You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid?
a) fit a (spline) surface, differentiate exactly
b) differentiate numerically in cartesian coordinates, project to angle
c) something else
The function at hand is a displacement vector ($u : \mathbb{R}^2 \to \mathbb{R}^2$ ) and I need the circumferential strain tensor component $\varepsilon_{\theta\theta} = \frac{\partial u_\theta}{\partial \theta}$. (Also the radial strain would is somewhat interesting.)
background: In 2d strain tensor has four components: $\varepsilon_{i,j}$, where i and j is either taken from {x,y} (cartesian) or {$\theta$,r} (polar). The first index defines the direction of displacement and the second index the differentiation direction, i.e. $\varepsilon_{i,j} = \frac{\partial u_i}{\partial j} $
u here is the 2d displacement vector. In cartesian coordinates above is easy, in polar not so much.