By "mock-parametric" interpolating curves I understand a class of curves that connect a discrete sequence of points with a predefined degree of smoothness and, that correspond to a non-parametric function in a local coordinate system defined by $p_i$ as the origin and, $\frac{p_{i+1}-p_i}{\|p_{i+1}-p_i\|}$ as the local x-axis between $p_i$ and $p_{i+1}$

The reason, why I am interested in that kind of splines is that they seem to combine the simple control of ordinary spline interpolation with some of the flexibility of parametric splines; also the formulation of some optimization goals seems to be easier to formulate: least deviation from the polyline translates to minimizing the Chebyshev norm across all local function-definitions and, because the curvature of functions can be approximated by the 2nd derivative, it seems to be easier to find interpolating curves that come close to optimality w.r.t. conditions on curvature.

Question:has that kind of mock-parametric spline interpolation already been investigated? I am especially interested in the formulation and algorithmic ensuring of the smoothness conditions, as well as in results about the quality of the resulting curves, when compared to parametric spline-interpolation.