Intersection of straight line with (lagrange) polynomial

[bigfooted]

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.

 

[fresh_42]

The usual procedure would be to parameterize both of them and look for the intersections. I can't see how the normals would help here.