Let S0 and S1 be two spaces of polynomial spline curves s : [a,b] → IRd, of order k0 and k1 with k1< k0, defined over knot sequences (ξ0/i) and (ξ1/i), respectively. Fix s0 € S0, and corresponding to each knot ξ 0/I assign a tolerance ǫi ≥ 0. In this paper we present an algorithm for computing the best convex constrained L2 approximant s1 := argmin {││s ─s0 ││ 2: s € S1 ∩ K} with K := ∩ {s : │s(t0/i) ─ s0 (t 0/i)│ ≤€j}, by the method of alternating projections.