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Bezier, what is t?

  1. Apr 18, 2008 #1

    I have a little problem with bezier curve. There is a bezier curve, with n degree, with some control points. And the problem is, that i want to know the t, and i know the y of that point. So there's a curve, in this curve there is a point, i know that place, and i wanna know, what is t in this bezier.

    With this algorithm(which i attached) i know the point, if i know the t

    but reverse?

    i know, that the t can be more numbers. I want just one of them...
  2. jcsd
  3. Apr 19, 2008 #2
    The Y(t) coordinate of the 2-dimensional B(t) equation you posted is a polynomial of degree n; what you want are the solutions of the equation Y(t) - Ygiven = 0, that is, the roots of the polynomial Y(t) - Ygiven, where Ygiven is your known Y value. The expression for Y(t) is, of course, the same as in the image you posted, substituting each point P with the Y value of that point. Among the multiple values for t, you want the one(s) that are in the interval [0,1].

    If n is small (like 2) there are known methods. Otherwise, a possibility is to use an iterative method like Newton's, since the derivative of the polynomial should be easy to write down. See, for example, the Wikipedia entry. Or maybe the polynomial turns out to be easy to factor, in which case the roots can be taken immediately from the factored expression.
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