Dimensions of biquadratic and bicubic spline spaces over hierarchical T-meshes

This paper discusses the dimensions of biquadratic C 1 spline spaces and bicubic C 2 spline spaces over hierarchical T-meshes using the smoothing cofactor-conformality method. We obtain the dimension formula of biquadratic C 1 spline spaces over hierarchical T-meshes in a concise way. In addition, we provide a dimension formula for bicubic C 2 spline spaces over hierarchical T-meshes with fewer restrictions than that in the previous literature. A dimension formula for bicubic C 2 spline spaces over a new type of hierarchical T-meshes is also provided.

[1]  Falai Chen,et al.  On the instability in the dimension of splines spaces over T-meshes , 2011, Comput. Aided Geom. Des..

[2]  Tom Lyche,et al.  Polynomial splines over locally refined box-partitions , 2013, Comput. Aided Geom. Des..

[3]  Bert Jüttler,et al.  Bases and dimensions of bivariate hierarchical tensor-product splines , 2013, J. Comput. Appl. Math..

[4]  Bernard Mourrain,et al.  On the problem of instability in the dimension of a spline space over a T-mesh , 2012, Comput. Graph..

[5]  LiXin,et al.  Polynomial splines over hierarchical T-meshes , 2008 .

[6]  Thomas J. R. Hughes,et al.  On linear independence of T-spline blending functions , 2012, Comput. Aided Geom. Des..

[7]  Ahmad H. Nasri,et al.  T-splines and T-NURCCs , 2003, ACM Trans. Graph..

[8]  Bernard Mourrain,et al.  On the dimension of spline spaces on planar T-meshes , 2010, Math. Comput..

[9]  David R. Forsey,et al.  Hierarchical B-spline refinement , 1988, SIGGRAPH.

[10]  Bert Jüttler,et al.  TDHB-splines: The truncated decoupled basis of hierarchical tensor-product splines , 2014, Comput. Aided Geom. Des..

[11]  Jiansong Deng,et al.  Dimensions of spline spaces over T-meshes , 2006 .

[12]  Jiansong Deng,et al.  Dimensions of biquadratic spline spaces over T-meshes , 2008, J. Comput. Appl. Math..

[13]  Meng Wu,et al.  Dimension of spline spaces with highest order smoothness over hierarchical T-meshes , 2011, Comput. Aided Geom. Des..

[14]  Tom Lyche,et al.  T-spline simplification and local refinement , 2004, ACM Trans. Graph..

[15]  Ren-hong Wang.  Multivariate Spline Functions and Their Applications , 2001 .

[16]  Hendrik Speleers,et al.  THB-splines: The truncated basis for hierarchical splines , 2012, Comput. Aided Geom. Des..

[17]  Jiansong Deng,et al.  Polynomial splines over hierarchical T-meshes , 2008, Graph. Model..

[18]  B. Simeon,et al.  A hierarchical approach to adaptive local refinement in isogeometric analysis , 2011 .

[19]  Bert Jüttler,et al.  On the completeness of hierarchical tensor-product B-splines , 2014, J. Comput. Appl. Math..

[20]  L. Schumaker.  On the Dimension of Spaces Of Piecewise Polynomials in Two Variables , 1979 .

[21]  Jiansong Deng,et al.  On the dimension of spline spaces over T-meshes with smoothing cofactor-conformality method , 2012, Comput. Aided Geom. Des..