## Finding a curve of some approximate arc length (with uniform or zero curvature) with a specified distance to a set of points in 3-space

Imagine I define a set of $N$ points in 3-space, $P$, and I would like to define a straight-line or curve, $C$, with uniform or zero curvature, that has some desired distance, $M$, to each of these points. The curve $C$ should also have an approximate contour or arc length of $R$.

More specifically, let $(x_1, x_2, ...,x_i,...,x_N)$ be the set of distances between the points in $P$ and the straight-line or curve, and let $M$ be some desired distance between the curve and each point in $P$. Here we'd like to minimize something like the sum: $\Sigma (M - x_i)^2 + (L - R)^2$, where $L$ is the actual arc length of the curve and $R$ is the desired arc length.

Provided the set of points $P$, how do I approximate $C$?

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