Algebraic cubature by linear blending of elliptical arcs

Abstract

We construct a cubature formula of algebraic degree of exactness n with n 2 /2 + O(n) nodes, on the bidimensional domains generated by linear blending of two arcs of ellipses corresponding to the same angular interval. The construction is based on recent results on "subperiodic" trigonometric quadrature. Our formula generalizes several recent cubature formulas on standard circular sections. Among its numerous possible applications, we quote for example integration of functions with singularities, and integration on nonstandard circular sections arising in optical design or in meshfree methods with compactly supported radial bases. AMS subject classification: 65D32.Keywords: algebraic cubature, subperiodic trigonometric quadrature, product Gaussian quadrature, linear blending of elliptical arcs, singular integrand, nonstandard circular/elliptical sections, obscured and vignetted pupils, meshfree methods.

Polynomial fitting and interpolation on circular sections

“…It is not difficult to show, by a possible reparametrization with a suitable angle shift when A i and B i are not orthogonal, that these curves are arcs of two ellipses centered at C 1 and C 2 , respectively (cf. [14]). Consider the compact domain X ¼ fðx; yÞ ¼ nðs; hÞ ¼ sPðhÞ þ ð1 À sÞQ ðhÞ; ðs; hÞ 2 ½0; 1 Â ½a; bg;…”

“…Several instances of standard as well as less standard sections of disks (and ellipses) can be treated by arc blending, as it has been shown in [14] in the framework of cubature. All the examples below can be reproduced by the Matlab software package WAM in [19].…”

“…This family has been extensively studied in [14], in view of the construction of product Gaussian formulas. Differently from [14], here we do not need that the underlying transformation is injective. Let…”

“…The case of solids of rotation was considered in [20]. Gaussian subperiodic trigonometric rules and subperiodic transformations were the basis for the construction of algebraic cubature formulas on several circular sections, obtained by linear blending of arcs [8,15,16,14] (such as circular lenses among others), and on circular lunes by suitable subperiodic trigonometric diffeomorphisms [18]. Applications arise, for example, in the framework of optical design, cf.…”

Product Gaussian Quadrature on Circular Lunes

“…A Matlab implementation of the latter is provided in [3], by exploiting fast and stable computation of modified Chebyshev moments and the algorithms for orthogonal polynomials in the OPQ suite [9] by W. Gautschi. It is worth recalling that Gaussian quadrature for the weight function (1.1), can also be embedded in the more general framework of "sub-range" Jacobi polynomials, cf.…”