# Intersection of straight line with (lagrange) polynomial

 P: 285 Hi, To calculate the intersection of two straight lines the cross product of the line vectors can be used, i.e. when the lines start in points p and q, and have direction vectors r and s, then if the cross product r x s is nonzero, the intersection point is q+us, and can be found from $p+t\cdot r = q+u\cdot s$. using $t=\frac{(q-p)\times s}{r \times s}$ I was wondering how to derive such a relationship for the intersection between a straight line and a second order polynomial. Specifically, I'm interested in second order Lagrange (and 3rd order Hermite) polynomials: $x=\Psi_1x_1+\Psi_2x_2+\Psi_3x_3$, with $\Psi_i=\prod_{M=1,M ≠ N}^{n}\frac{\xi-\xi_M}{\xi_N-\xi_M}$ where $\xi=0..1$ and $x_1$ is the starting point, $x_2$ the midpoint and $x_3$ the endpoint My guess is that standard techniques to find the intersection first transform the second order polynomial to the unit plane where the polynomial reduces to a line, then find the intersection and then transform back, but a (quick) search didn't give me anything.