Parametric curve iteration

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SUMMARY

The discussion focuses on using Newton's method to determine the parameter value at which a parametric curve intersects a line. The curve is defined as a function p: R -> R^3, while the distance to the line is represented by the function d: R^3 -> R. The composition of these functions, d.p: R -> R, allows for the identification of minimum distances, which correspond to intersection points. The key takeaway is that by rethinking the implementation of Newton's method, one can effectively find the parameter value for the intersection.

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  • Understanding of parametric curves and their representation
  • Familiarity with Newton's method for finding roots
  • Basic knowledge of calculus, specifically derivatives
  • Experience with programming concepts for implementing mathematical algorithms
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  • Study the mathematical principles behind parametric curves and their intersections.
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  • Learn about alternative numerical methods for root-finding, such as the bisection method or secant method.
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Mathematicians, software developers, and engineers working on computational geometry or graphics programming, particularly those interested in curve intersection problems.

dgm
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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...
 
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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.
 

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