Abstract
A novel 4-point interpolating subdivision scheme is presented, which generates the family of C2 limiting curves and its limiting function has support on [−7/3,7/3]. It behaves better than classical 4-point binary and ternary schemes with the same approximation order in many aspects that it has smaller support size, higher smoothness, and is computationally more efficient. The proposed nonstationary scheme can reproduce the functions of linear spaces spanned by {1,sin(αx),cos(αx)} for 0<α<π/3. Moreover, some examples are illustrated to show that the proposed scheme can also reproduce asteroids, cardioids, and conic sections as well.
Highlights
Subdivision schemes have become one of the most important, significant, and emerging modelling tools in computer applications, medical image processing, scientific visualization, reverse engineering, computer aided geometric designing, and so forth, because of their simple, elegant, and efficient ways to create smooth curves from initial control polygon by subdividing them according to some refining rules, recursively
These refining rules take the initial polygon to produce a sequence of finer polygons converging to a smooth limiting curve
In the field of nonstationary subdivision schemes, binary and ternary schemes are present in the literature [1,2,3,4,5,6]
Summary
Subdivision schemes have become one of the most important, significant, and emerging modelling tools in computer applications, medical image processing, scientific visualization, reverse engineering, computer aided geometric designing, and so forth, because of their simple, elegant, and efficient ways to create smooth curves from initial control polygon by subdividing them according to some refining rules, recursively. Some properties of the proposed scheme (like support size and symmetry of basic limit function, approximation order, convergence, and smoothness) have been investigated. Besides the convergence and approximation order, the two important and interesting properties of basic limit function of a subdivision scheme are the support size and the smoothness. They are commonly manipulated with each other in the sense that a higher degree of smoothness generally requires a larger support, leading to a more global influence of each initial data value on the limit function.
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