Closed B-spline functions:

From the equations given above we may write a more concise form as

P_ij (u, v) = U^T [M_s] [P_ij] [M_s]^TV

1 ≤ i ≤ 2, 1 ≤ j ≤ 3; (i - 1) ≤ u ≤ I, (j - 1) ≤ v ≤ j          

By equating the above equation with the bicubic equation, the equivalent [B_ij] matrix becomes

[B_ij] = [M_H]^{- 1} [M_S] [P_ij] [M_S]^T [M_H]^{T - 1}

Note down that the above procedure can be extended to an n × m cubic B-spline surface.

A similar process may be followed for a closed cubic B-spline patch. The difference comes into the form of the B-spline functions. For 4 × 4 cubic B-spline patch closed in the u direction, or both directions, the following three equations may be written respectively:

P(u, v) = [N_{0, 4} ((u + 4) mod 4, N_{0, 4} ((u + 3) mod 4), N_{0, 4} ((u + 2) mod 4),

and P(u, v) = [N_{0, 4} ((u + 4) mod 4), N_{0, 4} ((u + 3) mod 4), N_{0, 4} (u + 2) mod 4],

The closed B-spline functions have been evaluated as 5 × 6 closed cubic B-spline patch, the above process is repeated however with the functions [N_{0, 4} ((u + 5) mod 5), N_{0, 4} (u + 4) mod 5], N_{0, 4} ((u + 3) mod 5), N_{0, 4} ((u + 2) mod 5), N_{0, 4} ((u + 1) mod 5)] and [N_{0, 4} ((u + 6) mod 6), N_{0, 4} (u + 4) mod 6], N_{0, 4} ((u + 3) mod 6), N_{0, 4} ((u + 2) mod 6), N_{0, 4} (u + 1) mod 6)]. The control point matrix [P] is a 5 × 6 matrix in the case of the open patch.