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Array points densification without classical interpolation (non-unique solutions)

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I would like to 'densify' the points within an array of x and y points. For instance I would like to turn:
size([x y]) = (200,2)
into a new set of points such that:
size([xi yi]) = (10000,2)
I can't do it with interp1 because the curve doesn't have unique solutions (see below), so it needs to perform as a 'local' interpolation of some sort. I found a workaround with the interpm2 function, but it's slower than I would like to. Any ideas?

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