Parametric curve iteration

[dgm]

I haven't taken calculus yet, but I'd like to determine the closest parameter value for when a parametric curve crosses a line (for a program I'm writing). The curve is such that I need to use calculus in some way or another - it is unsolvable algebraically.

Newton iteration will find a point very close to where the parametric curve makes an intersection, but it won't give me a parameter value... is there a similar method that I can use in my case?

Any help is appreciated. :)

Peace & love...

 

[Tac-Tics]

It seems like this should be a simple case of Newton's method. You have a function (the curve) p : R -> R^3. Then, you have the distance to a line, which is a function d:R^3->R. Their composition is d.p : R->R, where (d.p)(t) is interpreted as the distance between the line and the curve at the given parameter. You want to find a minimum of d.p, and since (d.p)(t) >= 0 for all t, you know that all zeroes are minimums. Newton's method finds zeroes, so it is the right tool. It seems like you just need to rethink your program to figure out the correct way of using it.