Algebraic cubature by linear blending of elliptical arcs

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.