We construct (integral) interpolating subspline curves for given data points and the knot vector. The algorithm is very close to B spline approximation. The idea is to blend interpolating Lagrangian splines using B spline techniques. Everything is connected in an affinely invariant way with the control points and the knot vector. We are able to show that our scheme produces high quality subsplines, which include known procedures like Overhauser or quintic interpolation schemes. In addition we may sweep to B splines and return in a very lucid way. Examples show the power of the method. The given procedure allows generalisations to rational subsplines and to tensor product interpolating surfaces.