Interpolating Random Points

Question

I have a list of (x,y) co-ordinates that need to be interpolated. The co-ordinates are not necessarily part of a function. Therefore, polynomial interpolation will not work. Is there a way to use some other form of interpolation that doesn't need the co-ordinates to be part of a function, i.e. a form of interpolation that gives out coefficients of a pair of parametric equations, one for x-coordinates and one for y-coordinates?

Answer 1

Add some parameter $t$ to your data, so it becomes $(t_i, x_i, y_i)$ instead of $(x_i, y_i)$. The simpliest way is to take $t_i = i$. Now perform interpolation of $(t_i, x_i)$ and $(t_i, y_i)$, you'll get $x = P(t)$ and $y = Q(t)$, so your interpolation will have parametric form $$ (x, y) = (P(t), Q(t)),\quad t \in [0, n] $$