So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ \sin f(t) \sin (t) & \sin f(t) \cos (t) & \cos f(t) \\ \end{vmatrix} \cdot \begin{vmatrix} X_1 + u(X_2 - X_1) \\ Y_1 + u(Y_2 - Y_1) \\ Z_1 + u(Z_2 - Z_1) \\ \end{vmatrix} = \vec v(t) $$
where
$$ u = \frac{t-\lambda_1}{\lambda_2 - \lambda_1}$$
for $ 0^\circ \le t \lt 180^\circ$
Is there any way of knowing max and min values for each axis without differentiating and solving for x' = 0, y' = 0, and z' = 0? The problem with this approach is that I am trying to do these calculations through programming (and kind of quickly). The only thing I can think of is loop through your range (0-180) and see where your result is closest to 0. 1st issue: this is an approximation. 2nd issue: This is a lot of processing power. I wouldn't care so much that it is an approximation if I could come up with it fast, but I can't. Any ideas on a faster method or alternative approaches would be appreciated!