Nyquist Criterion in higher dimensions
i do not have a reference for you, but you are sampling among every one of the dimensions and the Nyquist Criterion must be satisfied for each one.

so fix $v$ to an arbitrary value $v_0$, then check if $\tilde{f}(u,v_0)$ is bandlimited some bandlimit, we'll call $B_u$.  find the max value of $B_u$ for all possible $v_0$, double that and that is the minimum sample rate along the $u$-axis.

then swap the roles of $u$ and $v$, fix $u$ to $u_0$, find the bandlimit of $\tilde{f}(u_0,v)$ for every possible value of $u_0$.  find the maximum (over all $u_0$) bandlimit $B_v$, double that and that is the minimum-sample rate along the $v$-axis.

if your curve can move at an arbitrary angle w.r.t. the $u$-axis and $v$-axis, then pick the maximum of $B_u$ and $B_v$ and sample both axes at double of that rate (or slightly higher).  then, along the path length, you must sample at a minimum of that rate, per unit of arc length, to satisfy Nyquist in two dimensions.