I have 4 known points $\textbf{P}_i \quad (i = 1, 2, 3, 4)$. One of these points is simply $\textbf0$.
I have 8 unknown points: $\textbf{F}_j$ and $\textbf{R}_{jk} \quad (j = 1,2; \quad k = 0,1,2)$.
I also known that points $\textbf{C}_{ji}$ are located on respective planes created by points $\textbf{R}_{jk}$.
I also know $u_{ji}$ and $v_{ji}$ and don't know $t_{ji}$. They are parameters in a parametric descriptions of planes (u, v) and lines (t).
I can write total of 48 equations:
$ \textbf{C}_{ji} = \textbf{R}_{j0} + u_{ji}(\textbf{R}_{j1} - \textbf{R}_{j0}) + v_{ji}(\textbf{R}_{j2} - \textbf{R}_{j0}) $
$ \textbf{C}_{ji} = \textbf{P}_i + t_{ji}(\textbf{F}_j - \textbf{P}_i) $
Which in this formulation leave me with 56 unknowns - coming from my original 8 points, $\textbf{C}_{ji}$ and $t_{ji}$.
Is it true then that this problem has no unique solution? Or what are the equations that I am missing?
What are your incidence constraints? Judging from the equations, it seems as though $C_{ji},R_{j0},R_{j1},R_{j2}$ are coplanar, while $C_{ji},P_i,F_j$ are collinear, right?
If you omit the $\textbf{C}_{ji}$, and instead directly equate the two right hand sides, you have $24$ equations in $32$ variables, which is the same thing. You are indeed left with a linear space of solutions of dimension $8$ or greater.
Unless you have further constraints you failed to mention. For example, if the parameter directions for $u$ and $v$ in each plane were orthogonal, that would add two more constraints, albeit non-linear ones.