Abstract
For the space of \(C^3\) quintics on the Powell–Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge and have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the \(L_\infty \) norm with a condition number independent of the geometry and have a well-conditioned Lagrange interpolant at the domain points and a quasi-interpolant with local approximation order 6. We show an \(h^2\) bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases, we derive \(C^0\), \(C^1\), \(C^2\), and \(C^3\) conditions on the control points of two splines on adjacent macrotriangles.