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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.

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  • $\begingroup$ Are Bezier curves a kind of "mock parametric splines" according to your definition? $\endgroup$
    – Dirk
    Commented May 30, 2017 at 20:34
  • $\begingroup$ @Dirk no, Bezier curves are parametric curves in the sense that they are calculate via different set of basis functions, that are weighted differently for the individual coordinates, whereas what I am looking for are pieces of graphs of ordinary f(x), that are rotated and translated to yield a smooth, continuous planar curve. What I am asking for also doesn't generalize to 3D curves or to surfaces, whereas that is the case for Beziers and other kinds of splines. $\endgroup$
    – Manfred Weis
    Commented May 31, 2017 at 4:53

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The "Wilson-Fowler" spline has the properties that you describe. Each segment is defined by a cubic polynomial in a rotated coordinate system. These splines were quite common in the days before parametric vector-valued ones became the standard. For example, they were used in most versions of the APT system for driving numerically controlled machine tools.

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