AbstractThis paper presents a systematic scheme for controlling the local behaviour of C2interpolating curves, based on the cubic B2‐splines and the quartic S‐splines. Both splines have an additional control point obtained by knot‐ insertion or degree‐elevation in each span of the conventional uniform cubic interpolating B‐splines. The shape designer can choose the desired range of locality for each span and get the corresponding additional control point as a barycentric combination of interpolation points within the range, without solving any variational problem and simultaneous equations. The scheme is consistent over the entire curve subject to some typical end conditions. The class of the curves derived includes the conventional cubic interpolating B‐splines. Examples demonstrate the behaviour of the new interpolating curves and the capability of