I am working with a nearly Archimedian Spiral (change in radius is linear). It progresses in time, and I have taken the newest point along the spiral at .4s time intervals. (That may not be too important here though).
So with this image in mind (an (x,y) plot), I want to smooth out the curves. The kinks in the current image indicate the measurements that I took. I have the radial vectors for those points, but I want to find the radial vectors (or (x,y) coordinates) along the newly fit curve. Ultimately, I need a continuous set of data for this spiral.
Let me consider the most general problem where you have $n$ ordered data points of coordinates $(x_i,y_i)$.
Compute for each data point the distance parameter $d_i$ defined as $$ d_{i+1}=d_i+\sqrt{(x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2} \qquad (i=1,\cdots,n-1)$$ with $d_1=0$.
After that, for each data point $i$, $(x_i,y_i,d_i)$. So, build two parametric cubic splines, one for $x=f(d)$ and another for $y=g(d)$, $d$ varying from $0$ to $d_n$. The splines will exactly go through all the data points.
Now, for a choosen value of $d$ (which is a continuous variable), you can compute the corresponding $x$ and $y$ and get a smooth description of the whole curve.