Gaussian quadrature for C¹ cubic Clough–Tocher macro-triangles

A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud (1956). The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C1 cubic Clough–Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results in a factor of three reduction in the number of quadrature points needed to integrate the Clough–Tocher spline space exactly.