One popular method that I came across for interpolation of a set of points is by using cubic Bezier curve segments with $C^1$ and $C^2$ continuity conditions at the junction point (or node) between two segments. The result is a set of Bezier curve segments that are smoothly connected. Is there a way to be able to control how much these segments are close to the piecewise linear function connecting every two consecutive nodes? I know there is something called tension spline that can follow such piecewise linear interpolation by changing a tension value, but can we do the same thing with Bezier interpolation?
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You seem to know about splines with tension controls. Some of these (the Alan Cline variant) use exponential functions. Others, like the $\nu$-splines developed by Greg Nielson in around 1973, are piecewise cubic polynomials.
Since any piecewise cubic can be expressed as a string of Bézier curves, the short answer to your question is “yes”.
