A new B-spline representation for cubic splines over Powell-Sabin triangulations

We consider a C 1 cubic spline space defined over a triangulation with Powell-Sabin refinement. The space has some local C 2 super-smoothness and can be seen as a close extension of the classical cubic Clough-Tocher spline space. In addition, we construct a suitable normalized B-spline representation for this spline space. The basis functions have a local support, they are nonnegative, and they form a partition of unity. We also show how to compute the Bezier control net of such a spline in a stable way. We consider a C 1 cubic spline space defined over a Powell-Sabin refined triangulation.The space has some local C 2 super-smoothness and can be seen as a close extension of the classical cubic Clough-Tocher spline space.We construct a suitable normalized B-spline representation for this spline space.We also show how to compute the Bezier control net of such a spline in a stable way.

[1]  Hendrik Speleers,et al.  A normalized basis for quintic Powell-Sabin splines , 2010, Comput. Aided Geom. Des..

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

[3]  Larry L. Schumaker,et al.  Spline functions on triangulations , 2007, Encyclopedia of mathematics and its applications.

[4]  Hendrik Speleers,et al.  Multivariate normalized Powell-Sabin B-splines and quasi-interpolants , 2013, Comput. Aided Geom. Des..

[5]  Hendrik Speleers,et al.  A normalized basis for reduced Clough-Tocher splines , 2010, Comput. Aided Geom. Des..

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

[7]  Hendrik Speleers,et al.  Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems , 2012 .

[8]  Paul Sablonnière,et al.  Composite finite elements of class Ck , 1985 .

[9]  Larry L. Schumaker,et al.  Smooth Macro-Elements Based on Powell–Sabin Triangle Splits , 2002, Adv. Comput. Math..

[10]  Hendrik Speleers,et al.  Numerical solution of partial differential equations with Powell-Sabin splines , 2006 .

[11]  Stephen Mann,et al.  Cubic precision Clough-Tocher interpolation , 1999, Comput. Aided Geom. Des..

[12]  Praveen Kashyap.  Improving Clough-Tocher interpolants , 1996, Comput. Aided Geom. Des..

[13]  Paul Dierckx,et al.  On calculating normalized Powell-Sabin B-splines , 1997, Comput. Aided Geom. Des..

[14]  D. Sbibih,et al.  Superconvergent $$C^1$$C1 cubic spline quasi-interpolants on Powell-Sabin partitions , 2015 .

[15]  Hendrik Speleers,et al.  Quasi-hierarchical Powell-Sabin B-splines , 2009, Comput. Aided Geom. Des..

[16]  Hendrik Speleers,et al.  Construction of Normalized B-Splines for a Family of Smooth Spline Spaces Over Powell–Sabin Triangulations , 2013 .

[17]  A. Bultheel,et al.  Stable multiresolution analysis on triangles for surface compression , 2006 .

[18]  Ahmed Tijini,et al.  A normalized basis for C1 cubic super spline space on Powell-Sabin triangulation , 2014, Math. Comput. Simul..

[19]  Paul Dierckx,et al.  Algorithms for surface fitting using Powell-Sabin splines , 1992 .

[20]  A. Serghini,et al.  Polar forms and quadratic spline quasi-interpolants on Powell--Sabin partitions , 2009 .

[21]  A. Serghini,et al.  Normalized trivariate B-splines on Worsey-Piper split and quasi-interpolants , 2012 .

[22]  Huan-Wen Liu,et al.  A bivariate C1 cubic super spline space on Powell-Sabin triangulation , 2008, Comput. Math. Appl..

[23]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

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

[25]  Carla Manni,et al.  Quadratic spline quasi-interpolants on Powell-Sabin partitions , 2007, Adv. Comput. Math..

[26]  Hendrik Speleers,et al.  A Family of Smooth Quasi-interpolants Defined Over Powell–Sabin Triangulations , 2015 .

[27]  D. Sbibih,et al.  Superconvergent quadratic spline quasi-interpolants on Powell–Sabin partitions , 2015 .

[28]  Larry L. Schumaker,et al.  Smooth macro-elements based on Clough-Tocher triangle splits , 2002, Numerische Mathematik.

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

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

[31]  Gerald E. Farin,et al.  A modified Clough-Tocher interpolant , 1985, Comput. Aided Geom. Des..