Intersection of straight line with (lagrange) polynomial

  1. 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
    [itex]p+t\cdot r = q+u\cdot s[/itex].
    [itex]t=\frac{(q-p)\times s}{r \times s}[/itex]

    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:



    [itex]\Psi_i=\prod_{M=1,M ≠ N}^{n}\frac{\xi-\xi_M}{\xi_N-\xi_M}[/itex]

    where [itex]\xi=0..1[/itex] and [itex]x_1[/itex] is the starting point, [itex]x_2[/itex] the midpoint and [itex]x_3[/itex] 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.
  2. jcsd
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