Nonpolynomial (X, A, φ)-splines of the third order and the special case of Bφ-splines of class C² are studied. For such splines calibration relations are obtained, owing to which the coordinate splines on the original grid is represented in terms of the coordinate splines on a refined grid. A nonlinear mapping (ℝ⁴)⁹ → ℝ⁴ and locally orthogonal chains of vectors are used for this purpose. Bibliography: 22 titles.