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Table 2 Symmetry orbits inside of a triangle

From: An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements

Orbit \(n_{\mathrm{pts}}\) \(n_{\mathrm{dof}}\) Barycentric coordinates
\(S_3(\alpha =\tfrac{1}{3})\) 1 1 \({{\mathrm{Perm}}}(\alpha ,\alpha ,\alpha )\)
\(S_{21}(\alpha )\) 3 2 \({{\mathrm{Perm}}}(\alpha ,\alpha ,1-2\alpha )\)
\(S_{111}(\alpha ,\beta )\) 6 3 \({{\mathrm{Perm}}}(\alpha ,\beta ,1-\alpha -\beta )\)
  1. The number of points (unique permutations) contributed by each orbit is given by \(n_{\mathrm{pts}}\) while the number of degrees of freedom contributed to the quadrature rule is indicated by \(n_{\mathrm{dof}}\)