Symmetric Numerical Integration Formulas for Regular Polygons
Marcel Nooijen, G. te Velde, Evert Jan Baerends · SIAM Journal on Numerical Analysis · 1990
The theory of interpolatory integration formulas in n dimensions is combined with group theory to classify and generate symmetric integration formulas in a systematic way. The points are computed as the common zeros of a set of polynomials. These polynomials constitute a canonical basis for the real ideal belonging to the formula. For a symmetric formula they can be chosen to span a representation of the symmetry group. For regions with high symmetry and formulas of low degree this leads to only a few distinct possibilities and the existence of the corresponding formulas is easily checked. In general, however, the canonical basis is not completely determined by symmetry and degree, and a suitable choice has to be made. The application to regular polygons in two dimensions is discussed and formulas are presented for polygons with three, five, six, seven, and eight vertices, with varying degrees of exactness (up to order 27 for the hexagon).