Some context
I'm looking for an approach to print the stress trajectory of a part via composite 3D printing, however our composite 3D printer only allows for a turn radius of 3mm ($C_r = 3$). I believe if I can find a transform $T_r$ that acts on the stress field to give the radius of curvature at any given point I can constrain the resultant scalar field to be greater than $C_r$ then use inverse transform of the resultant scalar field to plot the composite path. I will write this as: $$T_r(F(x,y)) = R(x,y)$$
A mathematical definition
Given a vector field $F(x, y)$ for every point $(x, y)$ there exists a streamline $\gamma(s) = (x(s), y(s))$ such that $$\gamma^\prime(s) = F(\gamma(s))$$ I believe I can obtain the radius of curvature of the streamline to a specic point in space using the equation of curvature of a parametric equation: $$\kappa = \frac{\det(\gamma^\prime, \gamma^{\prime\prime})}{||\gamma^\prime||^3}$$ however I am unsure how to solve this or if my mathematical logic applies.
Yes, your approach seems fine. So we have $\gamma'(t) = F(\gamma(t))$, from which we get $\gamma''(t) = DF(\gamma(t))(\gamma'(t))$. In particular, if $F(x,y) = (M(x,y),N(x,y))$, then $$\gamma''(t) = \begin{bmatrix} \frac{\partial M}{\partial x} & \frac{\partial M}{\partial y} \\ \frac{\partial N}{\partial x} & \frac{\partial N}{\partial y}\end{bmatrix}\begin{bmatrix} M(\gamma(t)) \\ N(\gamma(t)) \end{bmatrix},$$ so \begin{align*} \kappa &= \frac1{(M^2+N^2)^{3/2}}\left|\begin{matrix} M & \frac{\partial M}{\partial x} M + \frac{\partial M}{\partial y}N \\ N & \frac{\partial N}{\partial x} M + \frac{\partial N}{\partial y}N \end{matrix}\right| \\ &= \frac1{(M^2+N^2)^{3/2}}\left(M^2\frac{\partial N}{\partial x} + MN(\frac{\partial N}{\partial y}-\frac{\partial M}{\partial x}) - N^2\frac{\partial M}{\partial y}\right). \end{align*}