Bezier Curves 2: higher order polynomials
- Ryan Hirst
Bezier curve written as a polynomial: each point is weighted; the curve is the sum of weighted points. The weights vary with t. For example, at t=0, the curve is at A, and the contribution of points B, C, D... is zero. A influences the curve less and less as t increases. "Show All Point Weights" displays a vector in the direction of each point; the length of each vector show how far the curve is pushed in that direction. A tool for giving a nth order Bézier from n+1 control points: http://www.geogebratube.org/material/show/id/83845
At higher orders the curve becomes less flexible. Nearby points average each other out, and the range of influence of each point is very small. most of the time, most weights are near zero. We have the greatest control over the endpoints. There are three elements to the equation of the curve.
- Binomial coefficients Ck For a curve of order n, there are n+1 points. Multiply the points by the values of Pascal's Triangle: the (binomial) coefficients of .
- Control points Pk
- Powers of t and (1-t), Tk