Is it possible to express the coordinates of a point in a distorted cube? That is, given a distorted cube in 3D space with 8 points (the coordinates of which are not necessarily on the regular positions you'd expect them to be for a non-distorted cube), and a point within that cube, finding a universal way of expressing its position. And if this is possible, is there some established method for determining this alternative expression, so as to transfer it to a differently distorted cube?
The tetrahedron-equivalent of this would be to use a barycentric coordinate system, but that is for 4-point simplexes.
Yes, it is called trilinear interpolation: linear interpolation in three variables.
Let the vertices be called $\vec{p}_{000}$, $\vec{p}_{001}$, $\vec{p}_{010}$, $\vec{p}_{011}$, $\vec{p}_{100}$, $\vec{p}_{101}$, $\vec{p}_{110}$, and $\vec{p}_{111}$.
Let $u$ axis be between $\vec{p}_{0 v w}$ and $\vec{p}_{1 v w}$, $v$ axis be between $\vec{p}_{u 0 w}$ and $\vec{p}_{u 1 w}$, and $w$ axis between $\vec{p}_{u v 0}$ and $\vec{p}_{u v 1}$, for $u, v, w \in 0, 1$.
Then, $(u, v, w)$ is within the cuboid if and only if $0 \le u, v, w \le 1$, and corresponds to point $\vec{p}$: $$\begin{aligned} \vec{p} & = ((\vec{p}_{111} u + \vec{p}_{011} (1 - u)) v + (\vec{p}_{101} u + \vec{p}_{001} (1 - u)) (1 - v)) w \\ ~ & + ((\vec{p}_{110} u + \vec{p}_{010} (1 - u)) v + (\vec{p}_{100} u + \vec{p}_{000} (1 - u)) (1 - v)) (1 - w) \\ \end{aligned}$$
This is trilinear interpolation, because if we use $L(p_0, p_1, x)$ to denote linear interpolation between $p_0$ and $p_1$, $$L(p_0, p_1, x) = (1 - x)p_0 + x p_1 = p_0 + x (p_1 - p_0)$$ then $$\begin{aligned} \vec{p} & = L(L(L(\vec{p}_{000}, \vec{p}_{100}, u), L(\vec{p}_{010}, \vec{p}_{110}, u), v), L(L(\vec{p}_{001}, \vec{p}_{101}, u), L(\vec{p}_{011}, \vec{p}_{111}, u), v), w) \\ \end{aligned}$$