# A Class of Refinement Schemes With Two Shape Control Parameters

A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape...

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Main Authors: , , , Article English 2020-01-01 IEEE Access https://ieeexplore.ieee.org/document/9099567/
id doaj-55289eee8bc64f4faf9c5d6a8ee6f762 Article doaj-55289eee8bc64f4faf9c5d6a8ee6f7622021-03-30T02:15:29ZengIEEEIEEE Access2169-35362020-01-018983169832910.1109/ACCESS.2020.29975579099567A Class of Refinement Schemes With Two Shape Control ParametersGhulam Mustafa0https://orcid.org/0000-0002-0441-8485Rabia Hameed1Dumitru Baleanu2https://orcid.org/0000-0002-0286-7244Ayesha Mahmood3Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, PakistanDepartment of Mathematics, The Government Sadiq College Women University, Bahawalpur, PakistanDepartment of Mathematics, Çankaya University, Ankara, TurkeyDepartment of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, PakistanA subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.https://ieeexplore.ieee.org/document/9099567/Combined refinement schemescontinuous curvesinterpolationapproximationshape parametersnon-uniform parameters DOAJ English Article DOAJ Ghulam Mustafa Rabia Hameed Dumitru Baleanu Ayesha Mahmood Ghulam Mustafa Rabia Hameed Dumitru Baleanu Ayesha Mahmood A Class of Refinement Schemes With Two Shape Control Parameters IEEE Access Combined refinement schemes continuous curves interpolation approximation shape parameters non-uniform parameters Ghulam Mustafa Rabia Hameed Dumitru Baleanu Ayesha Mahmood Ghulam Mustafa A Class of Refinement Schemes With Two Shape Control Parameters A Class of Refinement Schemes With Two Shape Control Parameters A Class of Refinement Schemes With Two Shape Control Parameters A Class of Refinement Schemes With Two Shape Control Parameters A Class of Refinement Schemes With Two Shape Control Parameters class of refinement schemes with two shape control parameters IEEE IEEE Access 2169-3536 2020-01-01 A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented. Combined refinement schemes continuous curves interpolation approximation shape parameters non-uniform parameters https://ieeexplore.ieee.org/document/9099567/ AT ghulammustafa aclassofrefinementschemeswithtwoshapecontrolparameters AT rabiahameed aclassofrefinementschemeswithtwoshapecontrolparameters AT dumitrubaleanu aclassofrefinementschemeswithtwoshapecontrolparameters AT ayeshamahmood aclassofrefinementschemeswithtwoshapecontrolparameters AT ghulammustafa classofrefinementschemeswithtwoshapecontrolparameters AT rabiahameed classofrefinementschemeswithtwoshapecontrolparameters AT dumitrubaleanu classofrefinementschemeswithtwoshapecontrolparameters AT ayeshamahmood classofrefinementschemeswithtwoshapecontrolparameters 1724185516792872960