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In this paper restricted interpolations of data sets (xi, yj, zi, j), i = 0,…, n, j = 0,…, m, on the rectangular grid Δ × Σ = {x0 <… < xn} × {yo <… < ym} are handled using biquadratic C1 and bicubic C2 splines on refined grids. The constraints on the first order derivatives are piecewise constant with respect to the original grid Δ × Σ. If the refinement of the grids are chosen adaptively, the existence of restricted interpolants can be assured for bounds strictly compatible with the data.

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