Nyquist Criterion in higher dimensions I was just wondering if there were any "obvious" resources, papers, or textbooks I should look into regarding the nyquist-criterion in higher dimensions. For example, this would be relevant if there is some curve that is i.e. not 1-to-1 w.r.t. time: $f: t \in \mathbb{R}^1 \rightarrow f(t) \in \mathbb{R}^1 $, but instead for some other parameterization, we have a line $l: u \in \mathbb{R}^1 \rightarrow l(u) = \big(\tilde t(u),\tilde f(u)\big) \in \mathbb{R}^2$, it is 1-to-1, but not onto, as $(\tilde t,\tilde f)$ is on a 1-dimensional manifold (a line) in a 2-D plane.

Then I'd like to know what sampling scheme I need to be able to recreate the curve through reconstruction. Using this approach, you could use nyquist's theorem for general spatial surfaces and manifolds as well.

This might be a math question, so I can migrate this. But I figured some experts on this sub-stack might know.

EDIT: several small edits for clarity/correctness.

EDIT 2: Fixed parameterization!!! 

EDIT 3: I want to emphasize that although I am parameterizing by $t$, it may be more useful to think of $t$ as a spatial dimension, since time can only move forward.
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.