Ray-Polynomial Intersection

1. May 28, 2010

Harmony

I was trying to compute the intersection between a light ray traveled from a known location at a known angle with a polynomial line located some distance away, eg. y^5 = x+3.

The intersection i am looking for is the one that is nearest to the origin of the light ray.

I don't intend to solve the equation analytically, and wish to find the intersection using numerical technique instead. I am thinking of root finding algorithm, but i faced a few problems:

1) The root finding technique need either a starting guess (ie Newton Raphson) or an interval where the root lies in. (ie Brent's Method) How can i choose which starting guess, or interval to use, and still converge at the intersection that is closest to the origin?

Manually finding the guess using graph is useless in this case since i have thousands of rays to consider, each with different starting position and angle. (though the polynomial line that the ray will hit on is the same)

2) I am thinking of using Laguerre's method, and use the initial starting point of my ray as an initial guess. Would it converge for sure at the intersection closest to the origin of the ray?

3) Can i turn this into an optimization problem instead? ie. Minimize the distance between the ray origin and the intersection, with two constraint: a) The point lies at the ray line. b) The point lies at the polynomial.

4) How can i proceed with this problem if this is a piecewise polynomial equation? ie cubic spline