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A general local Cm(m ≥ 0) tetrahedral interpolation scheme bypolynomials of degree 4m + l plus low order rational functions from the given data is proposed. The scheme can have either 4m + l order algebraic precision if C2m data at vertices and Cm data on faces are given or k + E[k/3] + 1 order algebraic precision if Ck (k ≤ 2m) data are given at vertices. The resulted interpolant and its partial derivatives of up to order m are polynomials on the boundaries of the tetrahedra.