The Bernstein Form of a Bézier Curve

This chapter discusses a Bezier curve in terms of a nonrecursive formula. Barycentric combinations are invariant under affine maps. The curves that correspond to the two different orderings look the same; they differ only in the direction in which they are traversed. The process of forming the Bezier curve from the Bezier polygon leaves barycentric combinations invariant. Therefore, one can construct the weighted average of two Bezier curves either by taking the weighted average of corresponding points on the curves, or by taking the weighted average of corresponding control vertices and then computing the curve. This linearity property is essential for many theoretical purposes.