In freeform surface modelling, developable surfaces have much application value. But, in 3D space, there is not always a regular developable surface which interpolates the given boundary of an arbitrary piecewise smoo...In freeform surface modelling, developable surfaces have much application value. But, in 3D space, there is not always a regular developable surface which interpolates the given boundary of an arbitrary piecewise smooth closed curve. In this paper, tensor product Bézier surfaces interpolating the closed curves are determined and the resulting surface is a minimum of the functional defined by the L2-integral norm of the Gaussian curvature. The Gaussian curvature of the surfaces is minimized by the method of solving nonlinear optimization problems. An improved approach trust-region form method is proposed. A simple application example is also given.展开更多
Within the framework of the correlation theory of electromagnetic laser beams,the far field cross-spectral density matrix of the light radiated from an electromagnetic Hermite-Gaussian model source is derived.By utili...Within the framework of the correlation theory of electromagnetic laser beams,the far field cross-spectral density matrix of the light radiated from an electromagnetic Hermite-Gaussian model source is derived.By utilizing the convergence property of Hermite polynomials,the conditions of the matrices for the source to generate an electromagnetic Hermite-Gaussian beam are obtained.Furthermore,in order to generate a scalar Hermite-Gaussian model beam,it is required that the source should be locally rather coherent in the spatial domain.展开更多
In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I （a,b）, a function G E S（w）= （f： fxlf（x）lw（x）dx 〈 ∞ satisfying the ...In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I （a,b）, a function G E S（w）= （f： fxlf（x）lw（x）dx 〈 ∞ satisfying the conditions G（2J）（x） ：〉 O, x E （a,b）, j = 0, 1 , and growing as fast as possible as x→ a- and x → b-, plays an important role. But to find such a function G is often difficult and complicated. This implies that to prove convergence of Gaussian quadrature formulas, it is enough to find a function G E S（w） with G ≥ 0 satisfying展开更多
文摘In freeform surface modelling, developable surfaces have much application value. But, in 3D space, there is not always a regular developable surface which interpolates the given boundary of an arbitrary piecewise smooth closed curve. In this paper, tensor product Bézier surfaces interpolating the closed curves are determined and the resulting surface is a minimum of the functional defined by the L2-integral norm of the Gaussian curvature. The Gaussian curvature of the surfaces is minimized by the method of solving nonlinear optimization problems. An improved approach trust-region form method is proposed. A simple application example is also given.
基金supported by the National High Technology Research and Development Program of China (No. 2007AA04Z181)the National Natural Science Foundation of China (No. 61077012)+1 种基金the Research Fund of this University (No.2010ZYTS031)the Excellent Doctorial Candidate Training Fund in this University
文摘Within the framework of the correlation theory of electromagnetic laser beams,the far field cross-spectral density matrix of the light radiated from an electromagnetic Hermite-Gaussian model source is derived.By utilizing the convergence property of Hermite polynomials,the conditions of the matrices for the source to generate an electromagnetic Hermite-Gaussian beam are obtained.Furthermore,in order to generate a scalar Hermite-Gaussian model beam,it is required that the source should be locally rather coherent in the spatial domain.
基金Project supported by the National Natural Science Foundation of China (Nos. 11171100,10871065,11071064)the Hunan Provincial Natural Science Foundation of China (No. 10JJ3089)the Scientific Research Fund of Hunan Provincial Education Department (No. 11W012)
文摘In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I （a,b）, a function G E S（w）= （f： fxlf（x）lw（x）dx 〈 ∞ satisfying the conditions G（2J）（x） ：〉 O, x E （a,b）, j = 0, 1 , and growing as fast as possible as x→ a- and x → b-, plays an important role. But to find such a function G is often difficult and complicated. This implies that to prove convergence of Gaussian quadrature formulas, it is enough to find a function G E S（w） with G ≥ 0 satisfying
