How can I find the function of a polynomial curve using a set of points using Riesenfeld method and Bézier méthod?

Question

Let's take for example 4 points : (0,5), (1,0), (3,2) and (4,9). What is the corresponding function using the two methods?

Answer 1

A Bezier curve based on four points would be cubic and look like:

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$x(s)=b_{x0} (1-s)^3+3 b_{x1} s(1-s)^2 + 3 b_{x2} s^2 (1-s) + b_{x3} s^3 $ $y(s)=b_{y0} (1-s)^3+3 b_{y1} s(1-s)^2 + 3 b_{y2} s^2 (1-s) + b_{y3} s^3 $

Assuming that with Riesenfeld you mean a subdivision curve, it is created iteratively and would only be a polynomial curve as the limit of those iterations.

Answer 2

Do you want the curve to pass through the points, or merely be controlled by the points?

I know Rich Riesenfeld personally, and I still don't know what you mean by "the Riesenfeld method". Maybe you're talking about b-spline curves. A b-spline curve is composed of several Bézier curve segments that are glued together end-to-end. Since you have only 4 points, a cubic b-spline defined by these points would have only a single segment, so it would actually be a Bézier curve.