubic Splines on Triangulations of Separable Quadrangulations , 2002 .

[24]  Frank Zeilfelder,et al.  Local Lagrange interpolation by bivariate C 1 cubic splines , 2001 .

[25]  Frank Zeilfelder,et al.  Bivariate spline interpolation with optimal approximation order , 2001 .

[26]  G. Nürnberger.  Approximation by Spline Functions , 1989 .

[27]  Frank Zeilfelder,et al.  Interpolation by Bivariate Splines on Crosscut Partitions , 1997 .

[28]  Peter Lancaster,et al.  Curve and surface fitting - an introduction , 1986 .

[29]  N. Dyn,et al.  Adaptive thinning for bivariate scattered data , 2002 .

[30]  W. Dahmen,et al.  Scattered data interpolation by bivariate C1-piecewise quadratic functions , 1990 .

[31]  Günther Nürnberger,et al.  Bivariate spline interpolation at grid points , 1995 .

[32]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[33]  S. Rippa,et al.  Data Dependent Triangulations for Piecewise Linear Interpolation , 1990 .

[34]  Charles K. Chui,et al.  Swapping Edges of Arbitrary Triangulations to Achieve the Optimal Order of Approximation , 1997 .

[35]  Larry L. Schumaker,et al.  Macro-elements and stable local bases for splines on Clough-Tocher triangulations , 2001, Numerische Mathematik.

[36]  P. Sablonnière,et al.  Triangular finite elements of HCT type and classCρ , 1994, Adv. Comput. Math..