Edit: I rephrased the question to make it clearer, sorry!
I'm trying to solve for the intersection of two surfaces in three spatial dimensions and time. Consider each of these surfaces as some quadrilateral. For convenience, orient one corner "down". Any point on each surface is represented by bilinear interpolation:
$P_1(a_1,b_1) = x_1 + a_1*BL_1 + b_1*BR_1 + a_1*b_1*(TL_1 - BR_1)$ $P_2(a_2,b_2) = x_2 + a_2*BL_2 + b_2*BR_2 + a_2*b_2*(TL_2 - BR_2)$
Here, $x_1$ is the coordinate in x,y,z,t space of the bottom corner, $BL_1$ the bottom left edge, $BR$ the bottom right, and $TL$ the top left.
Then, an intersection is given by
$x_1 + a_1*BL_1 + b_1*BR_1 + a_1*b_1*(TL_1 - BR_1) = x_2 + a_2*BL_2 + b_2*BR_2 + a_2*b_2*(TL_2 - BR_2)$
How do I solve the system for the parameters $a1,a2,b1,b2$? I am looking for an analytic solution, if there is one.
Note: I ran a test using Mathematica for some arbitrary system, and found four roots - therefore, I think the system reduces to a quartic.