I have a set of data points given in $(x_i, y_i)$, they simply correspond to the positions in the $x-y$ plane over time $t$. They are positions of a particle in a plane. Since this curve is nonparametric, there is no way to analytically solve for the curvature of the curve.
One way to solve for the radius of curvature of the path traced by the data points is to 'fit' a circle to a number of points (at least 3, for a unique circle, or more.)
Link here: Numerical way to solve for the curvature of a curve
Furthermore, I can numerically compute the velocity and acceleration from the points $(x_i,y_i)$ using numerical methods.
Remember that the radius of curvature of a curve that can be explicitly expressed as $ y= f(x)$ is given by $c = \dfrac{(1+(f'^2))^{3/2}}{f'' }.$ If $x$ and $y$ are positions, then $f''$ could be viewed as the acceleration and $f'$ as the velocity.
My question is: From the nonparametric data points $(x_i,y_i)$ with a radius of curvature fitted numerically, does the acceleration and the radius of curvature have any relationship?