In this article,we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs.The C-type spline belongs to theωB-spline category of splines that are capable of blending polyn...In this article,we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs.The C-type spline belongs to theωB-spline category of splines that are capable of blending polynomial,trigonometric and hyperbolic functions.Commonly used basic shapes can be exactly represented by these types of splines.We derive explicit formulas for the convenience of modeling the basic curves.The entire blending curve is C^1-continuous.In comparison with the existing best blending method by rational G^2 splines,which are rational splines of degree 3,the proposed method allows simpler representation and blending of the basic curves,and it can represent numerous basic shapes including the hyperbolic types.We also design a subdivision method to generate blending curves;this method is precise for the basic curves and approximate for the blending sections.The subdivision process is efficient for modeling and rendering.It has also proven to be C^1-continuous by the asymptotically equivalent theory and the continuity of stationary subdivision method.In addition,we extend the proposed methods to cases involving the modeling and blending of basic surfaces.We provide many examples that illustrate the merits of our methods.展开更多
ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB...ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.展开更多
基金This work described in this article was supported by the National Science Foundation of China(61772164,61272032)Provincial Key Platforms and Major Scientific Research Projects in Universities and Colleges of Guangdong(2017KTSCX143)the Natural Science Foundation of Zhejiang Province(LY17F020025).
文摘In this article,we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs.The C-type spline belongs to theωB-spline category of splines that are capable of blending polynomial,trigonometric and hyperbolic functions.Commonly used basic shapes can be exactly represented by these types of splines.We derive explicit formulas for the convenience of modeling the basic curves.The entire blending curve is C^1-continuous.In comparison with the existing best blending method by rational G^2 splines,which are rational splines of degree 3,the proposed method allows simpler representation and blending of the basic curves,and it can represent numerous basic shapes including the hyperbolic types.We also design a subdivision method to generate blending curves;this method is precise for the basic curves and approximate for the blending sections.The subdivision process is efficient for modeling and rendering.It has also proven to be C^1-continuous by the asymptotically equivalent theory and the continuity of stationary subdivision method.In addition,we extend the proposed methods to cases involving the modeling and blending of basic surfaces.We provide many examples that illustrate the merits of our methods.
基金the National Natural Science Foundation of China(61772164,61761136010)the Natural Science Foundation of Zhejiang Province(LY17F020025).
文摘ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.
