Presents a way to construct orthogonal piece-wise polynomials on an arbitrary triangular domain via barycentric coordinates. Solution of a boundary value problem for Laplace equation; Methodology; Results and discussion.
In computer aided geometric design （CAGD）, B′ezier-like bases receive more andmore considerations as new modeling tools in recent years. But those existing B′ezier-like basesare all defined over the rectangular do...In computer aided geometric design （CAGD）, B′ezier-like bases receive more andmore considerations as new modeling tools in recent years. But those existing B′ezier-like basesare all defined over the rectangular domain. In this paper, we extend the algebraic trigono-metric B′ezier-like basis of order 4 to the triangular domain. The new basis functions definedover the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry,boundary representation, linear independence and so on. We also prove some properties of thecorresponding B′ezier-like surfaces. Finally, some applications of the proposed basis are shown.展开更多
In computer aided geometric design(CAGD),the Bernstein-Bézier system for polynomial space including the triangular domain is an important tool for modeling free form shapes.The Bernstein-like bases for other spac...In computer aided geometric design(CAGD),the Bernstein-Bézier system for polynomial space including the triangular domain is an important tool for modeling free form shapes.The Bernstein-like bases for other spaces(trigonometric polynomial,hyperbolic polynomial,or blended space) has also been studied.However,none of them was extended to the triangular domain.In this paper,we extend the linear trigonometric polynomial basis to the triangular domain and obtain a new Bernstein-like basis,which is linearly independent and satisfies positivity,partition of unity,symmetry,and boundary represen-tation.We prove some properties of the corresponding surfaces,including differentiation,subdivision,convex hull,and so forth.Some applications are shown.展开更多
Four new trigonometric Bernstein-like basis functions with two exponential shape pa- rameters are constructed, based on which a class of trigonometric Bézier-like curves, anal- ogous to the cubic Bézier curv...Four new trigonometric Bernstein-like basis functions with two exponential shape pa- rameters are constructed, based on which a class of trigonometric Bézier-like curves, anal- ogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an eliipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cu- bic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape pa- rameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C2 N FC3 continuous for a non-uniform knot vector, and C3 or C5 continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for G1 continuous joining two trigonometric Bézier-like patches over triangular domain arededuced．展开更多
In this paper, we propose a new set of orthogonal basis functions in the arbitrarytriangular domain. At first, we generalize the 1-D Sturm-Liouville equation tothe arbitrary triangular domain on a barycentric coordina...In this paper, we propose a new set of orthogonal basis functions in the arbitrarytriangular domain. At first, we generalize the 1-D Sturm-Liouville equation tothe arbitrary triangular domain on a barycentric coordinate, and derive a set ofcomplete orthogonal basis functions on this domain. Secondly, we analyze thesymmetry and periodicity property of these functions and classify them into fourclasses. At last, we show some of the visualization results of these basis functions.展开更多
基金Project supported by the Major Basic Project of China (No.G19990328) and National Natural ScienceFoundation of China.
文摘Presents a way to construct orthogonal piece-wise polynomials on an arbitrary triangular domain via barycentric coordinates. Solution of a boundary value problem for Laplace equation; Methodology; Results and discussion.
基金Supported by the National Natural Science Foundation of China( 60933008,60970079)
文摘In computer aided geometric design （CAGD）, B′ezier-like bases receive more andmore considerations as new modeling tools in recent years. But those existing B′ezier-like basesare all defined over the rectangular domain. In this paper, we extend the algebraic trigono-metric B′ezier-like basis of order 4 to the triangular domain. The new basis functions definedover the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry,boundary representation, linear independence and so on. We also prove some properties of thecorresponding B′ezier-like surfaces. Finally, some applications of the proposed basis are shown.
基金supported by the National Natural Science Foundation of China (Nos.60773179,60933008,and 60970079)the National Basic Research Program (973) of China (No.2004CB318000)the China Hungary Joint Project (No.CHN21/2006)
文摘In computer aided geometric design(CAGD),the Bernstein-Bézier system for polynomial space including the triangular domain is an important tool for modeling free form shapes.The Bernstein-like bases for other spaces(trigonometric polynomial,hyperbolic polynomial,or blended space) has also been studied.However,none of them was extended to the triangular domain.In this paper,we extend the linear trigonometric polynomial basis to the triangular domain and obtain a new Bernstein-like basis,which is linearly independent and satisfies positivity,partition of unity,symmetry,and boundary represen-tation.We prove some properties of the corresponding surfaces,including differentiation,subdivision,convex hull,and so forth.Some applications are shown.
基金We, wish to express our gratitude to the referees for their valuable re- marks for improvements. The research is supported by the National Natural Science Foun-dation of China （No. 60970097, No. 11271376）, Postdoctoral Science Foundation of China （2015M571931）, and Graduate Students Scientific Research Innovation Project of Hunan Province （No. CX2012Blll）.
文摘Four new trigonometric Bernstein-like basis functions with two exponential shape pa- rameters are constructed, based on which a class of trigonometric Bézier-like curves, anal- ogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an eliipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cu- bic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape pa- rameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C2 N FC3 continuous for a non-uniform knot vector, and C3 or C5 continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for G1 continuous joining two trigonometric Bézier-like patches over triangular domain arededuced．
文摘In this paper, we propose a new set of orthogonal basis functions in the arbitrarytriangular domain. At first, we generalize the 1-D Sturm-Liouville equation tothe arbitrary triangular domain on a barycentric coordinate, and derive a set ofcomplete orthogonal basis functions on this domain. Secondly, we analyze thesymmetry and periodicity property of these functions and classify them into fourclasses. At last, we show some of the visualization results of these basis functions.
