I have a cubic spline function, defined by vectors in R2 space $a$, $b$, $c$ and $d$, where $t$ is time:
$$f(t) = a(1-t)^3 + 3bt(1-t)^2 + 3ct^2(1-t) + dt^3$$
Given the four vectors $a, b, c, d$ we can draw a graph for $0 < t < 1$. Given an $x_1$ coordinate on that graph how could I get the $y$ coordinate of all points that have said $x$ coordinate?
Would the solution be to find the intersection between the polynomial function and line $x = x_1$? If so how would one approach solving this intersection algebraically?
If there isn't analytical approach to this problem or it's too complicated I would settle for a numerical solution.