Construction of Cubature Formulas of Degree Seven and Nine Symmetric Planar Regions, Using Orthogonal Polynomials

A method of constructing twelve-point cubature formulas with polynomial precision seven and nineteen-point and eighteen-point cubature formulas with polynomial precision nine is given for planar regions and weight functions, which are symmetric in each variable. The nodes are computed as common zeros of a set of linearly independent orthogonal polynomials.