Points of a Polynomial
The Bézier curve is a type of smooth curve that interpolates between a set of control points. These curves are especially useful in graphical design, where the designer can adjust the control points and see how the curve is affected in real time.
The Bézier curve BP1⋅⋅⋅Pn determined by n ordered points is a weighted average of the curves through the first n−1 points and the last n−1 points, where the weight depends on the parameter of the curve. A precise recursive definition is as follows:
BP1⋅⋅⋅Pnt = 1−t⋅BP1⋅⋅⋅Pn−1t+t⋅BP2⋅⋅⋅Pnt
or, more explicitly:
BP1⋅⋅⋅Pnt = ∑i=0nni⋅1−tn−i⋅ti⋅Pi
The Bézier curve is the graph of this parametric equation for t from 0 to 1.
The following plot contains a Bézier curve anchored by red points. Drag the various control points around to see the effect on the curve. Note that moving some of the control points can affect the other points as well.
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