Approximate a curve with a limited number of line segments and arcs of circles

Question

Is there any algorithm that would allow to approximate a path on the x-y plane (i.e. an ordered suite of points defined by x and y) with a limited number of line segments an

Answer 1

the C1 requirement demands the you must have alternating straights and arcs. Also realize if you permit a sufficient number of segments you can trivially fit every pair of points with a straight and use a tiny arc to satisfy slope continuity.

I'd suggest this algorithm,

1 best fit with a set of (specified N) straight segments. (surely there are well developed algorithms for that.)

2 consider the straight segments fixed and at each joint place an arc. Treating each joint individually i think you have a tractable problem to find the optimum arc center/radius to satisfy continuity and improve the fit.

3 now that you are pretty close attempt to consider all arc centers and radii (segments being defined by tangency) as a global optimization problem. This of course blows up if N is large.