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Homework Help: The Parametric Functions of a Bezier Curve

  1. Jan 1, 2012 #1
    1. The problem statement, all variables and given/known data
    Hello. I'm currently doing some work with bezier curves and have come across a certain format, which, from what I can tell, is called the parametric form of a bezier curve. I've run several searches and can't seem to find anything that explains how to obtain this form. The usual notation I see it written in is as follows:

    x(t)= x(t)/w(t), y(t)=y(t)/w(y)

    So basically, my question is, given I have all the control points of a cubic bezier curve, how do I find the parametric function (assuming that's what the notation above is called)? I've been trying to solve this myself for a while, and feel I won't get much further without help. If someone could explain this or post a link to something that does, I'd be more than grateful.

    2. Relevant equations
    The polynomial functions I currently have:

    x(t) = axt^3 + bxt^2 + cxt + x0

    y(t) = ayt^3 + byt^2 + cyt + y0

    The Bernstein basis functions of a cubic curve:

    t^3

    3t^2(1 - t)

    3t(1-t)^2

    (1-t)^3

    3. The attempt at a solution
    As I've said, I've run several searches. From what I can gather, this is called the parametric form of a bezier curve. The x(t) and y(t) functions may be the same as the polynomial functions I have, but I doubt that. I think the third function, w(t), is a "weight function" of the bezier curve. If this is correct, I guess it works as the "magnetic attraction" of bezier control points, so maybe it relates to the Bernstein basis functions somehow? But as I've said, I can barely find anything explaining these functions and most of this is guess work.
     
  2. jcsd
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